INTERPRETATION OFELECTROMAGNETIC SOUNDINGDATA - NUMERICAL METHODS

AND COMPUTER PROGRAMLISTINGS

BEN K.STERNBERG

U. OF W-ISCONSINGEOPHYSICAL EkPOLAR RESEARCHCENTER

DEPT OF GEOLOGY81 GEOPHYSICS

MADISON WIS. 53706

WISCONSIN REPORT No.77-1 MAY,1977SECTION B7 THROUGH B 10

Interpretation of Electromagnetic Sounding Data -- Numerical

Methods and Computer Program Listings

Ben K. Sternberg*

University of Wisconsin

Geophysical and Polar Research Center

Department of Geology and Geophysics

Madison, Wisconsin 53706

Wisconsin. Report Number 77-1 May, 1977 (section B7 through B0)

*Now at Continental Oil Company, Exploration Research Division,

Ponca City, Oklahoma- 74601

FORWARD

These reports* of our methods for the acquisition and analysis of

electromagnetic sounding data are from B.K. Sternberg's Ph.D. thesis.

In total, the system consists of an analogue transmitting and receiving

system, a computer-controlled digital recording system and its programs,

and numerical methods for analyzing the data. Most of the individual

parts are fairly standard. Even so, the system represents a lot of

effort in gathering pieces from many sources and adapting them to

work together. Since the programs are specific to our system, we give

the theoretical equations and their conversion to forms for numerical

evaluations. Listings of computer programs follow and are in either

BASIC or FORTRAN.

The equipment and programs are research gear and are not elegant.

Our technology ranges from about 1970 to 1976 and our cost is describ-

able as low budget. I believe that the material can be useful in two

ways to people who wish to do geoelectric measurements. First, if you

like the experiments, you can use our system for the basis of an advanced

system. Second, chunks such as a program or circuit here and there may

be useful and save a bit of work.

C.S. ClayWeeks HallFebruary 1978

* Steinberg, B.K., Electrical resistivity structure of the crust in the

southern extension of the Canadian Shield -- layered earth models.

* Steinberg, B.K., Interpretation of electromagnetic data -- numericalmethods and computer program listings.

* Steinberg, B.K., and L.R. Stanfield, Equipment for deep electromagnetic

soundings.

TABLE OF CONTENTS*

(section B7 through B10)

B7. Interpretation of magnetotelluric (IT) data...........

B7.1.1 Calculation of MT fields over a plane-layered

half space ...................

B7.1.2 Program MDL,ff. . ... ...................

B7.1.3 MDLMT sample calculation

B7.2.1 Calculation of MT fields over an inhomogeneous

earth..........................................

B7.2.2 Programs EPOL and IIPOL

B8. Interpretation of electromagnetic (EM) sounding data

(general, transition region) ..............

B8.1.1 Calculation of EM fields over a plane-layered

half space ...................

B8.1.2 Program EMFIN...........................

B8.1.3 Program EM-ITL listing . .

B8.1.4 EM-Iff)L sample calculations ...........

B8.1.5 Examples of apparent resistivity curves in the

EM transition interpretation region .......

B8.2.1 Non-linear least squares inversion of EM

sounding data...................

B8.2.2 Program EM-INV listing

B8.2.3 EM-INV input and listing ... ... ...

B8.2.4 EM-INV sample calculation.............

326

326

329

332

333

335

337

337

341

343

359

362

366

369

404

406

* Second part of Appendix B of: /

Sternberg, B.K., 1977, Electrical Resistivity Structure of the Crust

in the Southern Extension of the Canadian Shield, Ph.D. Thesis,

University of Wisconsin-Madison.

ii

B9. Acknowledgements.................... .. 414

BIO. References......................... 415

32(o

B7* INTERPRETATION OF MAGNETOTELLURIC (MT) DATA

B7.1.1 CALCULATION OF MT FIELDS OVER A PLANE-LAYERED HALF SPACE

Cagniard (1953) first suggested the use of natural sources to

measure the impedance of the subsurface of the earth. He assumed

that a plane EM wave is incident on a plane-layered half space.

Because of the large conductivity contrast between the earth and the

air, the waves at any angle of incidence are refracted such that

they are nearly vertical traveling in the earth. Since the

conductivity is only a function of z and the waves are traveling

along the z direction, the wave equations reduce to one dimension

± ikzand the solutions for E and H are of the form e- , where k is the

propagation constant. The impedance of each layer is:

Z=Ex EyBVo1o1Hy Hx

The impedance measured at the surface of the earth involves the

impedances of all the lower layers. The derivation of the surface

impedance is in many standard geophysics texts and is not repeated

here. From Keller and Frischknecht (1966), page 224:

-11ot(k 2 h 2 +coth'Ik

ok k

coth Lk h +coth -othk

-1kI II Ik [k2 12 ik 1coth kn2 coth (kih + coth1 k n-)

B7.1.2

32.7

Where: P = magnetic permeability usually taken equal to the

-7free space value =

4-ff X 10 ,

W = 2w x frequency,

k. = Jia P + v\7

= conductivity of layer i,

h. = thickness of layer i,1

Z = surface impedance,0

= a constant, related to the horizontal dimensions of

the source by 2w /v. Normally taken equal to zero,

i.e., an infinite source is assumed.

Apparent resitivity (p ) is defined as the resistivity of aa

uniform half space that has the same surface impedance as that

calculated by equation B7.1.2. From Keller and Frischknecht (1968),

page 217, the impedance of a half space is:

E 1\ 1/2Ex = [PWB7..

Hy .\I/

Therefore, apparent resistivity is:

1 E 2i

p _1 - x 2Z BT.I.4a I P H1 H

Program MDLMT was written to calculate equation B7.1.2. The

BASIC computer language, which is used on the PDP 8/E, does not

allow complex arithmetic, so it was necessary to write out this

32e

equation in terms of its real and imaginary parts. The program is

designed to interactively match calculated apparent resistivities,

for an assumed model, to a set of observed apparent resistivities.

The interactive commands are explained on lines 35-44 of the

program.

329

B7.1.2 Program MDLMT Listing.

I1 EM**** lC5-1AM ~~5 REM ***** CALjULATF MT APPAREN7. IESI3TI17ITV6 RFM ***** FOR N LAYEFrD EAqTH. OleqN9 REM ***** WRITTF 3Y 9. 3TElN3E13, NO. 9T.17 RFM *****23 0 IM F 13 ).,Trg9(1I ).,T 8(13)') (I -p-1 (13 ).#C2 (9)25 TJ=4*3.14159*.F-33727 S9=1.F-2029 32-...1

31 52=333 C2=435 REM ***** INPUT COMMAND *****

36 REM N - INPUT NO. OF LY="V337 RFM R - INPUT LAYEl NC. / INDUT L!YFR RSISTIVIITY38 REM C - INPUT LAYER NO. / IN71JT LAYFq THICIN SS (M)39 RFM CALC CALCULATE IM5CAlN,E43 RFM PAR - WIINT LAYF ? ?A'lAMETE"lS41 RFM C2 - INPUT NO. OF ZYCLFS (CFl%AF3)42 REM S2 - INPUT NC. OF O0INTS/CFCCF43 REM 2 - INPUT 3EGINNING FRF'IJFNZY44 REM S9 - INPUT SOURCE 0-4.4MFTE-45 RFM ***** INTERACTIVE INPUT5 INPUT Z$55 IF Z$="N" THEN 13060 IF Z$="R" T HEN 1165 IF Z$"CI THFN 1207 0 IF Z$ ="CALC" THEN 1 8783 IF Z7,="PAR" THEN 17085 IF ZS="C2" THEN 13086 IF ZS=*S2 THEN 13?89 IF Z$="32" THN 14091 IF Z$="Z9" THEN 14595 PRINT "ERROR"96 GOT0 5399 REM ***** INPUT LAYFR PARAMETERS133 INPUT N101 GOTO 53I 10 INPUT LIII INPUT R(L)

12 GOTo 53A

1270 INPUT L121I INPUT C2 (L)122 GO"TO 5,0

1 30 INPUT C21 31 GOTC 53132 INP UT 32

330

133 GOTO 50143 INPUT 321 42 GOTO 50145 INPUT S9147 GOTO 53169 RFM ***** PRINT LAYCR PARAMFTFIS1703 PRINT175 PRINT "RESISTIVITY THI[fpHNFSSt176 O.(N)=999999177 FOR1 I=l TC N1 83 PRINT R(I),C2 (I)I 03 NEXT I185 0GOTO 5211 87 REM ***** SFT UP FRFTJEN'IF.S1 90 F(I )=L03( 32)/2.33a591 95 J=2233 FOR L=l TO C2205 FOR H=1 TO 32213 F(J)=F(J-I1)+(/S)21 5 J=J+l223 NEXT X225 NEXT L230 FOR L=l TO J-235 F (L)=EXP (2.33259*F (L))237 NEXT L240 PRINT "FRFlUFNCY APP. RES, ?kiS3"245 REM ***** CLCULA TE IIMPESCA.NCE25.3 FOR M=l TO J-1255 T=2.*3 •o1415 9*F ( M260 FOR I= TO N261 S(I)=1/R(1)265 T9( I )=SQ"l(soR(S9**4 +(U*S(I)*W)**2))270 T8(I)=ATN(U*S(I)*W/3**)/277 NEXT I2 85 Zl=1293 Z2=0295 FOR I=l TO N-i330 T7=(T9(N-I)/T9(N+1-I))*Z1335 T6=(T0(N-I)-T8(N+-I))+Z2313 Al=( +T7*COS(T6))*(-I +T,7*CCS(T6))+(T7**-.2)*(SIN(T5))**,,2315 .2(-I +T7*COS(T6))**2+(T7**2),(3INCT6),**_323 A3=AI/5325 3I=(T7*3"IN(T6))*(-I+T7*COS (T6))-(I +T7*COS(76))*(T7*IN333 35=(-I +T7*CC3(T6))**2+(T7**2)*(SIN(T6))**2. I(T6))335 33=31/I533 E1FI=.5,LOG (32R(A3,,2 +33 **2 ))341I IF ABS(33) 1 I.E-303 GOTO 345342 B3 =1 .E-333345 E2=.5*(-.5, 3/$qR( 3,*,2 )+.5 ),(.5*33/SqR(33**2 )+.5)346 E3=.5*((-.5,.A3/S R(!43**2)+.5 )*(-.5*33/3Y--(t 33**2)+.5 ))

347 F2=E2*3 .14159-F3*3.14159+.5*.ATN(33/A3) "331350 T5=C2 (N- I )*T9([N-I)355 T4 =TP(N-I)3 64 h (T5*Cr(14)+EFI

741 F XEP(C G3-+. 10)*CC(-C))*(E3?(o)*"CS ()

-XC (-C ) _, C (-C))

375 2=(P(FYP )*S IN(C)+F't ? (-C )3 INC-C)).(E '(C)*3 IN(C)FE,) -C). IN -C )

38 P3 =(EXP (C )*COS (C )-EXP (-C )*CG3(-C) )**2381 P/4=(EXP(C)*-3 IN(C)-EXP(-C) *3 IN(-C))**2382 P3=P3+?4385 10P4 = (01+P-2)/?3393 1 E=P(F)T(C )*SIN(C )+X ?(-C)*31NC))*CIN(.(C)*CGS(C)

J-,: (-C )*C C s.-C ) )

391 P2=(EXP (C)*COS (C) + (-C) *' 03 (-C))(EX C )*31N(C)VEX? (-C)_*3_I (-&) )"

392 P5=(PI-P2)/P33 95 ZI:SQ.R(P4 **2 ?5**2)4,33 Zl(-.5*P4 /"tR(P4**2 ) + 5 )*( "5*?5/ S ll( 0 5 *)+ ' 5)

411 Z3=(-.5*P4 /SQ'R(P/4 * )+.5 )*(-.5*(5/ +(52 )+-5)402 Z2=Z2*3.14159-Z3*3.14159 +ATN(?5 /0)405 NEXT I41l3 ZI = ((CU,1) /T 9 (1 )*Zl

415 Z2=3.14159/28-T8(1)+Z2423 ZI=(ZI**2)/CJ*W425 Z2=Z2*183/3o14159433 ORINT F(M),ZI,Zl_435 NEXT M437 GOT0 53440 FNC

3-"2

B7.1.3 MDLMT Sample Calculation.

?N?2 -NO. Of LAYERS?R?1 LAYER NO.? 1000- NESS?R?2?100?D?I qt-- LAYER NO.?I 1000---T" ICK NESS

?PAR

THICKJNESS1000999999

[PRrNT MODEL PARAMETERSJ

[ CALCULATE " DECADES, I POINT/DECADE 'TARTNG-~AT 0-001 WERT4.

?CALC ,{&ACULA-TE MODELFREQUNY APP,* RES,

0.001 11.1370.01 103.640.100001 111*9431-.00001 141.96910.0001. 270.722100.002 835.8371000.02 1026.6410000*2 999,992

APPARENT RZSTWMEgSPHASE45.321346.00248.024253.269862.105861 .040744.172344.9999

ST IVITYRES I!I0OOQ100

?92

?S2?2?C2?7

B7.2.1 CALCULATION OF MT FIELDS OVER AN INHOMOGENEOUS EARTH

For electromagnetic plane wave propagation in a model which has

resistivity variations in two coordinate directions (x and z), the

solution can be separated into two independent cases: H

polarization (E field perpendicular to strike) and E polarization (E

field parallel to strike). The two wave equations to be solved are:

2H 2H

H polarization: Y-+ Y.k 2 H = 0 B7.2.1ax2 z

2E Ea_2_E Y+ a 2 k 2 E=0B..

E polarization:J- - k E 0 B7*2o22 2 y

WherIe k = a) 1/2 = (I) 1/2, magnetic permeability,

a = conductivity, w = 2f times the frequency, H =Y

magnetic field along strike, E = electric field alongY

strike.

We used a finite difference formulation of the wave equations

to solve for the E and H fields. The details of this calculation

are in Jones and Pascoe (1971) and Pascoe and Jones (1972).

Briefly, the numerical procedures are as follows.

A plane which is perpendicular to the strike direction is

divided into a grid with N x M cells. Within each cell there is

constant resistivity, but any arbitrary resistivity can be assigned

to each cell in the grid. A finite difference approximation to the

3754

wave equation holds within each cell:

2 21/(Ax) (Fi - 2F., + Fi_l,) + I/(Az) (Fi - 2F. , + Fi.l)

i+J12j 13 3 j i1ji9j+1 1,J $1.1

2-k..F.. =0 B7.2.3

1j 1,3

Where F.. = the E or H field at the i, j cell in the grid, Ax13

and Az are the dimensions of the cell, k.. = (ip a w) 1/2,GO*13 i

conductivity of the cell.

We must also satisfy the following boundary conditions:

(1) Within the grid, the tangential components of E and H are

continuous. (2) At the sides of the grid we force the fields to

equal the fields of a plane layered earth in which the layer

resistivities are equal to the resistivities at the side of the

grid. (3) The field at the bottom is forced equal to that from a

uniform half space with a resistivity equal to the resistivity at

the bottom of the grid. (4) A constant (plane wave) source is

assumed over the top of the grid. An air layer is placed between

the source and the earth.

Application of the finite difference wave equation to each cell

in the grid yields a set of simultaneous difference equations to be

solved. This matrix equation is solved by Gauss-Seidel iteration to

give the electromagnetic fields at the surface of the earth.

The size of the grid used in the model calculations was fixed

363

at 40 cells by 40 cells. The size of each cell depends on the

frequency (f) and the resistivity (p) within that cell. For

accuracy in the finite difference approximation, we always kept each

dimension of the cell less than one third skin depth (where skin

depth in km is approximately equal to 1/2 V/P7f . When a boundary

from a conductive to a more resistive region is crossed, a sudden

jump in the size of the cells is avoided by gradually increasing the

dimensions of each cell. The edges of the grid were made greater

than three skin depths distant from an inhomogeneity.

B7.2.2 PROGRAMS EPOL AND HPOL

Computer programs published by Jones and Pascoe (1971) were

used to perform the finite difference calculations discussed in

section B7.2.1. We also used refinements to this program given by

Pascoe and Jones (1972); Williamson, Hewlett, and Tafmneinagi (1973);

and Jones and Thompson (1974). Program EPOL calculates E

polarization (E parallel to strike) fields and HPOL calculates H

polarization (E perpendicular to strike) fields.

A problem was encountered when running these programs on the

University of Wisconsin's Univac 1110 computer because of the

relatively small word size of the 1110 (36 bits). We found that

many models (including the examples published by Jones and Pascoe,

1971, and Pascoe and Jones, 1972) overflowed the exponential

functions in subroutine BYCOND (Pascoe and Jones, 1972). We had to

maintain the following restriction on the vertical cell widths (Az i )

to keep the argument of the exponentials small enough. The actual

3-3(o

cell widths (Azi) equal k. times the scale factor in Pascoe and

Jones' notation.

(Z ki) X (scale factor in m X 102) X boundary-11

conductivity in mho/m X I0 X 4 X ff) X (frequency X

2 X )] 1/2 must be less than 88.

B8. INTERPRETATION OF ELECTROMAGNETIC (EM) SOUNDING DATA (GENERAL,

TRANSITION REGION)

B8..1 CALCULATION OF EM FIELDS OVER A PLANE LAYERED HALF SPACE

Wait (1966) has given the solution to the wave equation for EM

fields over a layered earth due to a horizontal electric dipole (see

also Dey and Morrison, 1973). The solution to the wave equation is

a Sommerfeld-type integral. Boundary conditions are applied at each

plane, horizontal interface to evaluate constants in the Sommerfeld

integral solution.

We evaluated the Sommerfeld integral by the digital filter or

convolution method (Koeffoed, Ghosh, Polman, 1972; Anderson, 1974;

Daniels, 1974). The Sommerfeld integral for the vertical component

of the magnetic field is:

B8. 1.1H IL sin (1 + RE.(I)) 4W (r) d?

0

and for the electric field (X component):

E -IL k 2 p (I R ()) J (r) d

P IL 2 22-kl2 2_ -A 2 + U12R.(+ 1 - cose [-hi+ 2P) + 141T LL X 2 1 2

0

2os0* + 2 2Z 1 )+(+R.,? )JJ 0 (Xr) dA+ 2 ) M A I +

2 (-2 +U 1 RL (

I+R.L~ ) J(r B..B8*.1.2

and for the electric field (Y component):

PllL __ __ _ 2Z1 ( )

PIL -sin (20) >2 E-kf2l 2Iy 221

2+ U1 R () + (I + R1 ( 0))j (hr) dA

, [-k2 2ZA2+ +U 1 R oosin (20) \I I I I 2 (70)

Sr A2 , =2

+ (I + R, (I))1 Jl(r) d

Where:

J

J

I

B8. 1.3

I = current in transmitter dipole,

L = length of transmitter dipole,

I --first order Bessel function,1

= zero order Bessel function,0

= variable of integration,

= angle between transmitter dipole and radius vector

to receiver site,

r = distance from center of transmitter dipole to

receiver site,

= wave number in layer n= ( io W I/P1n )2n0 n

) = resitivity of layer n,n

W = 21Ttimes frequency

-7=free space permeability = 41TX 10,

2 2+k 2n n

338

R.,P)- layered earth correction factor (general EM

reflection coefficient)

= (N - YI) / (N + Y1 )

N = ? / (iM A )0 0

Yn + N tanh (Unh n )1Y Nn+l nn

n +'Y tanh (Un YN+1 nhn

N = U I (ik o)n n0

Y = NN = U (i,o°0W)

h = layer thicknessn

Z ( ) layered earth correction factor (general EMn

reflection coefficient),

rzn+1+ M tanh (Unh n )n= n+1 n nn L Mn + z 1tanh (Unh n)

LM n+ 1 nn

M = UnP n n 1, 2 , ... Nn nn

Z =U PN N N

To insure convergence of these integrals, the homogeneous half

space expressions (Equations B3.5.1 and B3.5.2) are subtracted from

the integrands (Daniels, 1974).

If the substitutions r/d = ex and Xd = e - y are made in the0 0

,Sommerfeld integral, where d is an arbitrary unit of length, we0

obtain an integral which is in the form of a convolution integral:

Field = G SoK (P,h,Y) [ ex -y J l (eX-Y) ] dy B8. 1.3

Z40

Where: K= a kernel function which includes the layered earth

parameters (resistivity and thickness),

G = geometric factors.

The procedure for calculating theoretical fields, given an

assumed earth model, involves convolving the kernal function with a

known filter function (the term in square brackets) represented by a

48 or 61 point filter response. The filter response is obtained

from an integral for which the solution is known. The half space

response which was subtracted from the integrand to insure

convergence is now added back to the integrated result. We have

found that the convolution procedure in roughly an order of

magnitude faster than numerical integration of the Sommerfeld

integral. The accuracy for the filters we used was about three

significant digits.

The Sommerfeld integral expression is for an infinitesimal

dipole. The Sanguine antenna is actually quite long. We replaced

the long dipole antenna with a series of short dipoles where each

short dipole has a length equal to one-third the distance from its

center to the receiver site. We then summed the vector fields from

the series of short dipoles. The error with this approximation is

at most a few tenths of a percent for our sites.

341

B8.1.2. PROGRAM EMFIN

A computer program published by Anderson (1974) was used for

many of the layered earth EM calculations. Anderson's program

calculates all the EM field components from a finite grounded wire

source using either Gaussian Quadrature numerical integration or by

convolution with a digitial filter. The program may be used

interactively in order to match field data.

Some problems were encountered running this program on the

University of Wisconsin-Madison's Univac 1110. The required

modifications to this program are listed below:

(1) In subroutine CANC4:

Removed: X from complex declaration

Removed: EQUIVALENCE (FMX(1), FMAX)

Added: FMX(1) = REAL (FMAX)

FMX(2) = AIMAG (FMAX)

Removed: 210 WRITE (6,39)

GO TO 99

(2) In subroutine PPLOT

In the logical declaration, change DBOTGL to KBOTGL

(3) In complex function HZO(X)

Removed: EQUIVALENCE (B8, CB)

Added: B8 = B

GB = B

(4) With the regular Univac FORTRAN V complier, the Gaussian

342

Quadrature integration works correctly but convolution

integration often gives meaningless numbers. With the Univac

1110 ASCII * FORTRAN compiler, convolution integration works

correctly but Gaussian Quadrative integration often gives

meaningless numbers. It was necessary to use both the FORTRAN

V and ASCII * FORTRAN compilers in order to do both convolution

and Gaussian Quadrature integration.

(5) At small kR values (i.e., as we approach the DC interpretation

region), the convolution method gives more reliable results.

At large kR values (i.e., as we approach the MT interpretation

region), the Gaussian Quadrature methods gives more reliable

results.

We used several of Anderson's subroutines in another program -

EM-MDL. This program is much shorter than Anderson's general

purpose program and runs on a small computer (Datacraft 6024). EM-

MDL uses convolution integration to determine the E', Ey, and Hx y z

components of the Em field.

The E and E fields are combined to give the electric field inx y

an arbitrary direction. Subroutines were added to automatically

calculate apparent resistivities (see section 3.5.1 for a discussion

of the calculation of apparent resistivities in the EM transition

region). The program is designed for interactive calculations and

the interactive commands are given in lines 74 to 93 of the program.

34-3

B8.1.3 Program EM-MDL Listing.

I C PROGRAM EM-MDL2 C3 C INTERACTIVELY CALCULATE VERTICAL MAGNETIC (HZ) AND

4 C HORIZONTAL ELECTRIC (ER) ELECTRIMAGNETIC FIELDS OVER

5 C A LAYERED EARTH.6 C PROGRAM WRITTEN BY B. STERNBERG, 8/76.

7 C SUBROUTINES HZO, EXOP EYO; RECRIRI, RECIR2, HANKC,8 C F3,F7vF11, AND F12 WERE WRITTEN BY W.L. ANDERSON (USGS)9 C NTIS REPORT PB 238 199, 1974.

10 C11 COMPLEX SUM, SUM1 SUM2p ECONs TWO!12 COMPLEX HZO,EXOsEYO13 EXTERNAL HZO,EXOsEYO14 EXTERNAL HALFHZ,HALFER15 DOUBLE PRECISION EPS

16 REAL M9,L9,LS.L7# K(31),L17 REAL M92.L1O18 DIMENSION RAP( 100)19 INTEGER CONV( 100)20 DIMENSION AMP( 100)p,PHZ( 100). X2( 100 )t,F( 100),RE( 100). XIM( 100)21 DIMENS3.ION R(31 ), D( 30 )s D2( 30)22 DIMENSION D3(30)vAID(10)23 COMMON/MODEL/ K. Da M24 COMMON/SHARE/25 *EPS,26 *C2,C3sC4#27 *XX* YYs YY2, RHO. RHO2o DELRHO* B,2:3 *L., DEL, DEL2*2'? *METHOD,NZoNW30 COMMON/HALF/ X2s Y9, L8, M92, F9, Ns V9p X5o T431 INTEGER C6,S332 INTEGER SC. CS33 INTEGER 05,0634 DATA IER/2HER/P IHZ/2HHZ/, IS/1HS/35 DATA IN/1HN/s IY/lHY/36 DATA TWOI/(0. 0,2. 0)/37 I5=12138 OPEN 1539 05=02140 06=641 C42 C DEFAULT PARAMETERS43 Y9=30000.44 R9=30000.45 T9=90.46 X5=0.47 M=I48 R( 1 )= 10000.49 M92=1.50 L8=1.51 L10=0.52 T6=270.53 A1=100.54 B1=100000.55 C DATCOR FREQUENCIES

56575859606162636465666768 C6970 18007172 180173 9974 1007576 1017778 C79 C80 C81 C82 C83 C84 C*85 C86 C87 C88 C89 C90 C91 C92 C93 C94 C95 C96 C9798 3099 12100 10101102103104105106107108109110

IF( AJ. EQ.IF( AJ. EQ.IF( AJ. EQ.IF( AJ. EQ.IF( AJ. EQ.IF AJ. EQ.IF( AJ. EQ.IF( AJ. EQ.IF( AJ. EQ.IF( AJ. EQ.

'M) GO TO 13'R0) GO TO 14"D) GO TO 15"CALC') GO TO 16"PAR') GO TO 17'FREQ') GO TO 21-'SITE') GO TO 22"TX') GO TO 31"RX') GO TO 32"RMM') GO TO 33

34,q*

SC= 1IF( 1 )=0. 5F( 2 )=l. 5F( 3 )=2. 5F( 4 )=3. 97F( 5 )=5. 5F( 6 )=6. 98F( 7 )=8. 99F( 8 )=11. 95F( 9 )=15. 96F( 10 )=20. 93F( 11 )=27. 43

WRITE( 05s 1800)FORMAT(-" INPUT ID")READ( 15s 1801 ) (AID(J): J=I, 10)FORMAT( 10A6)WRITE( 05* 100)FORMAT(- CALCULATE ER OR HZ')READ( I5, 101) IEHFORMAT( A2)IF(( IEH. NE. IER). AND. ( IEH. NE. IHZ)) GO TO 99

***** INPUT COMMANDS*****M - INPUT NO. OF LAYERS IN MODEL.R - INPUT LAYER NO. / INPUT LAYER RESISTIVITY.

D - INPUT LAYER NO. / INPUT LAYER THICKNESS (M).

CALC - CALCULATE FIELDS FOR THE MODEL AND PRINT RESULTS PLUSA SUMMARY OF ALL INPUT PARAMETERS ON THE LINE PRINTER.

PAR- PRINT LAYER PARAMETERS.FREQ - INPUT BEGINNING FREQUENCY9 NO. OF DECADES, POINTS

PER DECADE.

SITE - INPUT RECEIVER LOCATION f RANGE, ORIENTATION.

TX - INPUT DIPOLE MOMENT, TRANSMITTER LENGTH.RX - INPUT COIL AREA IF HZ. RECEIVER DIRECTION IF ER.

RMM- iNPUT INITIAL MIN. AND MAX. RESISTIVITIES FOR APPARENTRESISTIVITY CALCULATION.

STOP - TERMINATE EXECUTION OF THE PROGRAM.

INIERACTIVE INPUT (JUMP TO INDICATED STATEMENTNUMBER TO EXECUTE THE COMMAND).'WRITE( 05,30)FORMAT(" INPUT COMMANDS FROM LISTO)READ( I5. 10) AJFORMAT( A6)

7349

111 IF(AJ. EQ. "'STOP') GO TO 95112 WRITE(05,11)113 11 FORMAT(' ERROR")114 GO TO 12115 C116 C INPUT LAYER PARAMETERS117 13 READ(I50-) M118 O TO 12119 14 READ(I5#-) L2120 READ(15,-)R(L2)121 GO TO 12122 15 READ( I5o-) L2123 READ( I5#-) D3(L2)124 GO TO 12125 C PRINT LAYER PARAMETERS ON TTY126 17 D3( M )=999999.127 WRITE(05, 19)128 19 FORMAT( - RESISTIVITY THICKNESS")129 DO 18 L3=1,M130 18 WRITE(05,20) R(L3)aD3(L3)131 20 FORMAT(2E12.3)132 GO TO 12133 C READ TRANSMITTER (TX) PARAMETERS134 31 WRITE(05a 130)135 130 FORMAT( " DIPOLE MOMENT ')

136 READ( 15,-) M92137 WRITE(05, 135)138 135 FORMAT( TX. LENGTH =)139 READ( 15,-) LIO140 GO TO 12141 C READ RECEIVER (RX) PARAMETER142 32 IF( IEH. EQ. IER) GO TO 141143 WRITE(15s140)144 140 FORMAT(' COIL AREA')145 READ( I5 -) L8146 141 IF( IEH. EQ. IHZ) GO TO 143147 WRITE(05v142)148 142 FORMAT(-' RECEIVER DIRECTION")149 READ( I5,-) T6150 143 GO TO 12151 C READ INITIAL MIN. AND MAX. RESISTIVITIES152 C FOR APPARENT RESISTIVITY CALCULATION.153 33 WRITE(05 151 )154 151 FORMAT(" MIN. AND MAX. RESISTIVITIES')155 READ( I5s-) A1B1156 GO TO 12157 C READ SITE LOCATION158 22 WRITE(05s 145)159 145 FORMAT(' RECEIVER LOCATION'/' RANGE ="7160 READ( I5,-) R9161 WRITE(05o147)162 147 FORMAT(" ORIENTATION =)163 READ(I5,-) T9164 X5=R9*SIN((T9-90. )*3. 14159/180. )165 Y9=R9*COS( ( T9-90. )*3. 14159/180. )

34b

166 WRITE( 05,931) Y9167 931 FORMAT(" Y=-'PE12.5)168 WRITE(059932) X5169 932 FORMAT(-' X=-',E12.5)170 GO TO 12171 C READ FREQUENCIES172 21 WRITE(05s200)173 200 FORMAT( -'BEGINNING FREQUENCY =o / NO. OF DECADES =1174 - - POINTS PER DECADE =)175 READ( 15,-) B2176 READ(I5,-) C2177 READ( 150-) S2178 F( 1 )=ALOG( B2 )/2. 30259179 S3=S2180 C6=C2181 J=l182 DO 210 J2=loC6183 DO 209 K2=1oS3184 J=J+1185 J3=J-1186 209 F( J )=F( J3 )+( /S2)187 210 CONTINUE188 SC=C2*S2189 DO 220 L2=IPSC190 220 F( L2 )=EXP( 2. 30259*F( L2))191 GO TO 12192 C193 C PRINT THE PARAMETERS ON THE LINE PRINTER194 16 WRITE(06, 150)195 150 FORMAT( IH11 - SUMMARY OF INPUT PARAMETERS-///)196 WRITE( 06a 1802) (AID( J )v,J=lo 10)197 1802 FORMAT(-' ID - - O10A6)198 WRITE(06P 148) IEH199 148 FORMAT(" CALCULATE -'9A2,-' FIELD-o)200 WRITE( 06s 155) M201 155 FORMAT(-' NO. OF LAYERS =.15)202 D3( M )=. 99999E+20203 DO 160 I=,,M204 160 WRITE(06s 165) IsR(I ),,IoD3(I)205 165 FORMAT( - RESISTIVITY(-' I1,- )=-'E12. 5s THICKNESS(-' IIs )=E12. 5)206 WRITE(06s 170) M92

207 170 FORMAT(- DIPOLE MOMENT =',E12.5)208 M9=M92/( 4.. *3. 14159)209 WRITE(06o 175) LIO210 175 FORMAT(-" TX LENGTH =-'#E12.5)211 L9=L 10212 IF( IEH. EQ. IER) GO TO 176213 WRITE(06, 180) L8214 180 FORMAT(-' COIL AREA =-'sE12.5)215 176 IF( IEH. EQ. IHZ) 00 TO 178216 WRITE(06P 177) T6217 177 FORMAT(-' RECEIVER DIRECTION ='.E12.5)218 T4=T6219 T5=T6*3. 14159/180.220 178 WRITE(06v 185) R9,T9,X5#Y9

347

221 185 FORMAT(" RECEIVER LOCATION,/- RANGE= ",E12.5/' ORIENTATION=-,222 * E12.5/" X='sE12. 5/" Y=-'E12. 5)223 WRITE(06s,191) A1,tBI224 191 FORMAT(" INITIAL MIN. APPARENT RESISTIVITY =",F12.5#225 & /" INITIAL MAX. APPARENT RESISTIVITY =-,F12.5)226 WRITE(O6s933)227 933 FORMAT(///)228 C229 C BREAK THE FINITE TRANSMITTER DIPOLE INTO N SMALLER DIPOLES230 C -(IF NECESSARY). EACH COMPONENT DIPOLE MUST HAVE A LENGTH

231 C - SMALLER THAN ONE THIRD THE DISTANCE TO THE RECEIVER SITE.232 IF( L9. EQ.0. ) L9=1. E-10233 229 N=INT(3*L9/Y9+1)234 L7=L9/N235 12=-i236 DO 230 L2=IPN237 12=I2+2238 230 X2(L2)=(X5-L7/2.*(N-I2))239 C240 232 IF(IEH. EQ. IHZ) GO TO 231241 IF(IEH. EQ. IER) GO TO 500242 C243 C CALCULATE HZ FIELD USING EMFIN SUBROUTINES.244 231 SIGI=I/R(1)245 DO 250 I=I,M246 D2(I)=D3(I)247 250 K(I )=( 1. /R(I))SIG1248 L=L9/2249 DO 300 I=I,SC250 DEL2=1.0/(39.4784E-7*SIG1*F(I))251 DEL=SQRT(DEL2)252 IF(M. EQ. 1) GO TO 253253 MI=M-1254 DO 251 12=1#M1255 251 D(12)=2.0*D2(I2)/DEL256 253 SUM=(0.0,0.0)257 DO 400 J=I#N258 XX=X2(J)259 YY=Y9260 YY2=YY**2261 XXX2=XX**2262 RHO2=XXX2+YY2263 RHO=SQRT(RH02)264 DELRHO=DEL*RHO265 B=RHO/DEL266 400 SUM=SUM+HZO(O)267 SUM=SUM/N268 RE( I )=REAL( SUM)269 300 XIM(I)=AIMAG(SUM)270 CALL POLAR( SC# RE, XIMo AMP, PHZ)271 CALL DRUM(SCjPHZ)272 C273 C CALCULATE APPARENT RESISTIVITY BY FINDING THE274 C ROOT TO THE NON-LINEAR HALF SPACE EQUATION.275 C

349

276 DO 3010 1=1 SC277 A=A1278 B=B1279 F9=F( I)280 V9=AMP( I )*M9*7. 8957*F9*L8*1. E-6281 CALL MRGFLS(HALFHZ, A.B.. 01.20 IFLAG)

282 CONV( I )=IFLAG283 RAPB=( A+B )/2.284 RAP( I )=RAPB285 3010 CONTINUE286 C287 WRITE( 06. 305)288 305 FORMAT(," FREQUENCY AMPLITUDE( H) PHASE COIL OUT (UV )Ie

289 & . PHASE APPARENT RESISTIVITY CONVERGENCEe)

290 DO 306 I=I#SC291 AMP( I )=AMP( I )*M9292 AMPB=AMP( I )*7. 8957*F( I )*L8293 PHZ( I )=PHZ( I )*180. /3. 14159294 PHASB=PHZ( I )+90.295 RE( I )=AMPL*COS( PHASB*3. 14159/180.296 306 WRITE( 06.,307 ) F( I ), AMP( I ), PHZ( I ), AMPBs PHASB* RAP( I )0 CONV( I)

297 307 FORMAT( 5E12. 5o 5X, E12. 5, lOX, 15)298 WRITE(06,933)299 GO TO 12300 C301 C302 C CALCULATE ER FIELD USING EMFIN SUBROUTINES.

303 500 SIGI=I/R(1)304 DO 1250 I=1#M305 D2( I )=D3( I )306 1250 K( I )=(I1. /R(I))/SIGI307 L=L9/2308 DO 1300 I=lsSC309 DEL2=I. 0/( 39. 4784E-7*SIG1*F( I))310 DEL=SQRT( DEL2)311 IF(M. EQ. 1) GO TO 1253312 MI=M-1313 DO 1251 12=1#M1314 1251 D(12)=2.0*D2(I2)/DEL315 1253 ECON=TWOI/(SIGI*DEL2)316 SUM1=( O. 0. 0. 0)317 SUM2=( 0. O,0. 0)318 DO 1400 J=.I#N319 XX=X2(J)320 YY=Y9321 YY2=YY**2322 XXX2=XX**2323 RHO2=X X X2+YY2324 RHO=SQRT( RH02)325 DELRHO=DEL*RHO326 B=RHO/DEL327 SUMI=SUM1+EXO( 0 )*ECON328 IF( T4. EQ. 90. AND. T9. EQ. 90. ) GO TO 1399329 IF(T4. EQ. 270. .AND. T9. EQ. 90. ) GO TO 1399330 00 TO 1398

-349

331 1399 SUM2=O.332 GO TO 1400333 1398 SUM2=SUM2+EYO( 0 )*ECON334 1400 CONTINUE335 TS=3. 14159/2336 IF( X5. NE. 0) T8=3. 14159-ATAN( Y9/X5)337 SUM=( SUM1/N )*COS( 3. 14159-T8-T5 )-( SUM2/N )*SIN( 3. 14159-T8-T5)338 RE( I )=REAL( SUM)339 1300 X IM( I )=AIMAG( SUM)340 CALL POLAR(SCPRE,XIMsAMPoPHZ)341 CALL DRUM(SCaPHZ)342 C343 C: CALCULATE APPARENT RESISTIVITY BY FINDING THE ROOT OF344 C THE NON-LINEAR HALF SPACE EQUATION.345 C346 DO 3020 I=1,.SC347 A=A1348 B=B1349 F9=F( I)350 V9=AMP( I )*M9351 CALL MRGFLS( HALFERI A, B,. 01.20, IFLAG)352 RAP( I )=( A+B )/2.353 3020 CONV( I )=IFLAG354 C355 WRITE( 06, 1305)356 1305 FORMAT(" FREQUENCY AMPLITUDE( MV/M) PHASE REAL le357 1" IMAGINARY APPARENT RESISTIVITY CONVERGENCE)358 DO 1306 I=IPSC359 AMP( I )=AMP( I )*M9*1000.360 PHZ( I )=PHZ( I )*180. /3. 14159361 RE( I )=RE( I )*M9*1000.362 XIM( I )=XIM( I )*M9*1000.363 1306 WRITE( 06, 1307) F( I )pAMP(I ),PHZ( I),RE( I )XIM( I ),RAP( I ),CONV( I)364 1307 FORMAT(5E12. 5*5X,E12. 5s lX, I5)365 WRITE(06,933)366 GO TO 12367 C368 95 STOP369 END370 SUBROUTINE DRUM(LPHZoPHZ)371 C MAKES A PHASE CURVE CONTINUES372 DIMENSION PHZ(LPHZ)373 PJ=O.374 DO 40 I= 2# LPHZ375 IF( ABS( PHZ( I )+PJ-PHZ( I-1 ) )-3. 14159265 )40, 40, 10376 10 IF( PHZ( I )+PJ-PHZ( I-1 ))20, 40, 30377 20 PJ=PJ+3. 14159265*2.378 GO TO 40379 30 PJ=PJ-3. 14159265*2.380 40 PHZ( I )=PHZ( I )+PJ381 RETURN382 END383 SUBROUTINE POLAR (Lo RE. XIMs AMP. PHZ)384 C COMPUTES POLAR COORDINATES385 DIMENSION RE(L)s XIM(L)rAMP(L),PHZ(L)

350

386 PI=3. 14159265387 DO 110 I=lsL388 AMP( I )=SQRT( RE( I )**2+XIM( I )**2)389 IF(XIM( I )) 10,20,30390 10 IF( RE( I1) )40P 50. 60391 20 IF(RE(I))70,80P60392 30 IF(RE( I )) 90, 100.60393 40 PHZ( I )=ATAN( XIM (I )/RE( I ) )-PI394 O0 TO 110395 50 PHZ( I )=-PI/2.396 GO TO 110397 60 PHZ(I)=ATAN(XIM(I)/RE(I))398 GO TO 110399 70 PHZ( I )= -Pl400 GO TO 110401 80 PHZ( I )=0.402 GO TO 110403 90 PHZ(I)=ATAN (XIM(I)/RE(I)) + PI404 GO TO 110405 100 PHZ( I )=PI/2.406 110 CONTINUE407 RETURN408 END409 SUBROUTINE MROFLS( F. At B. XTOLv NTOLs IFLAG)410 EXTERNAL F411 C412 C MODIFIED REGULA FALSI ALGORITHM.413 C FROM S.D. CONTE AND C. DE BOOR, ELEMENTARY NUMERICAL414 C ANALYSIS, MC GRAW HILLa 1972. CHAPTER 2.415 C416 IFLAG=O417 FA=F(A)418 SIGNFA=SIGN( 1. , FA)419 FB=F( B )420 C CHECK FOR SIGN CHANGE421 IF(SIGNFA*FB .LE. 0. ) GO TO 5422 IFLAG=3423 A=O424 B=O425 RETURN426 5 W=A427 FW=FA428 DO 20 N=IoNTOL429 C CHECK FOR SUFFICIENTLY SMALL INTERVAL

430 IF( ABS( B-A )/ABS( A+B) . LE. XTOL) RETURN431 9 W=(FA*B-FB*A)/(FA-FB)432 PREVFW=SIGN(1.#FW)433 FW=F(W)434 C CHANGE TO A NEW INTERVAL435 IF(SIGNFA*FW .LT. 0. ) GO TO 10436 A=W437 FA=FW438 IF( FW*PREVFW . GT. 0. ) FB=FB/2439 GO TO 20440 10 B=W

351

441 FB=FW442 IF(FW*PREVFW GT. 0. ) FA=FA/2.443 20 CONTINUE444 IFLAG=2445 RETURN446 END447 FUNCTION HALFHZ(RI)448 DIMENSION X2(100)449 REAL L8 M9450 COMMON/HALF/ X2p Y9, L8 M9o F9. No V9s X9o T4451 C452 C CALCULATE HOMOGENEOUS HALF SPACE RESPONSE FOR HZ453 C MAGNETIC FIELD. THE FINITE TRANSMITTER IS DIVIDED (IF454 C NECESSARY) INTO SMALLER DIPOLES WITH A LENGTH EQUAL TO ONE455 C THIRD THE DISTANCE TO THE RECEIVER SITE.456 C457 C WAVE NUMBER = G.458 G=SQRT(2*3. 14159*F9*4*3. 14159E-07/(2*R1))459 C COMPUTE THE REAL (Sl) AND IMAG (S2) PARTS460 C OF HALF SPACE RESPONSE.461 Sl=O462 S2=0463 DO 100 J=11N464 R8=SQRT(X2(J)**2+Y9**2)465 G2=G*R8466 Tl=3-3*E)P(-02)*COS(G2)467 T2=EXP(-G2)*(-3*C2*COS(C02)-3*G2*SIN(02))468 T3=-2*G2**2*EXP(-G2)*SIN(G2)469 T9=3. 14159/2-ATAN( X2( J )/Y9)470 S1=(TI+T2+T3)*SIN(T9)/RS**4 + S1471 Tl=-3*EXP( -G2 )*SIN( G2)472 T2=EXP( -02 )*( 3*G2*COS( 02 )-3*G2*SIN( 02))473 T3=2*G2**2*EXP(-02)*COS(02)474 100 C2=(Tl+T2+T3)*SIN(T9)/R8**4 + S2475 TI=M9*L8*RI*SQRT(S1**2+S2**2)/N476 T2=2*3.14159477 ZI=T1/T2478 HALFHZ=V9-Z1479 RETURN480 END481 FUNCTION HALFER(RI)482 DIMENSION X2(100)483 REAL L8, M9484 COMMON/HALF/ X2v Y9. L8, M9p F9, No V9s X9. T4485 C486 C CALCULATE HOMOGENEOUS HALF SPACE RESPONSE FOR ER ELECTRIC487 C FIELD. THE FINITE TRANSMITTER DIPOLE IS DIVIDED (IF488 C NECESSARY) INTO SMALLER DIPOLES WITH A LENGTH EQUAL TO489 C ONE THIRD THE DISTANCE TO THE RECEIVER SITE.490 C491 C WAVE NUMBER =0.492 G=SQRT(2*3. 14159*F9*4*3. 14159E-07/(2*R1))493 C COMPUTE THE REAL (Sl) AND IMAG (S2) PARTS OF HALF SPACE494 C RESPONSE.495 Sl=0

352

496497498499500501 C502503504505506507508509 C510511 C512513514515516517518519520521522 C-523 C524 C525 C526 C527 C528 C529530531532533

534535536537538539540541542543544545546547548549550

S2=0T5=T4*3. 14159/180.DO 100 J=I,NRS=SQRT( X2( J )**2+Y9**2)G2=G*R8EX REALTl=-EXP( -02 )*COS( G2 )-G2*EXP( -02 )*COS( 02)T2=-G2*EXP( -02 )*SIN( 02 )+2-3*( X2( J )/R8 )**2IF( X2(J). EQ. 0. ) X2(J)=I. E-10IF( X9. EQ. 0. ) X9=1. E-10T9=3. 14159-ATAN( Y9/X2( J))T8=3. 14159-ATAN( Y9/X9)Si=( ( TI+T2 )/R8**3 )*COS( 3. 14159-T8-T5) + $1EY REALSI=3*SIN(T9)*COS(T9)/RS**3 *SIN(3. 14159-TS-T5) + SEX IMAGINARYT1=-EXP( -02 )*SIN( 02 )-02*EXP( -02 )*SIN( 02)T2=G2*EXP( -G2 )*COS( 02)

100 S2=( ( T1+T2 )/R8**3 )*COS( 3. 14159-T8-T5) + 52TI=M9*R1*SQRT( S1**2+S2**2 )/NT2=2*3. 14159ZI=TI/T2HALFER=V9-Z 1RETURNENDCOMPLEX FUNCTION HZO(X)

HZ COMPONENT/Cl FOR A=O (GROUND CASE) WHEREPARAMETER

X = DIMENSION ARGUMENT.. NOTE: X-XX DISPLACEMENT USED IN RHO IFL>O; ELSE (L=O) X IS DUMMY PARM AND WHERE RHO IS GIVEN INCOMMON/SHARE/--PLUS OTHER PARAMETERS INCOMMON/MODEL/--SEE COMMON STATEMENTS BELOW--

DOUBLE PRECISION EPSoB8REAL Lo K( 31) tD( 30)COMMON/MODEL/K, D, MCOMMON/SHARE/

* EPS,* C2,C3,C4,* XX, VY, YY2, RHO, RHO2, DELRHO# Bp* L, DELo DEL2#* METHODo NZ. NWCOMPLEX

* TERM1 TERM2v CB, Cv* F4# HANKQp HANKCP N33# N3s 12,11EXTERNAL F4DATA

* N33/( 3. 0, 3. 0)/eN3/( 3. 0#,0. 0 )/a 12/(O. 02. 0 )/p I1/(O. 02-1. 0)/IF( YY. EQ. 0. 0) GO TO 9CB=BB8=BTERM1=( 0. O 0. 0)TERM2=DEL2*YY*( I1*( N3-( N3+CB*( N33+CB*I2 ) )*CEXP( CMPLX( -B,-B ) ) ) )/

* RHO**5IF( M. EQ. 1) GO TO 2

B3

551 C=2. O*YY/( DEL2*RHO)552 1 TERM1=C*HANKC( 1 ALOG( B )y F4 )/CB553 2 HZO=TERMI+TERM2554 3 RETURN555 9 HZO=(0. 0,0.0)556 0TO 3557 END558 COMPLEX FUNCTION F4(G)559 C--F4=G*F3( 6 )--SEE SUBPROGRAM F3.560 COMPLEX F3561 F4=0*F3(0)562 RETURN563 END564 COMPLEX FUNCTION F3(0)565 COMPLEX V1.F1sCONE566 DATA ONE/( 1. 0s0. 0 )/567 CALL RECUR1(GVl1F1)568 C=G569 F3=(VI*C*( ONE-Fl ) )/( (C+VI*Fl )*( C+Vl))570 RETURN571 END572 SUBROUTINE RECURI(o VltFl)573 C--BACKWARD RECURRENCE FOR COMPLEX Vl,Fl GIVEN DIMENSION ARGUMENT 3 AND:574 COMMON/MODEL/ PARAMETERS:575 C K(31) = NORMALIZED CONDUCTIVITY ARRAY (M VALUESoWHERE K(1)=1.0).576 C D( 30) =' LAYER THICKNESS ARRAY (M-1 VALUES) D=2*THICKNESS/DEL,577 C M = NUMBER LAYERS (M. GE. 1. AND. M. LE. 10)578 C SPECIAL CASE WHEN M=l1 (HOMOGENEOUS--D IGNORED)579 C580 C--NOTE: GPK,D ARE DIMENSION581 C582 C583 COMMON/MODEL/Ke Do M584 REAL K(31)*D(30)585 COMPLEX C, VMo V1,oF1EVDoONE586 DATA ONE/( 1. 0s0. 0 )/587 F1=ONE588 02=0*0589 VM=CSQRT( CMPLX( G2p 2. O*K( M)))590 IF(M. EQ. 1)GO TO 2591 J=M-1592 1 Vl=CSQRT( CMPLX( G2, 2. 0*K( J)))593 EVD=(O. p0. 0)594 C=-Vl*D(J)595 C2=REAL( C)596 IF(C2. LT.-80) GO TO 10597 EVD=CEXP( -VI*D( J))598 10 C=( ONE-EVD )/( ONE+EVD)599 F1=( VM*F1+V1*C )/( V1+VM*F1*C)600 IF(J. EQ. 1) GO TO 3601 J=1J-1602 VM=V1603 GOTO1604 2 VI=VM605 3 RETURN

354

606607608609610611612613614615616617618619620621622623624625626627628629630631632633634635636637638639640641642643644645646647648649650651652653654655656657658659660

ENDCOMPLEX FUNCTION HANKC( No Xi FUN)

C--COMPLEX HANKEL TRANSFORM OF ORDER N( =0, 1) AND ARGUMENT X( =ALOG( B))

C BY CONVOLUTION FILTERING WITH COMPLEX FUNCTION "FUN"--

CC PARAMETERS:CC N = 0 FOR JO-HANKEL TRANSFORM FILTER

C = 1 FOR J1-HANKEL TRANSFORM FILTER

C X = DIMENSION ARGUMENT (=ALOG( B) AT CALL) OF THE HANKEL TRANS

C IN FORM: INTEGRAL FROM 0 TO INFINITY OF FUN(G)*JN(GB) DG

C FUN( G )= EXTERNAL DECLARED COMPLEX FUNCTION NAME.

C NOTE: IF PARMS OTHER THAN G IS REQUIRED, USE COMMON IN

C CALLING PROGRAM AND IN SUBPROGRAM FUN.

C

C--THE RESULTING COMPLEX CONVOLUTION SUM IS GIVEN IN HANKC; THE HANKEL

C TRANSFORM IS THEN HANKC/B WHICH IS TO BE COMPUTED AFTER EXIT FROM

C THIS ROUTINE....C

C E. G: H=HANKC( N# ALOG( B ), FUN )/B WHERE CALLING PGM HAS THE STATEMENTS:

C EXTERNAL FUN

C COMPLEX FUN9HANKC,HCC

COMPLEX FUN

C--J1-FILTER ABSCISSA ? WEIGHT ARRAYS:DIMENSION Al( 48)DIMENSION AO(61)DIMENSION W1(48)DIMENSION WO(61)DATA Al/

*-4. 5307316E 00,-4. 3004731E 00,-4. 0702146E 00,-3. 8399561E 009

*-:3. 6096976E 00,-3. 3794391E 001-3. 1491806E 00m-2. 9189221E 00,

*-2. 6886636E 009-2. 4584051E 00,-2. 2281466E 00,-1. 9978881E 00.

*-1.7676296E 00,-1. 5373711E 00s-1.3071126E 00s-1. 0768541E 00.

*-8. 4659563E-01, -6. 163:3713E-019 -3. 8607863E-019 -1. 5582013E-01,

* 7. 4438369E-02o 3. 0469687E-01, 5.34955:37E-01, 7. 6521387E-01,

* 9. 9547237E-01, 1. 2257309E 001 1. 4559894E 00o 1. 6862479E 00t

* 1. 9165064E 00o 2. 1467649E 00, 2. 3770234E 00. 2. 6072819E 005

* 2. 8375404E 00 3. 0677989E 00, 3. 2980574E 00, 3. 5283159E 00

* 3. 7585744E 00. 3. 9888329E 00, 4. 2190914E 00, 4. 4493499E 00a

* 4. 6796084E 009 4. 9098669E 00, 5. 1401254E 00, 5. 3703839E 00.

* 5. 6006424E 00. 5. 8309009E 00, 6. 0611594E 00, 6. 2914179E 00/

DATA Wl/

* 3. 1010561E-06, 1. 8802098E-05 5. 4819540E-05. 9. 2891602E-068

* 1. 5523239E-04o 3. 0344652E-05a :3. 5:338744E-04, 1. 4798002E-048

* 7. 7342377E-04, 5. 3570857E-04a 1. 7170605E-03; 1. 6387239E-03,

* 3. 9247683E-03t 4. 5796508E-03t 9. 2111468E-03. 1. 2130467E-02,

* 2. 1938415E-02s 3. 0853660E-02i 5. 1973594E-02, 7. 4661566E-021

* 1. 1775455E-01 1. 6353574E-01, 2. 3127545E-01, 2. 7368461E-019

* 2. 8059285E-01, 1. 2875840E-01 -1.53.0437E-01 -4. 5659951E-01

*-3. 6077766E-02o 4. 2985683E-01,-2. 1506075E-01 -2. 3624312E-02,

* 8. 9316746E-02.-7. 4344203E-02, 4. 8572965E-02o-3. 0088872E-02a

* 1. 8846544E-02o-1. 2158687E-02o 8. 0708759E-03 ,-5. 4706275E-03,

* 3. 7554604E-03,-2. 5929707E-03o 1. 7909426E-03,-1. 2320277E-03,

31F

355

661662663664665666667

/ 668669670671672673674675676677678679680681682683684685686687688689690691692693694695696697698699700701702703704705706707708709710

711712713714715

3.2266260E-03,-2.8649137E-03, 2.5677680E-2. 1115187E-03,-1.9334662E-03) 1.7800248E-1.3468317E-03/

HANKC=(0. O,0. 0)IF(N. EQ. 1) GO TO IDO 2 I=1#61

2 HANKC=HANKC+FUN(EXP(-X+AO(I)))*WO(I)RETURN

1 DO 3 I=1#483 HANKC=HANKC+FUN(EXP(-X+AI(I)))*WI(I)RETURNENDCOMPLEX FUNCTION EXO(X)

C--EX COMPONENT/(ECON*C1) FOR A=O (GROUND CASE)C PARAMETERC X = DIMENSION ARGUMENT..NOTE: X-XX DI#C L>O; ELSE (L=O) X IS DUMMY. PARM ArC COMMON/SHARE/--PLUS OTHER PARAMETEC COMMON/MODEL/--SEE COMMON STATEMEFC

DOUBLE PRECISION EPS, B8REAL L..K( 31),D( 30)

-03,-2. 3202655E-03*-03, -1. 6465436E-03,

SPLACEMENT USED IN RHO IFN4D WHERE RHO IS GIVEN INERS INNTS BELOW--

* 8.4095286E-04,-5.6749747E-04, 3.7718405E-04,-1.5891835E-04/C--JO-FILTER ABSCISSA ? WEIGHT ARRAYS:

DATA AO/*-6.8348046E 00,-6.6045461E 00,-6.3742876E 00,-6. 1440291E 005*-5.9137706E 00,-5.6835121E 00,-5. 4532536E 00,-5.2229951E 00,*-4.9927366E 00,-4. 7624781E 00,-4. 5:22196E 00,-4. 3019611E 00o*-4.0717026E 00,-3.8414441E 00,-3. 6111856E 00,-3.3809271E 00,*-3. 1506686E 00,-2.9204101E 00,-2.6901516E 00,-2. 4598931E 00,*-2. 2296346E 00,-1. 9993761E 00, -1. 7691176E 00,-1. 5388591E 00,*-1. 3086006E 00,-1.0783421E 00j,-8.4808358E-01,-6. 17:32508E-01,*-3.8756658E-01 -1.5730808E-01, 7. 2950416E-02, 3. 0320892E-01,* 5.3346742E-01 7.6372592E-01, 9.9:33 98:442E-01, 1.2242429E 00,* 1.4545014E 00, 1.6847599E 00, 1.9150184E 00, 2. 1452769E 00,* 2.3755354E 00, 2.6057939E 00, 2.8360524E 00, 3.0663109E 001* 3.2965694E 00s 3.5268279E 00, 3. 7570864E 00, 3.9873449E 00,* 4. 2176034E 00, 4.4478619E 00, 4. 6781204E 00, 4.9083789E 00s* 5. 1386374E 00, 5.3688959E 00, 5.5991544E 00, 5.8294129E 00,* 6.0596714E 00, 6.2899299E 00, 6.52018:34E 00, 6.7504469E 00,* 6.9807054E 00/

DATA WO/* 7. 3260937E-04t 5. 6326423E-04i 1. 3727237E-04, 7. 5331222E-04,* 3. 5918326E-04, 1. 0500608E-03, 7. 1530982E-04, 1. 5160070E-030* 1.2841617E-03, 2.24979S5E-03, 2. 1906186E-0:3, 3. 4076782E-03,* 3. 6321245E-03, 5. 2376028E-03, 5. 9212519E-03, 8. 1315877E-030* 9.5527062E-03, 1. 2708615E-02, 1. 5305589E-02, 1. 9941086E-02,* 2.4396626E-02, 3. 133:3652E-02. 3. 86:33065E-02, 4.9127993E-028* 6.0824806E-02o 7. 6314344E-02t 9. 392:346E-02, 1. 1545027E-01e* 1. 3868663E-01 1.6248847E-01, 1. 8114:332E-01, 1.8424433E-01.* 1.5556741E-Olo 6.8592481E-02,-8.8339029E-02o-2.8819226E-01*-3.5565260E-01, -5.6288677E-02, 4.8186942E-o1,-5.1516453E-02,*-2.6102989E-01 2. 1416490E-01,-9. 4490687E-02, 2.6196370E-02#*-5. 1097828E-04# -6. 6032948E-03, 7. 519:3619E-03, -6. 7854344E-03t

5. 8044372E-03, -4. 9354894E-03, 4. 232:3106E-03, -3. 67:33648E-03

716 COMMON/MODEL/Ki Do M717 COMMON/SHARE/718 * EPSs719 * C2,C3oC4,720 * XXP YYo YY2s RHO, RH02, DELRHOo Bs721 * L, DELa DEL2,722 * METHOD. NZ# NW723 COMMON/FX/XR2724 COMPLEX725 $ F7, Fi1l HANKQ, HANKCs TERMI. TERM2, 11726 EXTERNAL F7oF11727 DATA Il/(0. 0,-1.0)/728 EXO=(0.Os O. 0 )729 TERM2=( 0. 00. 0)730 XD=XX731 IF(L. EQ. 0. 0) GO TO 10732 XD=X-XX733 10 XR2=XD**2/RHO2734 C4=1. 0-2. 0*XR2735 IF(M.EQ.1) GOTO 3736 1 ALOGB=ALOG( B)737 TERM1=HANKC( 0 ALOlOB. F 1 )/B738 IF( C4. NE. 0) TERM2=HANKC( 1. ALOGB, F7 )/B739 2 EXO=-TERMI/DEL+C4*TERM2/RHO740 3 TERMI=CMPLX(B#B)741 TERM2=I *DEL2*( 1. 0-3. 0*YY2/RH02+( 1. O+TERMI )*CEXP(-TERMI))742 $/( RHO*RH02)743 EXO=EXO+TERM2744 RETURN745 END746 COMPLEX FUNCTION F7(G)747 COMPLEX V1,F1,LIsI1,ONE,TWOvC748 DATA I1,ONEP TWO/(O. O.1. 0)9( 1. 090. 0),(2. 0,0. 0)!749 CALL RECUR2(GoV1F1L1)750 C=G751 F7=I1*V1*(LI-ONE)+(TWO*Vl*(ONE-F1 ))/((C+Vl*F1 )*(C+V1))

752 RETURN753 END754 COMPLEX FUNCTION F11(G)755 COMPLEX VlFloL1,o1,sONEsTWOmCsT1,T2756 COMMON/FX/XR2757 DATA I1,ONE,TWO/(0. 0,1. 0),( 1.0,0. 0),(2. 0.0.0)/758 CALL RECUR2(G,V1,FloL1)759 C=G760 TI=( TWO*VI*( ONE-Fl ))/((C+VI*Fl )*(C+V1))761 IF(XR2. EQ. 0. 0) GO TO 1762 T2=Il*V1*( LI-ONE )+T1763 1 F11=C*(T1-XR2*T2)764 RETURN765 END766 SUBROUTINE RECUR2( Gt Vlo F. Li)

767 C--BACKWARD RECURRENCE FOR COMPLEX Vl,FlsLl GIVEN DIMENSION ARGUMENT G A

768 COMMON/MODEL/ PARAMETERS:769 C K(31) = NORMALIZED CONDUCTIVITY ARRAY (M VALUES. WHERE K(1 )=1. 0).

770 C D( 30) = LAYER THICKNESS ARRAY (M-1 VALUES) D=2*THICKNESS/DEL.

C M = NUMBER LAYERS (M. GE. 1. AND. M. LE. 10)C SPECIAL CASE WHEN M=l ( HOMOENEOUS--D IGNORED)CC--NOTE: G,-K#D ARE DIMENSIONCC

771772773774775776777778779780781782783784785786787788789790791792793794795796797798799800801802803804805806807808809810811812813814815816817818819820821822823824825

DOUBLE PRECISION EPS, B8REAL Lo,K( 31 )p,D( 30)COMMON/MODEL/K, D MCOMMON/SHARE/

* EPS#* C2oC3sC4s* XX, YYp YY2s RHOs RHO2, DELRHO, B,* L DEL, DEL2s* METHODo NZo NWCOMPLEX

$ F7, F12, HANKQP HANKC, TERMI TERM2 I3EXTERNAL F7,F12DATA 13/( 0. 00-3. 0)

COMMON/MODEL/Ko Do MREAL K( 31 ),D( 30)COMPLEX C, YM Vl,F1#LI,E* ONEDATA ONE/( 1. 0.0. 0)1F1=ONEL1=ONE02=0*GVM=CSQRT( CMPLX( 02# 2. O*K( M)))IF( M. EQ. 1) GO TO 2J=M-1

1 VI=CSQRT( CMPLX( G2# 2. O*K( J)))E=(0. 0.°0. 0)C=-V1*D( J)C2=REAL(C)IF(C2.-LT.-80) GO TO 10E=CEXP( -VI*D( J))

10 C=( ONE-E )/( ONE+E)Fl=( VM*FI+VI*C )/( VI+VM*FI*C)E=K( J+I )*V1+K( J )*VM*L1*CIF( REAL( E). EQ. 0. 0. AND. AIMAG(E). EQ. 0. 0) E=( 1. OE-30# 1. OE-30)LI=( K( J )*VM*LI+K( J+l )*VI*C )/EIF( J. EQ. 1 ) GO TO 3

VM=V1GO TO 1

2 VI=VM3 RETURN

ENDCOMPLEX FUNCTION EYO( X)

C--EY COMPONENT/( ECON*CI) FOR A=O (GROUND CASE)C PARAMETERC X DIMENSION ARGUMENT.. NOTE: X-XX DISPLACEMENT USED INC L>Oi ELSE (L=O) X IS DUMMY PARM AND WHERE RHO IS GIC COMMON/SHARE/--PLUS OTHER PARAMETERS INC COMMON/MODEL/--SEE COMMON STATEMENTS BELOW--C

,57

4 RHO IFIVEN IN

Z59

826 EYO=(O.OO.0)827 IF(YY. EQ.0.0) GO TO 5828 XD=XX829 IF(L. EQ.0.0) GO TO 10830 XD=X-XX831 10 IF(XD. EQ.O.O) GO TO 5832 IF(M. EQ. 1) GO TO 3833 1 ALOGB=ALOG(B)834 TERMl=HANKC(1,ALOGB, F7)/B835 TERM2=HANKC(0, ALOGBs F12)/B836 2 EY0=-2.0*TERM1/RHO+TERM2/DEL837 3 TERM2=13*DEL2/(RH0*RHO2)838 EYO=XD*YY*(EYO+TERM2)/RHO2839 5 RETURN840 END841 COMPLEX FUNCTION F12(G)842 COMPLEX F7843 F12=G*F7(G)844 RETURN845 END$

EOF..

B8.1.4 EM-MDL Sample Calculations

Interactive Teletype Commands

Commands entered by operator are underlined.

I0 LNC f-- U [INSERT DATACRAIT FOWTRF RUN COMMANOS]

IlJ"!FL IL2 LAY EF. MOEEL TI.'TCP"LCLLA'I P OF .IT

F,I I" oGMOi/PAJEc IfT0 ' LI ci-

[NUMBER OF LAYERS]

1 [LAYER I.RESIST1VIJW]

R ~2 3 JLAMER 2, RESISTIVITY] [INPUT LAERED EAWT" MODEL.

[LAYER 1,THICKNES51

PAREISIc-71I'Y T11I C; Nj

0, L7 r,_+ V 4 . ] , E+ zC 1+ (7,• I+ y_7

71PEIPOL _LL,Y,EN'i =

" .. LENGITH =

FOCCOIL APE

CI 'IE-ECEIVE LOCfTI0ilFYNOF =

31E.CTPIN'Y TON

92Y= +

,, P, 0 17, C E+ c; I

MIN. AN MM"s.,. FEII\.INSIk

1222c :

FR ECELGINNIN'G FPI~f- ,.JCY =

NO. OF EICTG,IILS =

POIN'TS PEP IICLLI] -

.2218

.C,ALC,*-[c.ALCULATr, F1E.LD$ ANt PRINT ON~ LINE PR?INTE.R.)

'359

LINE PRINTFR OUTPUT.

INPUT PARAMETERS

ID - - 2 LAYER MODEL TESTCALCULATE HZ FIELDNO. OF LAYERS = 2RE.ISTIVITY( 1 )= . 10000E+04RESISTIVITY(2)= . 10000E+03DIPOLE MOMENT = .10000E+05TX L=NlITH = . 10000E+04COIL. AREA = .25000E+05RECEIVER LOCAT I ONRANGE= .31600E+04

ORIENTATION= .90000E+02X= . 00000E+01Y= 31600E+04INITIAL MIN. APPARENT RESISiINITIAL MAX. APPARENT RESIST

THICKNESS( 1 )= . 10000E+04THICKNESS( 2 )= .99999E+20

rIVITY =TIVITY =

50. 000001200. 00000

FREQLIE NCY.10000E-02. lOOOE--l* 10000E+O0. 10000E+01.10000E+026 10000E+03.10000E+04.10000E+05

AMFL I TUDE( H)" 79705E-04 -.

" 796-6E-)4 -.795142E-04 -

.77744E-04 -.

.63782E-04 -.

. 37687E-04 -.

.65705E-05 -.

.60642E-06 -.

PHASE69c43E+0026029E+0057901E+0039280E+0115649E+0234656E+0296649E+0290003E+02

COIL OUT (UV)S1573:3-01. 157:32E+00.15709E+01.15346E+02.12590E+03.74392E+03. 12970E+04 -.

.11971E+04 -.

PHASE APPARENT RESISTIVITY CONVERGENCE839302E+028,:740--E+02

89421E+0286072E+0274351E+0255344E+026649:3E+0130060E-02

000)OE+01OO0)CE+011 14:E+0313295E+0319211E+0353623E+0310884E+0499996E+03

* e

Hz FIELDS

SUMMARY OF

0

0000

(~J6'0

Ex FIELDS LINE PRINTER OULTPUT

SUMMARY OF INPUT PARAMETERS

ID - - 2 LAYER MODEL TESTCALCULATE ER F I ELDNO. 1F- LAYERS = 2RESISTIVITY( 1 )= .10000E+04 THICKNESS(1 )=RESISTIVITY(2)= .10000E+03 THICKNESS(2)=DIPOLE MOMENT = 10000E+05TX t-E.NGTH = ,10000E+04RECEIVER DIRECTION = .27000E+03RECE 1 VER LOCAT I ONRANI--E= 31600E+04ORIEN- *ATION= ..90000E+02X= .00000E+01Y= ..31600E+04INITIAL_ MIN. APPARENT RESISTIVITY = 50.INITIAL MAX. APPARENT RESISTIVITY = 1200.

.10000E+040 99999E+20

Receiver

Re caei vc-(KY>) dipole,

Orientcition

0000000000

Trans rni t-er(Tx) dipole

FREQUENCY10000E-0210000E-0110000E+0)

* 1000(E+01* 10000E+02.10000E+03. 10000E+04. 10000E+05

AMPLITUDE( MV/M)126?*/E-01

.12700E-011272:3E-01

.13314E-0121987E-01

.74634E-01

. 10396E+00

.10088E+00

12485E. 122:32E

11441E.89585E33656E

.39061E-.30344E.37729E

PHASE REAL-01 12699E-01-+00 . 1270C)E-0 1:+01 .12720E-0101 .13V52E-01

+02 18:301E-01:+02 .57951E-01:1+O . 10381E+00-03 .10088E+00

I MA1 I NARY2767:3E-05271 1:::E-0425404E-0320733E-02121:=:5E-0147031E-0155029E-0266426E-06

APPARENT RESISTIVITY";17:3E+C)32 17 'E +0:3

25I'/2E+03

.2965:3E+:3698,--/:E+03

. 10374E+04

.99999E+03

CONVERGENCE00000000

()A6%

tl.

. .. . . .. -. 0 . - -- l 4 Ab V dk- 6 9 .16, .0- N.F %0 .

B8.1.5 EXAMPLES OF APPARENT RESISTIVITY CURVES IN THE EM TRANSITION

INTERPRETATION REGION

E and H EM transition region apparent resistivities werex z

calculated for the same model that was used for the DC and MT sample

calculations. We will compare the EM apparent resistivity curves at

ranges of 316 a, 1,000 m, 3,160 m, and 10,000 m with the DC and MT

apparent resistivity curves.

At short range (316 m) the E electric field apparentx

resistivities are approximately equal, over the entire frequency

range, to the DC apparent resistivity. The H apparentz

resistivities at 316 m range cannot be calculated over most of the

frequency range because the signal is closely equal to the free-

space coupling value and, therefore, independent of the earth

properties.

At intermediate ranges (1,000 m and 3,160 m), the EM curves are

asymptotic to the MT curve at high frequencies. The EM curve equals

the DC apparent resistivity or tle free-space coupling signal at

lower frequencies.

At very long range (10,000 m), both the E and H ap parentx z

resistivity curves are approximately equal to the MT curve over the

entire frequency range.

If the earth model were a homogeneous half space with a

resistivity equal to the top layer resistivity (1,000 ohm m), then

the apparent resistivity would be equal to 1,000 ohm m at all

frequencies. The decrease in apparent resistivity at low

3(03

frequencies, which is observed at all ranges greater than 316 m,

indicates the presence of the deep, less-resisttve layer.

The apparent resistivity curve at very long range (10,000 m)

shows the greatest variation from high to low apparent resistivity,

but the apparent resistivity curves at much shorter raLige also

clearly indicate the presence of the deep, less-resistive layer.

The advantage of working close to the transmitter is that the

measurement is more localized and we do not have to assume laterally

homogeneous layers over large distances.

For more complicated models, the behavior of the apparent

resistivity curve qualitatively indicates the structure just as for

the two-layer case. For instance, with a three-layer model

consisting of low-high-low resistivity, the apparent resistivity

curve would be low-high-low. The extent to which the apparent

resistivity curve approaches the true resistivities depends on the

range and also, in this case, on the thickness of the middle layer.

1000

IO1

wOUu . I ' ---

DCkE M ,o/

Ole

s 1000 rm

ow aw I I . . . .II

I00

000

t00

FREQUENCY, IN HFRTZ

Fig. B8.1.5. Comparison of MT, DC, and EM dipole-dipole apparentresistivities for electric (E ) field.x

3(04

_- Dr.., l EM-" i-/f

/ MT

/.e .. oo

do.ao

co

02

cc

i1-

4C

I0,000

FREQUENCY) IN HERTZ

Fig. B8.1.6.resis tivities

Comparison of MT, DC, and EM dipole-dipole apparentfor H field.

z

3(65

- 1#* do,,I -

//

//

1131(pm.00

L _ _--- --00,,"000

1000

u~100

~.1000

0

P

U,'Fn5 100

I--.1o

w oo

z

0.

100

B8.2.1. NON-LINEAR LEAST SQUARES INVERSION - MARQUARDT'S ALGORITHm

We used Marquardt's algorithm (1963) for refining an initial

guess in order to obtain a solution (earth model) to a non-linear

equation (Equations B8.1.1, B8.1.2, or B8.1.3), given a set of

observed Hz, Ex, or E fields. The sum of squares differencesz y

between the observed and calculated fields is minimized with this

algorithm. Both the observed magnitude and phase of the H, E , or

E field are used.y

The difference between observed and calculated values is

weighted by the estimated error in the observed data. We will

minimize chi square, defined as:

[Fci mantd + w. j phase (B8.2.7)2FCi -Fo 12F C. F 0

x w. magnitude + Y.

Where: F = observed field,0.1

Fc. = fields calculated from equation B8.1.3,1

w. = estimated error in the observed data.I

It is helpful to visualize an N dimensional paraie.t(c space in

2which we plot x ( the least squared error) versus the model

parameters (resistivities, thicknesses, and time uncertainty). The

non-linear least squares algorith a is designed to locate a minima on

2the X surface.

One method of finding a minima is the method of steepest

descent. We search for a minima in the direction defined by the

negative gradient of X2 . Successive searches (calculation of the

gradient) brings us closer and closer to the iiiaiina. This method is

guaranteed to converge, but convergence becomes very slow when the

solution is near the minima. Another method of searching for a

minima is the Taylor series method. The linear terms of a Taylor2

series expansion are used to approximate X • The correction to the

current point in parameter space is then found by linear least

squares. Since the problem is actually non-linear, we iterate

toward the solution by making successive linear approximations.

This method generally converges rapidly when it is near a minima,

but may diverge if it is not close.

Marquardt's algorithm makes a linear interpolation between the

correction vector indicated by the steepest descent method and tliat

indicated by the Taylor series method. During the first few

iterations, when we are far from the minima, the correction vector

is made essentially equal to that of the steepest descent method in

order to insure convergence. As we approach the minima (sum of

squares decreases), the algorithm places more weight on the Taylor

series method in order to speed convergence.

To insure reasonably rapid convergence, it is important to have

all axes of parameter space scaled to roughly the same range. Since

resistivities and depths typically range over several decades, it is

natural to use the logarithm of these quantities. The time delay is

scaled by log (1000-1000 At).

3cos

To perform Marquardt's calculation, we used a computer program

from the University of Wisconsin, Madison Academic Computing Center

called GAUSSHAUS (Meeter and Wolfe, 1966; Wertz, 1968). This

program also calculates correlation coefficients between parameters

and estimates error bounds for the layer parameters assuming linear

statistics around the minima. Several recent papers on resistivity

and EM inversion methods nave found these statistics useful in

estimating the uncertainty of the model (e.g., Inman, 1975; Glenn et

al., 1973; Ward et al., 1976). We found that for the data discussed

here, the linear statistics generally lead to error estimates of an

order of magnitude or more even when the range of models which fit

the data is less than a factor of two. Furthermore, we found that

the parameter correlation coefficients are generally quite large, so

it is difficult to apply statistics for a single variable, which are

assumed independent of all other variables. We estimate instead the

uncertainty in the model parameters by searching parameter space for

models which fit the data within a prescribed amount.

B8.2.2 Program EM-INV Listing.

I c2 C PROGRAM EM-INV3 C4 C USE NON-LINEAR LEAST SQUARES INVERSION TO DETERMINEI C THE EARTH MODEL FOR WHICH THE CALCULATED EM FIELDS BEST

6 C MATCHES A SET OF OBSERVED EM FIELDS.7 C8 C PROGRAM WRITTEN BY B. STERNBERG, 11/76.9 C SUBROUTINES HZO. EXO. EYO, RECURI.RECUR2, HANKC,

10 C F3,F7oF11, AND F12 WERE WRITTEN BY W. L. ANDERSON (USGS)

11 C NTIS REPORT PB 238 199, 1974.12 C SUBROUTINES GASAUS (GAUSHAUS)s GASS59, MATIN, AND13 C SYMQR ARE FROM THE UNIVERSITY OF WI'-CON'SIN COMPUTING

14 C CENTER; D. A. MEETER AND P. J. WOLFE (GAU-HAUS)p 196515 C AND H. J. WERTZ (GASAUS)# 1968.16 C PROGRAM WRITTEN FOR A DATACRAFT 6024 (UW GEOPHYSICS).17 C18 DIMENSION T52(5 ), R92( 5)o,T92( 5), X52(5)o Y92(5)19 REAL M92( 5 ),L92( 5 ), L82( 5 ).NFREQ2(5p 15)20 INTEGER IEH2( 5 ),NF2( 5), IPH2( 5)21 DIMENSION TH( 15 ), Y( 70), SCRAT( 1000)22 COMMON/CONST/ M92, L92, L82, IEH2. IPH2, NF2, NFREQ2, NDATA5, T52,23 - R92, T92. X52, Y92, M524 DIMENSION R(5),D(5)25 DIMENSION TO(5)

26 INTEGER RFIX(5),DFIX(5),TFIX(5)27 INTEGER 0628 DIMENSION AID(10)29 DIMENSION BOUNDL( 9 ), BOUNDU( 9)30 REAL K7(5)jD6(7)oL31 DOUBLE PRECISION EPS32 COMMON/PARAMS/ Ms RFIX, R, DFIX, D, TFIX,TO. NDATA33 COMMON/CTEST/ RMAX34 COMMON/GAS/ NOBo Y, NP, TH, BOUNDL, BOUNDU, EPS1, EPS2, EPS3,35 - MAXITo GAMMA, FNUPSCRAT36 COMMON/SDEV/ SDY( 5, 30)37 COMMON/MODEL/ K7, D6, M638 COMMON/SHARE/39 * EPS:40 C2.,C3,C4#41 * XX, YY, YY2, RHO. RHO2, DELRHO, B,42 * Lo DEL. DEL2,43 * METHOD, NZ, NW44 COMMON/FX/ XR2

45 COMMON/GASPAR/ DUMIES( 7 ), PIVOTM46 DATA IER/2HER/, IHZ/2HHZ/, IS/1HS/47 DATA IY/lHY/, IT/'H*/, IB/1H /48 06=649 17=750 C51 C LOCATION AND IDENTIFICATION INFORMATION.

52 READ( 17,40) (AID(J),J=l, 10)53 40 FORMAT( 10A6)54 WRITE(06,41 ) (AID(J),J=1,10)55 41 FORMAT(1H18-" LOCATION - ID - - /,10A6//)

370

56 C57 C DETERMINE CURRENT DATE AND TIME.58 :ASSE59 TOK 1260 BLU $INFO61 TAM ITIME62 :END63 ITIME=ITIME/1064 WRITE(6, 101) ITIME65 101 FORMAT(*' TIME='j 17- SECS AFTER MIDNIGHT')66 :ASSE67 TOK 268 BLU $INFO69 TEM IMON70 TAM IDAT71 TIM IYR72 :END73 WRITE(6o 102) IMONP IDAT, IYR74 102 FORMAT(" DATE= , 3A3/)75 C76 C READ THE INITIAL GUESSES FOR THE EARTH PARAMETERS WE

77 C WISH TO ESTIMATE - - LAYER RESISTIVITIES, LAYER THICKNESSESP

78 C AND ZERO TIME UNCERTAINTY.79 C A * IN COLUMN 1 MEANS THE PARAMETER IS FIXED.80 C81 C NO. OF LAYERS82 READ( 17, 111) M83 M5=M84 111 FORMAT( II)85 DO 120 I=1,oM86 120 READ( 17, 112) RFIX( I ),R(I)87 112 FORMAT(A1sE12.5)88 MI=M-189 DO 121 I=IvM190 121 READ( 17s 112) DFIX( I ),D(I)91 READ( 17s 111) NDATA92 NDATA5=NDATA93 DO 90 I=1#NDATA94 90 READ( 17# 112) TFIX( I)aTO(I)95 C96 C STORE THE LOGARITHMS OF THE VARIABLE MODEL

97 C PARAMETERS IN THE WORKING ARRAY.

98 NP=O99 ERR=O100 DO 130 K=I,M101 IF( R( K). LT. 0. 01 . OR. R( K). GT. 1. OE+8) ERR=1102 IF(RFIX(K) .EQ. IT) GO TO 130103 RFIX(K)=IB104 NP=NP+1105 TH( NP )=ALOG( R( K))106 130 CONTINUE107 MI=M-1108 DO 140 K=I#M1

109 IF( D( K). LT. 0. 01 . OR. D( K). GT. 1. E+8) ERR=I110 IF(DFIX(K) .EQ. IT) GO TO 140

DFIX( K )=IBNP=NP+ 1TH( NP )=ALOG( D( K))

140 CONTINUEDO 145 K=I#NDATAIF( TO( K). LT. -1. 0 . OR. TO( K). GT. +1. 0) ERR=IIF (TFIX(tK). EQ. IT) GO TO 145TFIX( K )=IBNP=NP+ 1SCALE THE TIME UNCERTAINTY TO SAME ORDER OF MAGNITUDEAS THE OTHER PARAMETERS.TH( NP )=ALOG( 1000. -( 1000. *TO( K)))

145 CONTINUE

111112113114115116117118119120 C121 C122123124 C125 C126127128129130131132 C133 C134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165

ETERS.

UND" )

371

READ THE UPPER AND LOWER BOUNDS FOR THE MODEL PARAMEDO 185 K=I#NPREAD( 17, 113) BOUNDL( K), BOUNDU( K)

113 FORMAT(2E12. 5)IF( BOUNDL(K). LT. 0. 01 . OR. BOUNDL(K). 3T. 1. E+P) ERR=lIF( BOUNDU(K). LT. 0. 01 . OR. BOUNDU( K). GT. 1. E+8) ERR=I

185 CONTINUE

PRINT THE INITIAL GUESSES.WRITE( 06P301)

301 FORMAT( "/ INITIAL MODEL PARAMETERS-)WRITE( 06, 155) M

155 FORMAT(-' NO. OF LAYERS = "P15)D( M )=. 99999E+20WRITE( 06, 159)

159 FORMAT( 25XIe INITIAL GUESS LOWER BOUND UPPER BOIK6=0DO 160 I=IoMIF( RFIX( I ). EQ. IB) GO TO 190BL=R( I )BU=R( I )GO TO 160

190 K6=K6+1BL=BOUNDL( K6)BU=BOUNDU( K6)

160 WRITE(06o 165) RFIX( I ), I,R( I ),BL,BU165 FORMAT( 1X; Als RESISTIVITY( *aI1, )="3(5X) E12. 5))

MI=M-1DO 170 I=I,M1IF(DFIX( I ). EQ. IB) GO TO 171BL.=D( I)BU=D( I)GO TO 170

171 K6=K6+1BL=BOUNDL( K6)BU=BOUNDU( K6 )

170 WRITE(06, 172) DFIX( I ), I,D( I ),BLsBU172 FORMAT( 1X,,A1," THICKNESS( 'oIlp' )= p2X 3(5X#E12. 5))

DO 175 I=I#NDATAIF( TFIX( I ). EQ. IB) GO TO 176BL=TO( I )

166167168169170171172173174175176177178 C179 C180181182183 C184 C185 C186 C187 C188189190 C191 C192193 C194195196 C197198199 C200201202 C203204205 C206207 C208209210211 C212213214215216217218 C219 C220

BU=TO( I )GO TO 175

176 K6=K6+1P.L=BOUNDL( K6)BU=BOUNDU( K6)

175 WRITE( 06, 177) TFIX( I ), Is TO( I)s,BL, BU177 FORMAT(1X:A1," ZERO TIME( -,I1, " )=',2X,3(5X,E12. 5))

IF( ERR. EQ. 0) GO TO 178WRITE( 06, 179)

179 FORMAT(-' ONE OR MORE PARAMETERS OUT OF RANGE')STOP

178 CONTINUE

CONVERT THE BOUNDS TO LOG VALUES.DO 5 J=,NPBOUNDL( J )=ALOG( BOUNDL( J))

5 BOUNDU( J )=ALOG( BOUNDU( J))

READ IN THE PARAMETERS FOR N DATA SETS.

THE NUMBER OF DATA SETS INCLUDES THE NUMBER OF SITES

AND THE NUMBER OF FIELD COMPONENTS AT EACH SITE WHICH WEWILL FIT TO.READ( 17. 200) NDATA

200 FORMAT( 12)

READ IN ALL THE CONSTANT PARAMETERS FOR EACH DATA SET.

DO 1000 I=lPNDATACALCULATE ER OR HZ FIELDREAD( 17#201 ) IEH2( I)

201 FORMAT(A2)DIPOLE MOMENTREAD( 17,213) M92( I)

213 FORMAT(EI2.5)TX LENGTHREAD( 17,213) L92(I)IF ( IEH2( I ). EQ. IER) GO TO 241COIL AREAREAD (17,213) L82(I)

241 IF( IEH2( I ). EQ. IHZ) 00 TO 243RECEIVER DIRECTIONREAD(17,213) T52( I)RECEIVER LOCATIONs RANGE, ORIENTATION

243 READ(I17s,213) R92( I ), T92( I)X52( I )=R92( I )*SIN( ( T92( I )-90. )*3. 14159/180.Y92(I )=R92( I )*COS( ( T92( I )-90. )*3. 14159/180.FREQUENCIES TO BE CALCULATEDREAD( 17.249 )NF2( I)

249 FORMAT(12)NNF2=NF2( I)DO 250 J=lNNF2

250 READ(I7,213) NFREQ2(I,J)1000 CONTINUE

PRINT THE CONSTANT PARAMETERS FOR EACH DATA SETDO 2000 I=1PNDATA

-372

"Z73

221 WRITE(06s350)222 350 FORMAT(//' SUMMARY OF CONSTANT PARAMETERS)223 WRITE(06o 151) 1224 151 FORMAT(/" DATA SET', 15)225 WRITE( 06o 34:3 ) IEH2( I226 348 FORMAT( CALCULATE ,A2," FIELD")227 WRITE(06,370) M92(I)228 370 FORMAT(-' DIPOLE MOMENT ='PE12. 5)229 M92( I )=M92( I )/( 4*3. 14159)230 WRITE( 06,375) L92( I)231 375 FORMAT(' TX LENGTH =-,E12.5)232 IF( IEH2( I ). EQ. IER) GO TO 376233 WRITE( 06,380 ) L82( I)234 380 FORMAT(' COIL AREA =/,E12.5)235 376 IF( IEH2( I ). EQ. IHZ) 00 TO 378236 WRITE(06,377) T52(I)237 377 FORMAT( RECEIVER DIRECTION =',E12.5)238 T52( I )=T52( I )*3. 14159/180.239 378 WRITE(06,385) R92( I )o,T92( I )o X52( I ),Y92( I)240 085 FORMAT( RECEIVER LOCATION'./* RANGE= -,E12. 5/* ORIENTATION=,241 * E12.5/ X=',E12. 5/'" Y=,E12. 5242 WRITE(06,400)243 400 FORMAT(' FREQUENCIES )244 NNF2=NF2( I)245 WRITE(06P405) (NFREQ2(I,J)J=1,NNF2)246 405 FORMAT( 5E12. 5)247 2000 CONTINUE248 C249 C READ THE OBSERVED DATA VALUES250 C STORE IN THE FOLLOWING ORDER:251 C DATA SET 1 - MAG, DATA SET 1 - PHASE# ... DATA SET N - PHASE.252 C253 ERR = 0254 READ( 17,701) NOB255 701 FORMAT( I3)256 DO 710 I=I,NOB257 READ(17,712) Y(I)258 IF( Y( I ). LT. -1000 . OR. Y(IG). T. 1. E+6) ERR=I259 710 CONTINUE260 712 FORMAT(E12.5)261 C262 C READ STANDARD DEVIATIONS OF OBSERVED VALUES.263 C NORMALLY, THE MAGNITUE SDY IS A CONSTANT PERCENTAGE264 C OF THE OBSERVED Y (E.G. 5%). THE PHASE SDY IS265 C A CONSTANT (E. 0. 2 DEG. ).266 DO 713 J=I,NDATA267 NNF3=2*NF2( J)268 DO 713 I=1,NNF3269 READ(17,712) SDY(J,I)270 IF( SDY( J, I ). LT. -1000. OR. SDY( J, I ). GT. 1. E+6E) RR=1271 713 CONTINUE272 C273 C PRINT THE OBSERVED DATA VALUES AND STANDARD DEVIATIONS.274 C275 WRITE(06,720)

3-74

276 720 FORMAT(//" PRINT THE OBSERVED DATA VALUES-/)277 WRITE(06,725) (Y( I )# 1=1,NOB)278 725 FORMAT( 5E12. 5)279 WRITE( 06s726)280 726 FORMAT(/" PRINT STANDARD DEVIATIONS/)

281 DO 727 J=I,NDATA282 NNF3=2*NF2( J).283 WRITE(06,.725) (SDY(J# I), I=l#NNF3)284 727 CONTINUE285 IF( ERR. EQ. 0) GO TO 755286 WRITE( 06s 756)

287 756 FORMAT( - ONE OR MORE DATA VALUES OUT OF RANGE")

288 STOP289 755 CONTINUE290 C291 C WE WILL FIT TO Y( I )/SDY(I)292 K9=0293 DO 750 I=1,NDATA294 K8=0295 NNF2=NF2( I)296 DO 752 J=I#NNF2297 KS=KS+1298 K9=K9+1299 C MULTIPLY PERCENTAGE STANDARD DEVIATION TIMES MAGNITUDE.

300 SDY( Ia KS )=SDY( I. KS )*. 01*Y( K9)301 752 Y( K9 )=Y( K9 )/SDY( Io K8)302 DO 753 J=1PNNF2303 K9=K9+1304 K8=KS+1305 753 Y( K9 )=Y( K9 )/SDY( I, K8)306 750 CONTINUE307 C308 C READ AND PRINT THE GAUSHAUS CONSTANTS.

309 READ(17s810) MAXIT, GAMMA, FNU, EPSloEPS2. EPS:3, RMAX310 810 FORMAT(Il,7FIO.2)311 WRITE(06P820) MAXITa GAMMA, FNU. EPSI EPS2s EPS3oRMAX

312 820 FORMAT(/" GAUSHAUS PARAMETERS /

313 - 5Xf 'MAXIMUM NUMBER OF ITERATIONS ='s 15/314 - 5XPIGAMMA ='PElO.3/

315 - 5X#'FNU ='PF8.4/316 - 5X#'EPS1 ='jEIO.3/317 - 5X''EPS2 ='.EIO.3/318 - 5X#-EPS3 ='oEIO.3/319 - 5X "CHI SQUARE CONVERGENCE PARAMETER =',#F9. 5//)

320 C321 C CHAIN TO PROGRAM 2

322 CALL CHAIN( 6HPRGO02)323 END$OF..

1 NAME PRGO022 C3 C PROGRAM 2 - - CONTINUATION OF PROGRAM 1.4 C LEAST SQUARES INVERSION.

c6 EXTERNAL FOF7 C8 DIMENSION T52(5 ), R92( 5 ), T92( 5)o X52( 5)s,Y92( 5)9 REAL M92( 5 ),L92( 5 )o,L82( 5 )t NFREQ2(5, 15)

10 INTEGER IEH2( 5 ), NF2( 5), IPH2(5)11 DIMENSION TH( 15)s Y( 70 )1 SCRAT( 1000)12 DIMENSION F5( 100)13 COMMON/CONST/ M92, L92, L82, IEH2p IPH2, NF2, NFREQ2, NDATA5, T52014 - R92,T92,X52,Y92,M515 DIMENSION R(5)sD(5)16 DIMENSION TO(5)17 INTEGER RFIX(5)oDFIX(5),TFIX(5)18 DIMENSION BOUNDL(9),BOUNDU(9)19 REAL K7(5)oD6(7)tL20 DOUBLE PRECISION EPS21 COMMON/PARAMS/ M, RFIX. R, FIX, D) TFIX) TO. NDATA22 COMMON/CTEST/ RMAX23 COMMON/GAS!=,* NOB, Y, NP, THP BOUNDL, BOUNDU) EPSI) EPS2) EPS3124 MAXIT,GAMMA,FNUoSCRAT25 DIMENSION SDY(5o30)26 COMMONfSDEV/ SDY27 COMMON/MODEL/ K7, D6s M628 COMMON/SHARE/29 * EPSs30 * C2,C3,C4s31 * XX, Y, YY2o RHO, RH02, DELRHQ# B,32 * 1., DEL, DEL2j33 * METHODs NZ.,NW34 COMMON/FX/ XR235 COMMON/GASPAR/ DUMIES( 7 ). PIVOTM36 DATA IN/1HN/37 15="2138 OPEN I539 I06=,"2140 C41 C START THE TIMER.42 CALL BTIME43 100 CONTINUE44 C45 C PERFORM THE LEAST SQUARES INVERSION.46 C47 CALL GASAUS( 2. ,FOF NOBo Y. NP, TH, BOUNDUs BOUNDLi48 - EPSI, EPS2 EPS3, MAXIT, GAMMAo FNU, SCRAT)49 C50 C COMPUTE FINAL FIT AND PRINT.51 CALL FOF( TH,F5sNOB,NP)52 CALL PRTFIT( NDATA. NF2 NOB. NP. F5,. , SDYj NFREQ2)53 C54 WRITE( I06, 200)55 200 FORMAT(" CONTINUE ITERATIONS (Y N )?*)

37(o

56 READ( I5,201) IYN57 201 FORMAT( Al)58 IF( IYN. EQ. IN) GO TO 50059 C60 C READ THE GAUSSHAUS PARAMETERS (GAMMA AND NO. OF

61 C ITERATIONS) AND THE MODEL PARAMETERS (STORED IN TH(J)).

62 C FOR TO, INPUT (1000. -1000. *TO).

63 WRITE( 106,300)64 300 FORMAT(" GAMMA =)65 FEAD( I5,305) GAMMA66 305 FORMAT( E12. 5)67 WRITE(I06,330)68 330 FORMAT( r.NO. OF ITERATIONS =")

69 READ( 15,335) ANIT70 335 FORMAT( F5. 1)71 MAXIT=ANIT72 DO 310 J=I,NP73 WRITE( I06,320) J74 320 FORMAT( "'TH(,I1,) )75 READ(I5,305) TH(J)76 TH( J )=ALOG( TH( J))77 310 CONTINUE78 GO TO 10079 C80 500 CALL EXIT81 END82 SUBROUTINE FOF(TH,F5,NOB,NP)83 C84 C SUBROUTINE TO CALCULATE THE EM FIELD VALUES FOR THE EARTH MODEL.

85 C CALCULATE THE EM FIELDS FOR EACH OF THE N DATA SETS

86 C AND STORE SEQUENTIALLY IN ARRAY F5.

87 C88 DIMENSION TH(NP),F5(NOB)89 DIMENSION T52( 5 ), R92( 5 ), T92( 5), X52( 5 ), Y92( 5)

90 REAL M92( 5 ), L92( 5 ), L82( 5 ), NFREQ2( 5, 15)91 INTEGER IEH2(5), IPH2(5),NF2(5)92 COMPLEX SUMo SUM1, SUM2, ECON, TWOI93 COMPLEX HZO,EXO,EYO94 EXTERNAL HZO,EXO,EYO95 DOUBLE PRECISION EPS96 REAL M9,L9,L8, L7, K( 5 ), L97 DIMENSION SDY( 5, 30)98 INTEGER RFIX,DFIX,TFIX99 DIMENSION AMP( 15 ), PHZ( f5), X2( 15 ), F( 15 ), RE( 15), XIM( 15)

100 DIMENSION R(5),D(7),D2(7)101 COMMON/CONST/M92, L92, L82, IEH2, IPH2, NF2, NFRErj2 NDATA, T52 9 R92, T92,

102 - X52,Y92,M5103 COMMON/SDEV/ SDY

104 COMMON/PARAMS/ M6, RFIX( 5 ), R5( 5)o DFIX( 5), D5( 5),105 - TFIX(5),T0(5),NDATA5106 COMMON/MODEL/ K, Dj M107 COMMON/SHARE/108 *EPS,109 *C2,C3,C4,110 *XX, YY, YY2, RHO, RH02 9 DELRHO, B,

Z377

111 *La DEL, DEL2s112 *METHOD,NZ,NW113 INTEGER C6aS3114 INTEGER SCI CS115 INTEGER 05,06116 DATA IER/2HER/o IHZ/2HHZ/P IS/IHS/117 DATA IN/1HN/m IY/IHY/118 DATA TWOI/(O. 0s2.0)/119 06=6120 17=7121 M=M5122 NDATA5=NDATA123 C124 K3=0125 DO 3000 15=1..NDATA126 C127 IEH=IEH2( 15)128 M9=M92( I5)129 L9=L92( I5)130 L8=L82( I5)131 T5=T52( 15)132 R9=R92( 15)133 T9=T92( I5)134 X5=X52( I5)135 Y9=Y92( 15)136 SC=NF2( 15)137 DO 3010 J=I,SC138 3010 F(J)=NFREQ2(I5,J)139 C140 C CURRENT MODEL PARAMETERS141 CALL ALLPAR( TH)142 M6=M143 DO 4000 J=l,PM6144 R( J )=R5( J )145 4000 D( J )=D5( J )146 C147 C BREAK THE FINITE TRANSMITTER DIPOLE INTO N SMALLER DIPOLES148 C -(IF NECESSARY). EACH COMPONENT DIPOLE MUST HAVE A LENGTH149 C - SMALLER THAN ONE THIRD THE DISTANCE TO THE RECEIVER SITE.150 IF(L9. EQ. 0. ) L9=1.E-10151 229 N=INT(3*L9/Y9+1)152 L7=L9/N153 12=-i154 DO 230 L2=1#N155 12=I2+2156 230 X2(L2,)=( X5-L7/2. *(N-12))157 C158 232 IF( IEH. EQ. IHZ) 00 TO 231159 IF( IEH. EQ. IER) GO TO 500160 C161 C CALCULATE HZ FIELD USING EMFIN SUBROUTINES.162 231 SIG1=1/R(1)163 DO 250 I=1,M164 D2( I )=D( I)165 250 K(I)=(1. /R(I))/SIG1

378

166 L=L9/2167 DO 300 I=1.SC168 DEL2=1. 0/( 39. 4784E-7*SIG1*F( I))169 DEL=SQRT( DEL2)170 IF(M.EQ.1) GO TO 253171 MI=M-1172 DO 251 12=1,M1173 251 D( I2)=2. O*D2( 12)/DEL174 253 SUM=(O.0,0.O)175 DO 400 J=1#N176 XX=X2(J)177 YY=Y9178 YY2=YY**2179 XXX2=XX**2180 RHO2=XXX2+YY2181 RH0=SQRT( RH02)182 DELRHO=DEL*RHO183 B=RHO/LIEL184 4o0 SUM=SUM+HZO( 0)185 SUM=SUM/N186 RE( I )=REAL( SUM)187 300 XIM( I )=A I MAG( SUM)188 CALL POLAR( SC, RE, XIM, AMP, PHZ)189 CALL DRUM( SC, PHZ)190 C191 DO 306 I=1,SC192 AMP( I )=AMP( I )*M9193 AMP( I )=AMP( I )*7. 8957*F( I )*L8194 PHZ( I )=PHZ( I )*180. /3. 14159195 306 PHZ( I )=PHZ( I )+90196 C DIVIDE THE CALCULATED AMPLITUDE AND PHASE BY THE STANDARD197 C DEVIATION OF THE MEASURED AMPLITUDE AND PHASE AND198 C STORE IN ARRAY F5.199 J8=0200 DO 310 J=I,SC201 K3=K3+1202 J8=J8+1203 AMP( J )=AMP( J )/SDY( I5 sJ8)204 310 F5( K3 )=AMP( J)205 DO 320 J= #SC206 K3=K3+1207 J8=JS+1208 C PHASE CORRECTION BASED ON ZERO TIME UNCERTAINTY (-WT).209 PHZ( J )=PHZ( J )-( 2. *3. 14159*F( J )*TO( I5 )* 180. /3. 14159)210 320 F5( K3 )=PHZ( J )/SDY( 15s, J8)211 GO TO 2999212 C213 C CALCULATE ER FIELD USING EMFIN SUBROUTINES.214 500 SIG1=1/R(1)215 DO 1250 I=1#M216 D2( I )=D( I )217 1250 K( I )=( 1. /R( I )/SIG1218 L=L9/2219 DO 1300 I=1,SC220 DEL2=1. 0/( 39. 4784E-7*SIG1*F( I))

379

221 DEL=SQRT( DEL2)222 IF( M. EQ. 1) GO TO 1253223 M1=M-1224 DO 1251 12=1,M1225 1251 D( 12 )=2. 0*D2( I2 )/DEL226 1253 ECON=TWOI/( SIG *DEL2)227 SUM 1=( 0. 0, 0. 0)228 SUM2=( 0. 0, 0. 0)229 DO 1400 J=1*N230 XX=X2(J)231 YY=Y9232 YY2=YY**2233 XXX2=XX**2234 RHO2=XXX2+YY2235 RHO=SQRT( RH02)236 DELRHO=DEL*RHO237 B=RHO/DEL238 SUMI=SUMI+EXO( 0 )*ECON239 T4=T5*180. /3. 14159240 IF( T4. EQ. 90. . AND. T9. EQ. 90. ) GO TO 1399241 0 TO 1398242 1399 SUM2=0.243 GO TO 1400244 1398 SUM2=SUM2+EYO( 0 )*ECON245 1400 CONTINUE246 1'8=3. 14159/2.247 IF( XS. NE. 0) T8=3. 14159-ATAN( Y9/X5)248 SUM=( SUM1/N )*COS( 3. 14159-T3-T5 )-( SUM2/N )*SIN( 3. 14159-T8-T5)249 RE( I )=REAL( SUM)250 1300 XIM(I)=AIMAG(SUM)251 CALL POLAR( SC, REs XIMs AMPo PHZ)252 CALL DRUM( SC* PHZ)253 C254 DO 1306 I=1.,SC255 AMP( I )=AMP( I )*M9*1000.256 1306 PHZ( I )=PHZ( I )*180. /3. 14159257 C DIVIDE THE CALCULATED AMPLITUDE AND PHASE BY THE STANDARD258 C DEVIATION OF THE MEASURED AMPLITUDE AND PHASE259 C AND STORE IN ARRAY F5.260 J8=O261 DO 1310 J=I,SC262 K3=K3+1263 JS=J8+1264 AMP( J )=AMP( J )/SDY( 15 J8 )265 1310 F5(K3)=AMP(J)266 DO 1320 J=I.SC267 K3=K3+1268 C PHASE CORRECTION BASED ON ZERO TIME UNCERTAINTY (-WT).269 PHZ( J )=PHZ(J)-(2. *3. 14159*F( J )*TO( 15 )180. /3. 14159)270 J8=JS+1271 1320 F5( K3 )=PHZ( J )/SDY( I 5J8)272 C273 2999 CONTINUE274 3000 CONTINUE275 C

39o

276 RETURN277 END278 SUBROUTINE DRUM(LPHZv PHZ)279 C MAKES A PHASE CURVE CONTINUES280 DIMENSION PHZ(LPHZ)281 PJ=O.282 DO 40 I= 2# LPHZ283 IF( ABS( PHZ( I )+P.J-PHZ( I-1 ) )-3. 14159265 )40, 40s 10284 10 IF( PHZ( I )+F'J-PHZ( I-I ) )20. 40, 30285 20 PJ=PJ+3. 14159265*2.286 GO TO 40287 30 PJ=PJ-3. 14159265*2.288 40 PHZ( I )=PHZ( I )+PJ289 RETURN290 END291 SUBROUTINE POLAR (L,REoXIMPAMP,.PHZ)292 C COMPUTES POLAR COORDINATES293 DIMENSION RE(L)s XIM(L)sAMP(L)sPHZ(L)294 PI=3. 14159265295 DO 110 I=IPL296 AMP( I )=SQRT( RE( I )**2+XIM( I )**2)297 IF( XIM( I )) 10,20,30298 10 IF(RE( I ))40,50s60299 20 IF( RE( I ) )70, 80, 60300 30 IF( RE( I )) 90, 100. 60301 40 PHZ(I)=ATAN(XIM (I)/RE(I))-PI302 GO TO 110303 50 PHZ( I )=-PI/2.304 GO TO 110305 60 PHZ(I)=ATAN(XIM(I)/RE(I))306 GO TO 110307 70 PHZ( I )= -Pl308 GO TO 110309 80 PHZ( I )=0.310 GO TO 110311 90 PHZ(I)=ATAN (XIM(I)/RE(l)) + PI312 GO TO 110313 100 PHZ( I )=PI/2.314 110 CONTINUE315 RETURN316 END317 COMPLEX FUNCTION HZO(X)318 C-- HZ COMPONENT/Cl FOR A=O (GROUND CASE)319 C320 DOUBLE PREC IS ION EPS BS321 REAL LoK(5),D(7)322 COMMON/MODEL/K. Da M323 COMMON/SHARE/324 * EPSs325 * C2,C3sC4#326 * XX, YY, YY2s RHO, RH02, DELRHO, B.327 * L; DEL. DEL2,328 * METHOD. NZ# NW329 COMPLEX330 * TERM1oTERM2,CBPC,

33133233333433533633733833934034134234334434534634734834935035135235335435535635735835936036136236:3364365366

367368369370371372373374375376377378379380381382383

384385

* F4s HANKQo HANKCo N33o N3o 12s IlEXTERNAL F4DATA

* N33/( 3. 0 3. 0 )/, N3/( 3. O 0. 0 )/o 12/( 0. 0o 2. 0o)/ I1/( 0. 0,o-1, 0)1IF( YY. EQ. 0. 0) GO TO 9

CIB=BB8=BTERM1=( 0. O, 0. 0)TERM2=DEL2*YY*( I*( N3-( N3+CBi,( N:33+CB*12 ) )*CEXP( CMPLX( -, -B ))) )/

* RHO**5IF( M. EQ. 1 ) GO TO 2C=2. 0*YY/( DEL2*RHO)

1 TERMI=C*HANKC( 1 ALOG( B ), F4 )/CB2 HZO=TERM1+TERM23 RETURN9 HZO=( 0. O, 0. 0)

GO TO 3ENDCOMPLEX FUNCTION F4( G)

C--F4=G*F3( G )--SEE SUBPROGRAM F3.COMPLEX F3F4=G*F3( G)RETURNENDCOMPLEX FUNCTION F3( 0)COMPLEX VI,F.,Ce ONEDATA ONE/( 1. O 0. 0 )/CALL RECUR1(G, Vl.,Fl)C:=GF3=( VI*C*( ONE-FI ) )/( ( C+VI*F1 )( C+V1))RETURNENDSUBROUTINE RECURI(G, VI,Fl)

C--SACKWARD RECURRENCE FOR COMPLEX VlmFl GIVEN DIMENSION ARGUMENT 0 AND:COMMON/MODEL/ PARAMETERS:C

COMMON/MODEL/K. D, MREAL K(5),D(7)COMPLEX C. VM°VI, Fl,EVD, ONEDATA ONE/( 1. 0. 0. 0 )/F 1 =ONEG2=6*GVM=CSQRT( CMPLX( 02, 2. 0*K( M)))IF( M. EQ. 1) GO TO 2J=M-1

1 VI=CSQRT( CMPLX( G2, 2. O*K( J)))EVD=( 0. O. 0. 0)C=-Vl*D( .J)C2=REAL( C)IF(C2. LT. -80) GO TO 10EVD=CEXP( -Vl*D( J))

10 C=( ONE-EVD )/( ONE+EVD)FI=( VM*FI+VI*C )/( VI+VM*Fl*C)IF( J. EQ. 1) GO TO 3J=J-1

39Z

VM=V I30 TO 1

2 Vl=VM3 RETURN

ENDCOMPLEX FUNCTION HANKC( N, X, FUN)

C--COMF'LEX HANKEL TRANSFORM OF ORDER N( =0 1) AND ARGUMENT X( =ALOG( B))

C BY CONVOLUTION FILTERING WITH COMPLEX FUNCTION "FUN"--CC EXTERNAL FUNC COMPLEX FUN, HANKCsHCC

386387388389

390391392393394395396397398399400"401402403.404405406407408409-4104114124134144154164174184194204214224234244254264274284294304314324334344354364:37438439440

8399561E 00,9189221E 00s9:17:-:31E 00,0768541E 00,55320 13E-01,6521:387E-016862479E 00,6.72819E 00,528:3159E 00,4493499E 00O370:3839E 00a2914179E 00/

1010561E-06i 1. 8802098E-05, 5. 4819540E-05, 9. 2891602E-06,5523239E-04, 3 0344652E-05, 3. 5338744E-04, 1. 4798002E-04,7342377E-04s 5. 3570857E-04, 1. 7170605E-03, 1. 637239E-03,

9247683E-03, 4. 5796508E-03, 9. 2111468E-03p 1. 2130467E-02a1938415E-02, :3. 085:-_-660E-02p 5. 1973.594E-02, 7. 4661566E-0201775455E-01, 1. 6353574E-01, 2. 3127545E-01, 2. 7368461E-0188059285E-01, 1. 2875340E-01, -1. 5:30437E-01,-4. 5659951E-016077766E-()2, 4. 2985613E-01,-2. 1506075E-01,-2. 3624312E-02,9316746E-02, -7. 4344203E-02, 4. *:572965E-02,p-3. 00::-:372E-02,8846544E-02,-1. 2158687E-02, 8. 070'=759E-03,-5. 4706275E-03,7554604E-03, -2. 5929707E-03, 1. 790?426E-03s -1. 2320277E-0304095286E-04o-5. 6749747E-04, 3. 7718405E-04,-1. 5891835E-041TER ABSCISSA ? WEIGHT ARRAYS:

DATA AO/*-6. 8348046E 00,-6.*-5. 9137706E 00, -5.*-4. 9927366E 00 -4.*-4. 0717026E 00 -3.*-3. 1506686E 00, -2.*-2. 2296346E 00, -.

6045461E6835121E7624781E8414441E9204101E9993761E

00, -6.00, -5.00, -4.00, -:3.00, -2.00, -1.

*-1. 3086006E 00, -1. 078:3421E 00,-8.*-3. 875665SE-01 -1. 57:30808E-01, 7.

3742876E O0,-6. 1440291E 00,4532536E 00,-5. 2229951E 00,5322196E 00,-4. 3019611E 00,6111856E 00,-3. 3.09271E 00,6901516E 00,-2. 4598931E 00,7691176E 00,-I. 5388591E 00,4808358E-0 ,s-6. 1782508E-01.2950416E-02, 3. 0320892E-01,

COMPLEX FUNC--JI-FILTER ABSCISSA ? WEIGHT ARRAYS:

DIMENSION Al( 48)DIMENSION AO(61)DIMENSION W1(48)DIMENSION WO(61)DATA All

*-4. 5307316E 00,-4. 3004731E 00,-4. 0702146E 00,-3.

*-3. 6096976E 00,-3. 3794:391E 00,-3. 1491806E 00,-2.*-2. 68 86636E 00,-2. 4584051E 00,-2. 22::1466E 00,-1.

*-1. 7676296E 00,-1. 537371 1E 00, -1. 30711*'6E 00, -1.*-8. 4659563E-01, -6. 1633713E-01, -3. 8607863E-01, - 1.

* 7. 4438369E-02, 3. 0469687E-01, 5. 3495537E-01, 7.* 9. 9547237E-01, 1. 2257309E 00, 1. 4559894E 00, 1.* 1. 9165064E 00, 2. 1467649E 00, 2. 3770234E 00, 2.* 2. 8375404E 00, 3. 0677989E 00, 3. 2980574E 00, 3.*:3. 7585744E 00, 3. 981:329E 00, 4. 2190914E 00, 4.* 4. 6796084E 00, 4. 9098669E 00, 5. 1401254E 00, 5.* 5. 6006424E 00, 5. 8309009E 00, 6. 0611594E 00, 6.DATA WI/

*3.•* 1.• 7.* 3.

" 2.•* 1.* 2.*-3.•*N 8.•* 1.* 3.*JO" 8.

I-'.--. l1C -F T L

* 5.3346742E-O1 7.* 1.4545014E 00, 1.* 2. 3755354E 00, 2.* 3.2965694E 00s 3.* 4.2176034E 00, 4.* 5. 1386374E 00, 5.* 6.0596714E 00a 6.* 6.9807054E 00/DATA WO/

* 7.3260937E-04, 5.* 3. 5918:326E-04o 1.* 1.2841617E-03, 2.

3. 632124.jE-03,5.9.55*7062E-03i 1.2.4396626E-02, 3.6.0824806E-02, 7.

*.1.3868663E-01, 1.* 1.5556741E-01, 6.*-3.5565260E-01-5.*-2.6102989E-01 2.*-5.1097828E-04,-6.* 5. 8044372E-03t,-4.

6372592E-01,6847599E 00,6057939E 00,5268279E 00,4478619E 00,:3688959E 00s2899299E 00,

9.9398442E-01,1.9150184E 00,2. :o60524E 00;3. 7570864E 00,4. 6781204E 00,5. 5991544E 00,6. 5201884E 00,

441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471 1472473474475476 C--E)477 C478479480481482483484485486487488489490491492493494495

:3. 2266260E-0:3, -2. 864'9137E -0:3, 2.2. 1115187E-03P -1. 9334662E-03, 1.

5$7760E-03.7800248E-03.

1.3468317E-03/HANKC=(0. 0,0.0)IF(N. EQ. 1) C0 TO 1DO 2 1=1,61HANKC=HANKC+FUN(EXP(-X+AO(I)))*WO(1)RETURNDO 3 1=1,48HANKC=HANKC+FUN(EXP(-X+Al(1)))*WI(I)RETURNENDCOMPLEX FUNCTION EXO(X)COMPONENT/(ECON*C1) FOR A=O (GROUND CASE)

1. 242429E2. 1452749E3. 0663109E3. 987:3449E4. 908:3789E5. 8294129E4. 7504469E

00.00.00000,00,00200,

7. 5331222E-04,1.5160070E-03,3. 4076782E-03,8. 1315877E-03Pi. 9941086E-02,4.9127993E-02,1. 1545027E-01,J, 8424433E-01,

-2. 8819226E-01,-5. 151645rE-02.

2. 6196370-020-6.7854344E-03,-3. 67:3*3648$E-03#-2. 3202655E-03,-1. 6465436E-030

DOUBLE PRECISION EPS, B8REAL L,K(5),D(7)COMMON/MODEL/KoDoMCOMMON/SHARE/

* EPSs* C2,C3,C4,* XX, YYoYY2pRHOoRHO2PDELRHOoB,* L,DEL,DEL2o* METHOD, NZ, NWCOMMON/FX/XR2COMPLEX

$ F7, Fll,HANKQ,HANKC, TERMI, TERM2, IIEXTERNAL F7,F11DATA I1/(0. 0, -1. 0)EXO=(0. 0,0. 0)TERM2=( 0. 0,0. 0)XD=XXIF(L. EQ.0. 0) 00 TO 10

6326423E-04i 1. 3727237E-04,0500608E-03, 7. 15: 09'2E-04,2497985E-03# . 1906186E-032376028E-C)03 5. 9212519E-03p2708615E-02, 1. 5305589E-02,13 3:'-"-V-.=,-- : E .....- 02, 3. 868:',065E-026314344E-02, 9. 39?28346E-026248847E-01, 1. 8114:332E-01S592481E-02, -8, 8339029E-02s6288677E-02, 4. 81.36942E-01,1416490E-0 ls-9. 4490687E-02)6032948E-03, 7. 5193619E-03)9354894E-03p 4. 232:3106E-03

2

13

x

364

496 XD=X-XX497 10 XR2=XD**2/RHO2498 C4=1. 0-2. 0*XR2499 IF(M. EQ. 1) GO TO 3500 1 ALOGB=ALOG( B)501 TERMI=HANKC( O, ALOrB, Fl 1 )/B502 IF( C4. NE. 0) TERM2=HANKC( 1, ALOGB, F7 )/B503 2 EXO=-TERMl/DEL+C4*TERM2/RHO504 3 TERMI=CMPLX(B,B)505 TERM2=I l*DEL2*( 1. 0-3. 0*YY2/RH02+( 1. O+TERM1 )*ICEXP(-TERMI))506 $/( RHO*RH02)507 EXO=EXO+TERM2508 RETURN509 END510 COMPLEX FUNCTION F7(G)511 COMPLEX V1,FI,L1,sIIONEsTWOoC512 DATA I1,ONE,TWO/(O. Or 1. 0),( 1.0,0. 0),(2. 0,0. 0)/513 CALL RECUR2(GViPFI,Li)514 C=G515 F7=I1*V1*( LI-ONE )+( TWO*Vl*( ONE-Fl ))/((C+Vl*Fl )*(C+Vi))516 RETURN517 END518 COMPLEX FUNCTION Fll(G)519 COMPLEX Vt.FlsLip IlsONEv TWOs Cs Ti, T2520 COMMON/FX/XR2521 DATA II,ONE,TWO/(0. 0, 1. 0),(1. 0O0. 0),(2.Or0.O)/522 CALL RECUR2(6, VliFI,L1)523 C=G524 T1=( TWO*Vl*( ONE-Fl ) )/( ( C+VI*Fl )*( C+Vl))525 IF(XR2. EQ. 0.0) GO TO 1526 T2=11*Vl*( LI-ONE )+Tl

527 1 F11=C*(Tl-XR2*T2)528 RETURN529 END530 SUE:ROUTINE RECUR2(Gt VioFl, Li)

531 C--BACKWARD RECURRENCE FOR COMPLEX V1,Fl,Li GIVEN DIMENSION ARGUMENT G p

532 COMMON/MODEL/ PARAMETERS:533 C534 COMMON/MODEL/Ki Do M535 REAL K(5),D(7)536 COMPLEX CP VM, V1,F1,LI,EoONE537 DATA ONE/( 1. 0, 0. 0 )/538 Fl=ONE539 L1=ONE540 G2=G*G541 VM=CSQRT( CMPLX( G2r2. O*K( M)))542 IF(M. EQ. 1) GO TO 2543 J=M-1544 1 Vl=CSQRT( CMPLX( G2, 2. 0*K( J)))545 E=( 0. 0. 0)546 C=-Vl*D(J)547 C2=REAL( C)548 IF(C2. LT. -80) GO TO 10

549 E=CEXP(-VI*D( J))550 10 C=(ONE-E)/(ONE+E)

385

551552553554555556557558559560561562563 C--E564 C565566567568569570571572573574575576577578579580581582583 10584585 1586587588 2589 3590591 5592593594595596597598599600'c

601602603 C604 C605

Fl=( VM*F1+V1*C )/( V1+VM*FI*C)E=K( J+1 )*VI+K( J )*VM*LI*CIF(REAL(E). EQ. 0. 0. AND. AIMAG(E). EQ. 0. 0) E=( 1. OE-30P 1. OE-30)L=( K( J )*VM*LI+K( J+l )*VI*C )/EIF( J. EQ. 1) O0 TO 3J=J-1VM=V1GO TO 1

2 VI=VM3 RETURN

ENDCOMPLEX FUNCTION EYO( X)

-Y COMPONENT/( ECON*C1) FOR A=O (GROUND CASE)

DOUBLE PREC IS ION EPSP B8REAL L, K(5)o D( 7)COMMON/MODEL/Ko D° MCOMMON/SHARE/

* EPSP* C2sC3oC4s* XX, YY, YY2p RHO. RHO2, DELRHOt B** LP DELP DEL2#* METHODs NZo NWCOMPLEX

$ F7, F12, HANKQO HANKC, TERMI, TERM2, 13EXTERNAL F7,F12DATA 13/( 0. 0.°-3. 0)/EYO=( C. O. 0. 0)IF( YY. EQ. 0. 0) GO TO 5XD=XXIF(L. EQ. 0. 0) GO TO 10XD=X-XXIF( XD. EQ. 0. 0) GO TO 5IF(M. EQ. 1) 00 TO 3ALOGB=ALOG( B)TERMl=HANKC( 1 ALOGB, F7 )/BTERM2=HANKC( 0 ALOG5, F12 )/BEYO=-2. 0*TERM 1/RHO+TERM2/DELTERM2=13*DEL2/( RHO*RH02)EYO=XD*YY*( EYO+TERM2 )/RHO2RETURNENDCOMPLEX FUNCTION F12(0)COMPLEX F7F12=0*F7( 0)RETURNENDSUBROUTINE GASAUS( PRNFLG, MODELt NOB, OBSs NP, PAR. UBND, BLWRsI EPSIP EPS2P EPS3, MIT, FLAM, FNU, SCRAT)

THIS VERSION MODIFIED FOR THE UNIVAQ. 1108 BY THE AUTHORDIMENSION SCRAT( 1)

DIMENSION OBS( NOB ), PAR( NP )s UBND( NP), BLWR( NP)

H. J. WERTZ AEROSPACE CORPIRATIONNSCRAT = NP*( 7+2*NP+NOB) +2*NOB

3840

606 11 = 1607 12 = I1 + NP * NP608 13 = 12 + NP * NP609 14 = 13 + NP610 I5 = 14 + NP

611 16 = 15 + NP612 17 = 16 + NP613 I8 = 17 + NP614 19 = I8 + NP615 I10 = 19 + NP616 111 = 110 + NOB617 112= III + NOB

618 CALL GASS59( PRNFLGo MODELs NOS, OBS, NP, PAR, UBND, BLWR,

619 1 EPSIo EPS2o EPS3, MIT, FLAMs FNU, SCRAT(I1), 'CRAT(I2),

620 3 SCRAT( I3), SCRAT( 14), SCRAT( I5)o SCRAT( 16), SCRAT( 17)o SCRAT( I8)

621 4 ,SCRAT( 19),SCRAT( I10),SCRAT( Ill ),SCRAT(112))622 RETURN623 END624 SUBROUTINE GASS59( PRNFLG, FOF, NBO, Y, NQ, TH, UBOUND, LBOUND, EPIS

625 1 s EP2So EP3S, MIT# FLAMs FNU, D, A, Q P, E, PHI, TB, TBS9 WTSs626 2 F,R,DELZ)627 C628 C VERSION OF JULY 3, 1968629 C H. J. WERTZ AEROSPACE CORP.630 C L.A.o CALIF 90045

631 c THE STRUCTURE OF THIS PROGRAM AND THE LOGIC FOR CONTR

632 C THE PARAMETER OA (FLAM) WAS TAKEN DIRECTLY FROM THE U

633 C OF WISCONSIN FROGRAM 3AUSHAUS WRITTEN BY D.A. MEETER

634 C

635 DIMENSION TH( NO). UBOUND( N)), LBOUND( NO), Y( NBO)

636 DIMENSION Q(Nf). P(NQ). E(NQ), PHI(N,). TB(NQ)

637 DIMENSION F(NBO)o R(NBO)638 DIMENSION A(Nr.NQ), D(NONO)o DELZ(NBO,NQ)639 DIMENSION WTS(NQ)t TBS(NQ)640 REAL LBOUND

641 DOUBLE PRECISION SUMD642 COMMON/GASPAR/DUMIES( 7), PIVOTM

643 LOGICAL INERRIs INERR2644 DIFZ=. 001645 NP=NQ646 NOB = NBO647 EPS1 = EPIS648 EPS2 = EP2S649 NPSQ = NP * NP650 GA = FLAM651 EPS3 = EP3S

652 NSCRAT = NP*( 7+2*NP+NOB )+2*NOB653 PRINT 1000, NOB# NP, NSCRAT654 1 CONTINUE655 C656 C TEST USER PARAMETER INPUTS657 ERRFLG = 0

658 IF(NP . GE. I . AND. NP . LE. 50 . AND. NOB GE. NP) GO TO 3659 ERRFLG = 1. 0660 PRINT 8600

-5B7

661 8600 FORMAT(51H ERROR NP . LT. 1 . OR. NP .OT. 50 .OR. NOB .LT. 1NP

662 3 IF(MIT .GE. 1 AND. MIT .LE. 999) GO TO 4663 ERRFLG = 1.0664 PRINT 8601665 8601 FORMAT( 37H ERROR MIT . LT. 1 . OR. MIT .YT. 999

666 4 IF(FNU GT. 1.0 AND. GA .GE. O) GO TO 5667 ERRFLG = 1.0668 PRINT 8602669 8602 FORMAT( 39H ERROR FNU .LE. 1.0 .OR. GAMMA .LT. 0670 5 CONTINUE671 DO 19 I=Io NP672 IF( UBOUND(I) .OT. LBOUND(I)) GO TO 6673 ERRFLG = 1.0674 PRINT 8603, I, I675 8603 FORMAT( 16H ERROR BNDUPR(, I2, 12H). LE. BNDLWR(, 2, 1H )676 6 IF( TH( I ) . LT. UBOUND( I) . AND. TH( I O) . T. LDOUND( I)) GO TO 19677 ERRFLG = 1. 0678 PRINT 8604, I679 8604 FORMAT(11H ERROR P(,I2,Z8H) IS INITIALLY OUT OF BOUNDS,680 19 CONTINUE681 DO 8 I = 1 NP6.92 IF(TH(I) NE. 0) GO TO 8683 ERRFLG = 1.0684 PRINT 8607, I685 8607 FORMAT(11H ERROR P(912,94H) IS ZERO INITIALLY. (ALL GUESSES MUS686 IT BE NONZERO IF PARTIALS ARE NOT CALQULATED BY THE USER)687 8 CONTINUE688 9 IF(ERRFLO .EQ. 1) GO TO 410689 C690 C PASSED PARAMETER CHECK691 NIT = 1692 SSQ = 0693 CALL FOF(TH, Fe NOBP NP)694 DO 90 I = 1. NOB695 R(I)=Y(I) -F(I)696 90 SSQ=SSQ+R(I)*R(I)697 KOUNT = 1698 SSQINT = SSQ699 SSQB4 = SSQ700 PRINT 1003a SSQ701 IF(SSQ .EQ. 0) GO TO 410702 C703 C BEGIN ITERATION704 C705 100 CONTINUE706 C707 PRINT 1004PNITsKOUNT708 GA = GA * SQRT(SSO/SSQB4)/FNU709 SSQB4 = SSQ710 INTCNT = 0711 C712 C COMPUTE PARTIAL$ BY DIFFERENCING713 C714 DO 130 J=I.NP715 TEMP = TH(J)

P(J) = DIFZ * TH(J)TH(J)= TH( J )+P(J)Q( j )=0CALL FOF(TH, DELZ( 1J )s NOBs NP)KOUNT = KOUNT + 1DO 120 I = 1 NOBDELZ( I,J)= DELZ( ItJ)-F( I)

120 Q( J )=Q( J )+DELZ( I, J )*R( I)Q( J)= Q( J )/P( J)

0=)

130 TH(J) = TEMPDO 150 I = 1, NPDO 151 J=1#ISUMD = 0DO 160 K = 1, NOB

L60 SUMD = SUMD + DELZ(K. I) * DELZ(K,J)TEMP = SUMD / (P( I )*P(J))D( Jo I )=TEMP

151 D(I,J)=TEMPD(Is I ) = AMAX1(D(I I) 1. E-20)

L50 E(I) = SQRT(D( I I))GO TO 180

716717718719720721722723724725 C726727728729730731 1732733734735736 1737738 C739 1740741742743744 20745746747748749750751752753' 2754755756757 2758759760761762 C763 C764 6765766767768769770

(T*R (STEEPEST DESCENT)

D=XT*X (MOMENT MATRIX)DO 200 I = 1 NPWTS( I )=0DO 200 J = 1 ISUM=D( IJ )A( J, I )=SUMA( IP J )=SUMCALL SYMQR( A. NP. NPs Pt DELZ)IF( PRNFLG . GE. 4. 0) PRINT 1006IF(PRNFLG .GE. 4.0) PRINT 2001. (P(I). I = 1, NP)ISUB=1K2=1.DO 201 I=I#NPDO 203 J=IPNPK1=J+ISUB-1IF(Kl. LE. NOB) GO TO 2003K2=K2+ 1K1=K 1-NOBGO TO 2004CONTINUEWTS( J )=WTS( J )+ABS( DELZ( Kb K2 ) )*P( I)ISUB=ISUB+NPCONTINUEMASTER = 1

COMES HERE IF NECESSARY TO INCREASE GACONT INUEDPROD = 1. 0DO 153 1 = 1 NPIF(WTS(I) . GT. WTS(MASTER)) MASTER = IDPROD = DPROD * D(I, I) " (1.0/NP)DO 153 J=I# IA( I; J )=D( It J )/( E( I )*E( J))

-3B8

80

)0

004

_M03203

201

'66

389

771 153 A(J,I )=A(I,J)772 DPROD = AMAXI( DPRODP 1. E-20)773 C A= SCALED MOMENT MATRIX774 WTSMST = WTS( MASTER)775 DO 155 I=1#NP776 P( I )=Q( I )/E( I777 PHI( I )=P( I)778 WTS( I ) = WTSMST/AMAX1(WTS( I ), 1. E-20)779 155 A( I, I ) = A( I, I) + GA * WT$( I) * DPROD / D( I, I)780 C781 1=1782 CALL MATIN(A., NP, P. 1, DET)783 C P/E = CORRECTIQN VEQTOR784 IF(PIVOTM LE. 0) GO TO 279785 STEP=1. 0786 SUM1=0.787 SUM2=0.788 SUM3=O.789 DO 231 I=IoNP790 SUMI=P( I )*PHI( I )+SUM1791 SUM2=P( I )*P( I )+SUM2792 SUM3= PHI( I) * PHI( I) + SUM3793 231 PHI(I)=P(I)794 TEMP = SUM1/SQRT( SUM2*SUM3)795 TEMP = 57. 295 * ACOS( AMINI( 1. 0, TEMP))796 PRINT 1041a DETs PIVOTM# TEMP797 C798 C COMES HERE IF ANGLE IS LESS THAN 30 DEGREES OR IF GA799 170 DO 218 I = 1 NP800 218 P(I) = PHI(I) * STEP / E(I)801 INERRI = FALSE.802 INERR2 = TRUE.803 C804 C ADD CORRECTION AND TEST FOR OUT OF BOUNDS805 DO 220 I = 1, NP806 TB( I )=TH(I) + P(I)807 IF( TB( I ). GT. UBOUND( I) . OR. TB( I ). LT, LBOUND( I)) INERRI=. TRUE.808 220 CONTINUE809 PRINT 7000810 PRINT 2006: (TE(I)i I = 1, NP)811 IF(INERR1 AND. INERR2) GO TO 663812 C813 C PARAMETERS ARE IN BOUNDS, COMPUTE SUM-OF-SQUARES814 SUMB=O815 CALL FOF(TB, F: NOB, NP)816 KOUNT = KOUNT + 1817 DO 230 I=I#NOB818 R(I )=Y( I )-F(I)819 230 SUMB=SUMB+R( I )*R( I)820 PRINT 1043, SUMB821 661 IF(SUMB . LE. (1. O+EPS1 )*SSQ ) GO TO 662822 C823 C COMES HERE IF CURRENT TEST PARAMETER SET POESNPT REDU824 663 IF( TEMP . GT. 30. 0 . AND. GA . NE. 0) GO TO 664825 665 STEP=STEP/2. 0

390

826827828829830831

832833 C834 C835 6,836837838 66839840841 C842 C843844845 C846847848 C849 C850851852853854855 2856857858 2859

860861 C862 C863864865866867868 2869870871872873874875876 287787887988O

664

62

9

240

'65

t70

ATTEMPT TO KICK THE SOLUTION

IF(STEP .NE. 1.0) GO TO 271

DO 274 IDUBLE = 1. 5DO 272 I = 1, NP

TB(I) = TH(I) + P(I)IF( TB(I) .GT. UBOUND(I) .OR. TB(I) .LT. LBOUND(I)) GO TO 271

172 CONTINUESUMB = 0CALL FOF(TB, DELZ., NOB, NP)KOUNT = KOUNT + 1DO 273 I = 1, NOB

273 SUMB=SUMB+(Y(I)-DELZ(IpI))**2IF(( 1.O+EPS1 )*SUMB .GE. SSQ) GO TO 271

DO 275 I = 1. NP275 TH(1)I = TB( I)

DO 276 I = 1. NOBR( I )=Y(I )-DELZ(I. 1)

276 F( I )=DELZ( Isl)PRINT 7000

INTCNT = INTCNT + 1IF(INTCNT .GE. 36) 00 TO 2700GO TO 170GA=GA*FNUINTCNT = INTCNT + IIF(INTCNT GE. 36) GO TO 2700GO TO 666

CURRENT TEST SET ACCEPTED. SAVE CHANGE FOR KICKING.

CONTINUEDO 669 I=IPNPP(I) = TB(I) - TH(I)TH( I )=TB(I)sso = SUMBPRINT 1007

PRINT THE CURRENT PARAMETERS AND ELAPSED TIME.

CALL PRTPARCALL ETIME

PRINT 10401 GAs SUMB

IF(EPSI .LE. 0 .AND. EPS2 .LE. 0) GO TO 270

TEST FOR CONVERGENCE

DO 240 I = 1 NP

IF( ABS( P(I)) .GT. ABS( TH( I ) )*EPS2) GO TO 265CONTINUF

PRINT 1009. EPS2

GO TO 280

IF(ABS(SUMB-SSDB4) .GT. EPSl*SSQB4) GO TO 270PRINT 1010, EPS1

GO TO 280CONTINUECALL-CHISQ(NOB. NPaYoF.IADONE)IF(IADONE .NE. 0) GO TO 280

391

274271

3001

300C

270C271C

279

279(

PRINT 2006, (TB(I), I =1. NP)PRINT 1043a SUMBSSQ = SUMBCONTINUENIT=NIT+lIF(SSQ .LE. EPS3*SSQINT) GO TO 280

STOP THE RUN IF SENSE SWITCH NO. 2 IS PUSHED FORWARD.IF( ISENSE( 2). EQ. 0) GO TO 3000PRINT 3001FORMAT(/' ITERATIONS STOPPED BY OPERATOR'/)GO TO 2801IF(NIT-MIT)100, 100#2801

881882883884885886887 C888 c889890891892893894 C895896897898 c899 c900 c901902903904905 C906907908909910911912913914915916917918919920921922923924925926927928929930931

932933934935

GO TO 280PRINT 1005, DETj PIVOTM

PRINT 2790D FORMAT(5X,26HMINIMUM PIVOT IS LE. ZERO

280 CONTINUE2801 CONTINUE500 CALL FOF (TH, Fs NOB, NP)

DO 501 J = 1 NPTEMP = TH(J)P(J) = DIFZ * TH(J)TH(J) = TH(J) + P(J)CALL FOF(TH# DELZ(loJ)o NOB, NP)DO 502 I = 1, NOB

502 DELZ(Is J) = DELZ(Is J) - F(I)501 TH(J) = TEMP

DO 503 I = 1, NPDO 503 J= 1 ISUMD = 0DO 504 K = 1, NOB

504 SUMD = SUMD + DELZ(K, I) * DELZ(K. J)SUMD = SUMD / (P( I )*P(J))A(J, I) = SUMD

503 A(I, J) = SUMD4:33 CONTINUE

IDF=NOB-NPDO 281 I=I,NPDO 281 J=1,NP

281 D(IsJ)=A(IsJ)CALL MATIN(Do NP, P. It DET)DO 7692 I=I#NP

7692 E(U) = SQRT(AMAXI(D(I,I)p 1. E-36))DO 340 I=I.NPDO 340 J = Is NPD(J, I) = D(Js I) I (E(I)*E(J))

PRINT 2710FORMAT(//115HO**** THE SUM OF SQUARES CANNOT BE REr)YCED TO THE SUM

IOF SQUARES AT THE END OF THE LAST ITERATION - ITERATING STQP$, )

END ITERATION

L

)

392

936 340 D(CI, J) = D(J, I)937 IF(IDF EQ. 0) GO TO 392938 SDEV = SSQ / IDF939 SDEV = SQRT( SDEV)940 DO 391 I=1,NP941 P( I )=TH( I )+2. O*E( I )*SDEV942 391 TB( I )=TH( I )-2. 0*E( I )*SDEV943 CALL PRTFPR( SDEVt IDFP NPs THo Po TB)944 C945 C SAVE DIAGONAL OF A946 392 DO 505 1 = 1. NP947 505 Q( I ) = A( I, I)948 CALL SYMQR( Ap NP; NPs Po DELZ)949 C RESTORE A (ONLY LOWER TRIANGLE USED IN SYMQR)

950 DO 506 I = 1, NP951 A( Is I ) = Q( I )952 DO 506 J = 1, I953 506 A(I, J) = A(JsI)954 CALL PRTCOR( NP, Ds E P)955 C RESTORE D FOR COVARIANCE MATRIX COMPUTATION

956 DO 408 I=I,NP957 DO 408 J=I,NP958 408 D( I,J)=A(Is,-)959 CALL MATIN(D# NP, P. O. DET)960 410 CONTINUE961 RETURN962 1000 FORMAT(," GASAUS ENTERED -/

963 - 110, fOBSERVATIONSe/964 - 110. PARAMETERS"/965 - 110, , SCRATCH REQUIRED-)

966 1003 FORMAT(/25HOINITIAL SUM OF SQUARES = E12. 4)

967 1004 FORMAT(//// ------------------------ ITERATION NO. o, 141

968 - 20XF"NO. OF FUNCTION CALLS =-1#14)969 1005 FORMAT(/14H DETERMINANT = E12.4,6XP12HMIN. PIVOT = E12.4)

970 1006 FORMAT(/52HOEIGENVALUES OF MOMENT MATRIX - PRELIMINARY ANALYSIS )

971 1007 FORMAT(/32HOPARAMETER VALUES VIA REGRESSION )972 10090FORMAT(/62HOITERATION STOPS - RELATIVE CHANGE IN EACH PARAMETER LE973 ISS THAN E12.4)974 10100FORMAT(/62HOITERATION STOPS - RELATIVE CHANGE IN SUM OF SQUARES LE

975 ISS THAN E12.4)976 10400FORMAT(/9HO GAMMA =EIO. 3,40Xo33HSUM OF SQUARES AFTER REGRESSION =

977 1E15.7)978 1041 FORMAT(/14H DETERMINANT = E12. 46XP12HMIN. PIVOT = E12.48

979 1 12X, 25H ANGLE IN SCALED COORD. = F5. 2. 8H DEGREES980 1043 FORMAT( 28HOTEST POINT SUM OF SQUARES = E12. 4)981 2001 FORMAT(/10E12.4)982 2002 FORMAT( 14H0 EIGENVECTORS983 2006 FORMAT( 10E12. 4)984 7000 FORMAT(30HOTEST POINT PARAMETER VALUES985 END986 SUBROUTINE MATIN( Ap NVAR, B. NBo DET)987 DIMENSION A(NVAR, 1 ). B(NVAR, 1)988 COMMON/GASPAR/DUMIES( 7), PIVOTM989 PIVOTM = A(1,1)990 DET = 1. 0

394-

1046 NI=N-11047 N2=N-21048 C COMPUTE THE TRACE AND EUCLIDIAN NORM OF THE INPUT MATRIX

1049 C LATER CHECK AGAINST SUM AND SUM OF SQUARES OF EIGENVALUES1050 ENORM=O.1051 TRACE=O.1052 DO 110 J=I,N1053 DO 100 I=JoN1054 100 ENORM=ENORM+A( I# J )**21055 TRACE=TRACE+A( J, J)1056 110 ENORM=ENORM-. 5*A( J, J )**21057 ENORM=ENORM+ENORM1058 GAMMA( 1 )=A( 1, 1)1059 IF(N.-) 280,270,1201060 120 DO 260 NR=I,N21061 E=A( NR+I,tNR)1062 S=O.1063 DO 130 I=NR,.N21064 130 S=S+A( I+2. NR )**21065 C PREPARE FOR POSSIBLE BYPASS OF TRANSFORMATION1066 A( NR+1o NR )=0.1067 IF(S) 250s25091401068 140 S=S+B*B1069 SGN=+1.1070 IF( B) 1501 160s 1601071 150 SGN=-1.1072 160 SORTS=SQRT(S)1073 D=SGN/( SQRTS+SQRTS )1074 TEMP=SQRT(. 5+B*D)1075 W(NR)=TEMP1076 A( NR+1, NR )=TEMP1077 D=D/TEMP1078 B=-SGN*SQRTS1079 C D IS FACTOR OF PROPORTIONALITY. NOW COMPUTE AND SAVE W VECTOR.

1080 C EXTRA SINGLY SUBSCRIPTED W VECTOR USED FOR SPEED.1081 DO 170 I=NRPN21082 TEMP=D*A( I+2o NR)1083 W( I+1 )=TEMP1084 170 A( I+2o NR )=TEMP1085 C PREMULTIPLY VECTOR W BY MATRIX A TO OBTAIN P VECTOR.

1086 C SIMULTANEOUSLY ACCUMULATE DOT PRODUCT WP,(THE SCALAR K)

1087 WTAW=O.1088 DO 220 I=NR,N11089 SUM=O.1090 DO 180 J=NR, I1091 180 SUM=SUM+A( I+1oJ+I )*W(J)1092 Ii=i+11093 IF(NI-II) 210a 190, 1901094 190 DO 200 J=II,NI1095 200 SUM=SUM+A( J+ 1o I + 1 )*W( J)1096 210 P( I)=SUM1097 220 WTAW=WTAW+SUM*W( I)1098 C P VECTOR AND SCALAR K NOW STORED. NEXT COMPUTE Q VECTOR

1099 DO 230 I=NRPNI1100 230 Q( I)=P( I )-WTAW*W( I)

395

1101 C NOW FORM PAP MATRIXm REQUIRED PART

1102 DO 240 J=NR#N11103 QJ=Q( dj)1104 WJ=W( )1105 DO 240 I=JoN11106 240 A( I+l,J+l )=A( I+1,J+1 )-2. *(W( I )*QJ+WJ*Q(I))1107 250 BETA(NR)=B1108 BETASQ(NR)=B*B1109 260 GAMMA(NR+1 )=A(NR+1oNR+1)1110 270 B=A(N,N-1)1111 BETA(N-1)=B1112 BETASQ( N- I )=B*B1113 GAMMA( N )=A(NoN)1114 2:30 BETASQ( N )=O.1115 C ADJOIN AN IDENTITY MATRIX TO BE POSTMULTIPLIED BY ROTAT;ONS.

1116 DO 300 I=I,N1117 DO 290 J=I#N1118 290 VEC( It J)=0.1119 300 VEC( I I )=1.1120 M=N1121 SUM=O.1122 NPAS=11123 0 TO 4001124 310 SUM=SUM+SHIFT1125 COSA=I.1126 G=GAMMA(1)-SHIFT1127 PP=G1128 PPBS=PP*PP+BETASQ(1)1129 PPBR=S';RT(PPBS)1130 DO 370 J=I,M1131 COSAP=COSA1132 IF(PPBS .NE. 0. ) GO TO 3201133 SINA=O.1134 SINA2=O.1135 COSA=1.1136 00 TO 3501137 320 SINA=BETA(J)/PPBR1138 SINA2=PETASQ(J)/PPBS1139 COSA=PP/PPBR1140 C POSTMULTIPLY IDENTITY BY P-TRANSPOSE MATRIX

1141 NI=J+NPAS1142 IF(NT GE. N) NT=N1143 330 DO 340 I=I,NT1144 TEMP=COSA*VEC( I, J )+SINA*VEC( I J+I)

1145 VEC( I, J+l )=-SINA*VEC( I) J )+COSA*VEC( I, J+l)1146 340 VEC(IjJ)=TEMP1147 350 DIA=GAMMA(J+1)-SHIFT1148 U=SINA2*(G+DIA)1149 GAMMA(J)=G+U

1150 G=DIA-U

1151 PP=DIA*COSA-SINA*COSAP*BETA(J)1152 IF(J .NE. M) GO TO 360

1153 1:ETA(J)=SINA*PP1154 BETASQ(J)=SINA2*PP*PP1155 GO TO 380

1156115711581159116011611162116311641165116611671168116911701171117211731174117511761177117811791180118111821183118411851186118711881189119o11911192119311941195119611971198119912001201120212031204120512061207120812091210

360 PPBS=PP*PP+BETASQ( J+1)PPBR=SQRT( PPBS)RETA( J )=SINA*PPBR

370 BETASQ( J )=SINA2*PPBS3f0 GAMMA( M+I )=G

C TEST FOR CONVERGENCE OF LAST DIAGONAL ELEMENTNPAS=NPAS+ 1IF(E:ETASQ(M) .GT. 1. E-21 ) 30 TO 410

390 EIG(M+I) = GAMMA(M+1) + SUM400 BETA( M )=0.

BETASQ( M )=0.M=M-1IF(M .EQ. 0) GO TO 430IF(BETASQ(M) .LE. 1. E-21 ) 30 TO 390

C TAKE ROOT OF CORNER 2 BY 2 NEAREST TO LOWER DIAGONAL IN VALUEC AS ESTIMATE OF EIGENVALUE TO USE FOR SHIFT

410 A2=GAMMA(M+1)R2=. 5*A2R1=. 5*GAMMA( M)R12=R1+R2DIF=R1-R2TEMP=SQRT( DIF*DIF+BETASQ( M))Rl=R12+TEMPR2=R 12-TEMPDIF=ABS( A2-R1 )-ABS( A2-R2)IF(DIF .LT. 0. ) GO TO 420SHIFT=R2GO TO 310

420 SHIFT=R1GO TO 310

430 EIG( 1 )=GAMMA( 1 )+SUMC INITIALIZE AUXILIARY TABLES REQUIRED FOR REARRANGING THE VECTORS

DO 440 J=1#NI POSV( J )=jIVPOS( J )=J

440 IORD(J)=JC USE A TRANSPOSITION SORT TO ORDER THE EIGENVALUES

M=NGO TO 470

450 DO 460 J=1,,MIF(EIG(J) .GE. EIG(J+1)) GO TO 460TEMP=EIG( J)EIG( J )=EIG( J+l)EIG( J+1 )=TEMPITEMP=IORD( J)IORD( J )=IORD( J+1)IORD( J+1 )=ITEMP

460 CONTINUE470 M=M-1

IF(M .NE. 0) GO TO 450IF(NI EQ. 0) GO TO 500DO 490 L=1N1NV=IORD( L)NP= I POSV( NV)IF(NP EQ. L) GO TO 490

:397

1211 LV=IVPOS(L)1212 IVPOS( NP )=LV1213 IPOSV( LV )=NP.1214 DO 430 1=1,N1215 TEMP=VEC( I# L)1216 VEC( I,L)=VEC( I,NP)1217 4:30 VEC( I,NP)=TEMP1218 490 CONTINUE1219 500 ESUM=O.1220 ESSQ=O.12*21 C BACK TRANSFORM THE VECTORS OF THE TRIPLE DIAGONAL MATRIX1222 DO 550 NRR=IPN1223 K=N11224 510 K=K-11225 IF(K .LE. 0) GO TO 5401226 SUM=O.1227 DO 520 I=K,N11228 520 SLIM=SUM+VEC( I+1o NRR )*A( I+1, K )1229 SLIM=SUM+SUM1230 DO 530 I=K#N11"231 530 VEC( I+1,NRR)=VEC( I+1,NRR)-SUM*A( I+1,K)1232 GO TO 5101233 540 ESUM=ESUM+EIG( NRR)1234 550 ES:-=O=ESSQ+E I G( NRR )**21235 TEMP-ABS( 12. *TRACE)1236 IF( ( ABS( TRACE-ESUM )+TEMP )-TEMP NE. 0. ) IND=IND+l1237 TEMP=256. *ENORM1238 IF( ( ABS( ENORM-ESSI- )+TEMP )-TEMP . NE. 0. ) IND=INP+21239 C IF(IND) PRINT 86, IND1240 86 FORMAT(3OHO**PRODAB-LE ERROR** INDICATOR= 12)1241 560 RETURN1242 END1243 SUBROUTINE ALLPAR(PARAM)1244 DIMENSION PARAM(1)1245 COMMON/PARAMS/ M, RFIX( 5),R(%5), DFIX(5),D(5)TFIX(c5),TO( 5)1246 - NDATA1247 INTEGER RFIX,DFIX,TFIX1248 DATA IT/1H*/1249 C1250 C THIS SUBROUTINE SPLITS THE SINGLE PARAMETER VECTOR, PARAMS1251 C USED BY THE FITTING PROGRAM INTO SEPARATE VARIABLEs, R,1252 C D, AND TO IN COMMON /PARAMS/. FIXED PARAMETER VALUES1253 C IN COMMON ARE NOT ALTERED.1254 C1255 IP=I1256 DO 100 K=I#M1257 IF(RFIX(K) .EQ. IT ) GO TO 1001258 R(K)=EXP(PARAM(IP))1259 IP=IP+11260 100 CONTINUE1261 C1262 MI=M-1126:3 DO 200 K=I,M11264 IF(DFIX(K) EQ. IT ) GO TO 2001265 D( K )=EXP( PARAM( IP))

126612671268 C12691270127112721273127412751276127712781279128012811282 C1283 C1284 C1285 C1286 C128712881289129012911292129312941295129612971298129913001301130213031304 C1305 C1306 C1307 C1308 C1309 C1310 C13111312131313141315131613171318 C13191320

IF( FCHISQ. LT. RMAX) GO TO 200RETURN

39E

IP=IP+l200 CONTINUE

DO 300 K=l1NDATAIF( TFIX( K). EQ. IT) 0 TO 300TO(K)=(-EXP(PARAM( IP ))+1000. )/1000.IP=IP+I

300 CONTINUERETURNENDSUBROUTINE PRTPAR

COMMON/PARAMS/ Ma RFIX( 5)s,R( 5)s DFIX(5),D(5).TFIX( 5)p,TO( 5),- NDATAINTEGER RFIX,DFIX,TFIXINTEGER 0606=6

THIS SUBROUTINE PRINTS CURRENTPARAMETER VALUES STORED INCOMMON /PARAMS/

WRITE( 06,10)

10 FORMAT(" ( DENOTES FIXED VALUE)'o7Xo- LAYER RESISTIVITY THICKNESS')MI=M-1DO 50 K=IsM1WRITE(06m20) Ks,R( K)s,RFIX( K), D( K).,DFIX(K)

20 FORMAT(35Xs I3o4X, E12. 6,A1.iX, E12. 6,A1)

50 CONTINUEWRITE(06,20) MjR(M)oRFIX(M)DO 60 I=1,NDATA

60 WRITE(06,55) TFIX(I ).,Ia,TO(!)55 FORMAT(/35XsA1,'TO(-'sI1')=',E12.5)

RETURNENDSUBROUTINE CHISQ(NOB. NP. FOBS. FCOMP9 IFLAG)DIMENSION FOBS(NOB),FCOMP(NOB)COMMON/CTEST/ RMAX

THIS SUBROUTINE TESTS TO SEE IF THE REDUCED CHI SQUARE

IS LESS THAN RMAX. SEE P.R. BEVINGTONs DATA REDUCTION

AND ERROR ANALYSIS FOR THE PHYSICAL SCIENCES, MCGRAW

HILL, 1969 FOR A DISCUSSION OF THE CHI SQUARE GOODNESSOF FIT TEST.

IFLAG=OFCHISQ=O.DO 100 I=lpNOB

100 FCHISQ=FCHISQ+(FOBS(I)-FCOMP(I))**2FCHISQ=FCHISQ/(NOB-NP)PRINT 150, FCHISQ

150 FORMAT(/- CHI SQUARE =,FIO.5)

a

399

1321 200 PRINT 300,RMAX1322 300 FORMAT(' ITERATION STOPS'/1323 - 5X," REDUCED CHI SQUARE LESS THAN ,F8. 3)1324 IFLAG=l1325 RETURN1326 END1327 SUBROUTINE PRTFPR(SD, IDF, NP, TH, CL, CL)1328 DIMENSION TH( NP ), CU( NP ), CL( NP)1329 COMMON/PARAMSi M, RFIX( 51 ), R( 5), DFIX( 5 ), D( 5 ), TFIX( 5 ), TO( 5 ),1330 - NDATA1331 REAL LO1332 INTEGER 0613:33 INTEGER RFIX,DFIX,TFIX1334 DATA UP/5HUPPER/, LO/5HLOWER/

1335 06=61336 C13:37 C THIS !SUBROUTINE PRINTS FINAL PARAMETER VALUES, WITH UPPER AND

1338 C LOWER CONFIDENCE LIMITS.13:39 C1340 WRITE( 06.9)1341 9 FORMAT( IHI)1342 WRITE(06,10)1343 10 FORMAT(//38X, FINAL PARAMETER VALUES AND ASSOCIATED STATISTICS')1344 C1345 WRITE(06, i00) SD, IDF

1346 100 FORMAT( '.-TANDARD DEVIATION OF RESIDUALS ='sE12. 49 ,'

1347 - 16,' .DEGREES OF FREEDOM')1348 C1349 WRITE( 06, 110)1350 110 FORMAT(//-' FINAL PARAMETER VALUES')1351 CALL ALLPAR( TH)1352 CALL FRTPAR1353 C1354 WRITE(06, 120) UP

1355 120 FORMAT(/ilXoA5o-' CONFIDENCE LIMITS (BASED ON LINEAR HYPOTHESIS)')1356 CALL ALLPAR( CU)1357 CALL PRTPAR1358 C1359 WRITE( 06, 120) LO1360 CALL ALLPAR( CL)1361 CALL PRTPAR1362 RETURN1363 END1364 SUBROUTINE PRTCOR( NP, C, Ei EV)1365 DIMENSION C(NP, NP),E(NP),EV(NP)1366 COMMON/PARAMS/ M, RFIX(5), R(5 ), DFIX( 5), D( 5), TFIX( 5), TO( p),1367 - NDATA1368 INTEGER RFIX,DFIX,TFIX1369 DATA IT/IH*/

1370 DATA IR/1HR/s ID/1HD/, ITO/2HTO/1371 C1372 C THIS SUBROUTINE PRINTS THE CORRELATION MATRIX OF THE

1373 C PARAMETERS TOGETHER WITH THE NORMALIZING ELEMENTS.1374 C EIGENVALUES ARE ALSO PRINTED1375 C

100

1376 NL=M1377 PRINT 91378 9 FORMAT(IHI)1379 PRINT 10

1380 10 FORMAT(//30XI-CORRELATION MATRIX OF THE PARAMETERS AND ASSOCIATED1381 -INFORMATION')1382 PRINT 20,(JoJ=1NP)1383 20 FORMAT(/'OPARAMETER PARAMETER CORRELATION WITH PARAMETER -/1384 e NUMBER TYPE e,1616/(20X,1616)1385 NLINE=l1386 NPAGE=I1387 IP=O1388 KR=l1389 KH=I1390 TH=I1391 100 IP=IP+l1392 IF(IP .GT. NP) GO TO 2001:393 110 IF(KR .GT. NL) GO TO 140

1394 IF(RFIX(KR) .NE. IT ) GO TO 1201395 KR=KR+11396 GO TO 1101397 120 INUM=KR1398 ITYPE=IR1399 KR=KR+I1400 GO TO 1801401 140 IF(KH .GT. NL-1) GO TO 1601402 IF(DFIX(.KH). NE. IT ) GO TO 1501403 KH=KH+I1404 O0 TO 1401405 150 INUM=KH1406 ITYPE=ID1407 KH=KH+l1408 GO TO 1801409 160 IF(TH .GT.NDATA) O0 TO 1801410 IF(TFIX(TH).NE. IT) GO TO 1611411 TH=TH+I1412 GO TO 1601413 161 INUM=TH1414 ITYPE=ITO1415 TH=TH+l1416 180 PRINT 185a IP, ITYPEP INUM,(C( IP,J),J=I, IP)

1417 185 FORMAT("0/,15oSX? A2v 12,3X, 16F6. 2/ (21X, 16F6.2))1418 NLINE=NLINE + 2 + (IP-1)/161419 IF(NLINE LT. 40) GO TO 1001420 IF(IP .EQ. NP) GO TO 2001421 NPAGE=NPAGE+I1422 PRINT 101423 PRINT 20s (J#J=I,NP)1424 NLINE=O1425 GO TO 1001426 C1427 200 NOUT=8+3* ((NP-I)/10)1428 IF(NLINE .LT. 40-NOUT) GO TO 210

1429 NPAGE=NPAGE+l1430 PRINT 10

401

1431 NLINE=O

1432 210 NLINE=NLINE+NOUT1433 PRINT 215

1434 215 FORMAT( ///"ONORMALIZING ELEMENTS-)1435 LOW=1

14:36 220 LUP=LOW+91437 IF( LUP . GT. NP ) LUP=NP1438 PRINT 230, (JaoJ=LOWPLUP)14:39 230 FORMAT( /'"OPARAMETER NUMBER, 5X, I5, 9110)1440 PRINT 240, (E(J).J=LOW,LUP)1441 240 FORMAT( * ELEMENT VALUE "#5X,10(1XsE9.3)1.442 LOW=LOW+ 101443 IF(LOW .LE. NP) GO TO 2201444 C1445 IF(NLINE .LT. 40-NOUT) GO TO 3001446 NPAGE=NPAGE+11447 PRINT 101448 300 PRINT 3101449 310 FORMAT(///-OEIGENVALUES OF MOMENT MATRIX")1450 LOW=lI1451 320 LUP=LOW+91452 IF(LUP GT. NP) LUP=NP145:3 PRINT 330, (JsJ=LOWPLUP)1454 330 FORMAT( "016X,9I10)1455 PRINT 340, (EV( J ), J=LOWp LUP)1456 340 FORMAT( 20X s10( 1X'tE9. 3)1457 LOW=LOW+101458 IF(LOW .LE. NP) GO TO 3201459 RETURN1460 END1461 SUBROUTINE PRTFIT( NDATA# NF2p NOB, NP F5 Y; "=-DY NFREQ2)

1462 DIMENSION F( 15 ), R( 15 ), SDR( 15 ), RC( 15 ) 0(15 )SDQ( 15 )a QC( 15)1463 DIMENSION NF2( 5 )# F5( NOB ), Y( NOB )s SDY( 5, 30)1464 REAL NFREQ2(5, 15)1465 DIMENSION X(2#30)1466 C1467 C THIS SUBROUTINE PRINTS AND PLOTS FINAL FIT1468 C WITH OBSERVED VALUES.1469 C1470 PRINT 91471 9 FORMAT( IHI)1472 PRINT 101473 10 FORMAT(//48Xp'"OBSERVED AND FITTED DATA VALUESe/)

1474 PRINT 20

1475 20 FORMAT( 22X "*****************MAGNITUDE********************'O1476 - 6X, " ** **** F*HASE************1477 NUMBER FREQUENCY'1478 - OBSERVED S.D. COMPUTED (OBS-COMP )/SD, 4X,1479 - OBSERVED S. D. COMPUTED (OB:S-COMP )/SD'-)1480 K3=01481 DO 100 I=1,NDATA1482 K4=01483 NNF2=NF2( I)1484 DO 50 J=INNF21485 K3=K3+1

40.

1486148714881489149014911492149314941495149614971498149915001501150215031504150515061507150815091510151115121513151415151516151715181519152015211522152315241525152615271528152915301531153215331534153515361537153815391540

K4=K4+1F( J )=NFREQ2( Is J)R( J )=Y( K3)SDR( J )=SDY( Is K4 )R( J )=R( J )*SDR( J)

50 RC( J )=F5( K:3 )*SDR( J)DO 60 J=1,NNF2K3=K3+1K4=K4+1Q( J )=Y( K3)SDQ( J )=SDY( It K4 )Q( j )=Q( j )*SDQ( j)

60 QC( J )=F5( K3 )*SDQ( J)DO 75 J=1,NNF2DR=( R( J )-RC( J ) )/SDR( J)Dl=( Q( J )-QC( J ) )/SDQ( J)

75 PRINT 80, Js F( J )p R( J)s SDR( J); RC( J )o DRo Q(J), SDI.(J), QC(J)o,DQ80 FORMAT( 15, F15. 6o F1O. 6sFl1. 6a F12. 6p F13. 5, 6X, F12. 6m Fl1. 6; F12. 6,F13. 5:

-/5

CALL CHISQ( NOBs NP, Y, F5o IFLAG)PRINT 200

200 FORMAT(//' PLOT MAGNITUDE - - OBSERVED=Ot THEORETICAL=+')DO* 210 J=INNF2X( 1, J)=R(J)

210 X(2# J)=RC(J)CALL PPLOT( Xs NNF2)PRINT 300

300 FORMAT( //" PLOT PHASE - - OBSERVED=Oo THEORETICAL=+')DO 220 J=1.NNF2X(1, J)=Q(J)

220 X( 2# J )=QC(J)CALL PPLOT( X, NNF2)

100 CONTINUECALL ETIMERETURNENDSUBROUTINE PPLOT( Xi NM)DIMENSION X( 2s 30 ), KK( 130)DATA IBLANKi IHIGHt ILOW/1H , IHO, 1H+/XN=1. OE+36XM=-XNIF (NM. LT. 1) GO TO 11DO 4 I=1.NMDO 4 J=l2IF( X(Jo I )-XM) 212. 1

1 XM=X(Jp I )2 IF(X(J, I )-XN) 3,4,43 XN=X(JoI)4 CONTINUE

CON=( XM-XN )/129.DO 14 I=I#NMDO 8 J=1, 130

8 KK(J)=IBLANKKX=( X( 1, I )-XN)/CON +1. 0KK( KX )=IHIGH

40:3

1541 KX=( X(2, I )-XN)/CON +1. 0

1542 KK( KX )=ILOW1543 PRINT 12s (KK(K),K=ls 130)1544 14 CONTINUE1545 11 CONTINUE1546 12 FClRMAT(/lX,130A1)1547 RETURN1548 END

1549 FUNCTION ACOS(X)

1550 ACOS=ATAN( SQRT( 1. 0-X**2)/X)1551 RETURN1552 END$EOF..

404

B8.2.3 EM-INV Input Card Listing.

I23

456789

1011121314151617181920212223

2425

26272829303132333435363738

394041424344454647

r-i

JOB BEN UI(ASSIGN 6=6ASSIGN 21=PRGO01SITE 16

$F

:3

1

1HZ1680000.24000.

25000.19000.1O0.

100.51.52.53.96825.56.982118.9860911.947615.960820.930220

29.282.6133.202.262.303.354.374.360.

D6 M41000 T9999

- ROCKY RUN CK. - - HZ

10000.1. E+71. E+715000.1. E+71500.

2000.180000.1000.11000.3001.

0.1000.30000.10.3500.

3000.500.

€

405

48 323.49 79. 150 63. 951 51.852 29. 953 9. 4754 -10. 255 -36. 256 -72. 057 -112.58 -159.59 2.560 2.561 2.562 2.563 2.564 2.565 2.566 5.67 10.68 20.69 1.570 1.571 1.572 1.573 1.574 1.575 1.576 2.577 5.78 10.79 10 . 01 10. 0 . 001 . 001 1. E-20

80 $EOJEoF..

B8.2.4 EM-INV Sample Calculation.

LOCAT ION - ID - -

SITE 16 - - ROCKY RUN CK. - - HZ

TIME= 84692 SECS AFTER MIDNIGHTDATE= 05-06-77

INITIAL MODEL PARAMETERSNO. Ok LAYERS = 3

INITIAL GUESS

RESISTIVITY(1)= 20000E+04RESISTIVITY(2)= .18000E+06RESISTIVITY(3)= .10000E+04THICKNESS(I)= 11000E+05

THICKNESS(2)= .30010E+04ZERO TIME(1)= O0000E+01

SUMMARY OF CONSTANT PARAMETERS

DATA SET I

CALCULATE HZ FIELD

DIPOLE MOMENT = 16800E+07

TX LENGTH = 24000E+05

COIL AREA = 25000E+05

RECEIVER LOCATION

RANGE= . 19000E+05

OR I ENl AT I ON= 10000E+03X= 329'*3E+04Y= 1S711E+05FREGILNCIES

50000E+00 15000E+01 .25000E+01

69321E+01 *.89e61E+01 .11948E+02

PRINI THE OBSERVED DATA VALUES

29*200E+02 .*82600E+02 .13300E+03

30Z0oE+03 35400E+03 .37400E+03

79100E+t)2 63900E+02 .51800E+02

102OOE-402 -. 36200E+02 -. 72000E+02

PRINI STANDARD DEVIATIONS

.25000E+o1 25000E+01 25000E+01

.25000E+01 .25000E+01 .50000E+01

.15000E+01 15000E+01 .15000E+01

. 150ooE+1 15000E+01 .25000E+01

LOWER BOUND.10000E+04.30000E+05.10000E+02.35000E+04.30000E+04.50000E+03

UPPER BOUND.10000E+05.10000E+08.10000E+08.15000E+05.10000E+08.15000E+04

.39682E+01 .55000E+01

.159&1E+02 .20930E+02

.20200E+03 26200E+03

. 36000E+03 .32300E+03.29900E+02 .94700E+01

-. 11200E+03 -. 15900E+03

.25000E+01 25000E+01

.10000E+02 20000E+02

.15000E+01 15000E+01

.50000E+01 10000E+02 6"

GAUSHAUS PARAMETERSMAXIMUM NUMEER OF ITERATIONS

= 10

GAMMA = .100E-01

FNU = 10.0000

EPSI . 100E-02

EPS2 = .IOOE-02

EPS3 .100E-19

CHI SQUARE CONVERGENCE PARAMETER 1.00000

GASAUS ENTERED -

20 OBSERVATIONS6 PARAMETERS

274 SCRATCH REQUIRED

INITIAL SUM OF SQUARES = . 6407E+04

ITERATION NO. I NO. OF FUNCTION CALLS 1

DETERMINANT =

.6500E+l1 MIN. PIVOT m

.4378E+00 ANGLE IN SCALED COORD. =57.10 DEGREES

TEST POINT PARAMETER VALUES

.7972E+01 .1210E+02 .6889E+01 9263E+01 8012E+01 6884E+01

TEST POINT SUM OF SQUARES . 2782E+02

PARAMETER VALUES VIA REGRESSION(* DENOTES FIXED VALUE)

LAYER RESISTIVITY THICKNESS1 .289837E+04 105390E+05

2 .180000E+06 .301574E+04

3 .980990E+03

TO(I)= .23664E-01

RUN lIME = OOHRS 03MIN 17.512000SEC

SUM OF SQUARES AFTER REGRESSION .2782190E+02

GAMMA = IOOE-02

CHI SQUARE = 1.98728

ITERATION NO. 2 NO. OF FUNCTION CALLS 9

DETERMINANT = 2209E+04 MIN. PIVOT .1259E+00 ANGLE IN SCALED COORD. =63. 20 DEGREES

TEST POINT PARAMETER VALUES

7946E+01 .1210E+02 .5564E+O1 .9594E+01 .9314E+01 .6884E+01

TEST POINT SUM OF SQUARES = .3774E+02

C

DETERMINANT .7970E-01 MIN. PIVOT = .1573E+00 RNUL

TEST POINT PARAMETER VALUES

.7910E+01 . 1556E+02 .5765E+O1 9356E+01 .9553E+01 6884E+O1

BAKOND: FER 05 @ 026104

BAKOND: FER 05 0 026136

TEST POINT SUM OF SQUARES = .3380E+02

DETERMINANT = .2593E+10 MIN. PIVOT .3731E+00 ANOL

TEST POINT PARAMETER VALUES

.8004E+01 .1210E+02 .6874E+01 .9339E+01 .8040E+01 .6883E+01

TEST POINT SUM OF SQUARES =

.2526E+02

PARAMETER VALUES VIA REGRESSION(* DENOTES FIXED VALUE)

E IN SCALED COORD. =65.58 DEGREES

LE IN SCALED COORD. =75. 32 DEGREES

LAYER RESISTIVITY THICKNESS

I .299336E+04 .113722E+052 .180000E+06 .310402E+04

3 .966512E+03

TO(I)m .24110E-01

RUN TIME = OOHRS 07MIN 23. 178000SEC

SUM OF SQUARES AFTER REGRESSION = .2526057E+02GAMMA = .659E-03

CHI SQUARE = 1.80433

--- ITERATION NO. 3 NO. OF FUNCTION CALLS =

19

DETERMINANT = .2444E+07 MIN. PIVOT .3133E+00 ANGLE IN SCALED COORD. =49.53 DEGREES

TEST POINT PARAMETER VALUES

7997E+01 .1210E+02 .6657E+01 .9388E+01 .8278E+01 .6883E+01

TEST POINT SUM OF SQUARES = .2295E+02

PARAMETER VALUES VIA REGRESSION

(* DENOIES FIXED VALUE)

RUN TIME = OOHRS IOMIN 40.390000SEC

GAMMA = .628E-04

LAYER RESISTIVITY THICKNESSI .297066E+04 .119435E+05

2 .180001E+06 .393699E+04

3 .778110E+03

TO( I )m . 23993E-01

0

SLIM OF SQUARES AFTER REGRESSION = .2295033E+02

AMCMl- - .-- A W"VJ.%r- . ^^

CHI SQUARE = 1.63931

TEST POINT PARAMETER VALUES7989E+01 .1210E+02 .6440E+O1 9437E+01 .8516E+01 .6884E+01

TEST POINT SUM OF SQUARES = .2212E+02

ITERATION NO. 4 N

DETERMINANT = .5657E+04 MIN. PIVOT .1019E+00

TEST POINT PARAMETER VALUES

.7Y32E+01 1210E+02 .6308E+01 .7978E+01 .1019E+02

DETERMINAN = .5640E-01 MIN. PIVOT = .1036E+00

TEST POINT PARAMETER VALUES.7776E+01 1564E+02 .5650E+01 .8290E+01 .1009E+02

TEST POINT SUM OF SQUARES = .6044E+03

DETERMINANI = .5361E+10 MIN. PIVOT = .4108E+00

TEST POINT PARAMETER VALUES

.7973E+01 .1210E+02 .6427E+1 .9267E+01 .8523E+01

TEST POINT SUM OF SQUARES w .1963E+02

PARAMLTER VALUES VIA REGRESSION(* DENOTES FIXED VALUE)

40. OF FUNCTION CALLS = 28

ANGLE IN SCALED COORD.

6886E+O1

ANGLE IN SCALED COORD.

=75.75 DEGREES

=73.85 DEGREES

.6885E+O1

ANGLE IN SCALED COORD. =64.22 DEGREES

.6884E+01

LAYER RESISTIVITY THICKNESSI .290096E+04 .105806E+05

2 .180003E+06 .503002E+04

3 .618029E+03

TO(1)= .23779E-01

RUN TIME = OOHRS 14MIN 44.651000SEC

SUM OF SQUARES AFTER REGRESSION = .1962644E+02GAMMA = .588E-03

CHI ISQUARE = 1.40189

ITERATION NO. 5 NO. OF FUNCTION CALLS = 37

DETERMINANT = .9183E+06 MIN. PIVOT a 1902E+00 ANGLE IN SCALED COORD. =73. 73 DEGREES

TEST POINT PARAMETER VALUES

7834E+01 .1210E+02 .6340E+01 .8973E+01 8914E+01 6884E+01

TEST POINT SUM OF SQUARES = .1780E+02

-Fh0

PARAMETER VALUES VIA REGRESSION(* DENOTES FIXED VALUE) LAYER RESISTIVITY THICKNESS

1 .265479E+04 .788799E+04

2 .180004E+06 .743661E+04

3 .566921E+03

TO(lI1)w .2317SE-01

RUN TIME = OOHRS 18MIN 00.524000SEC

SUM OF SQUARES AFTER REGRESSION = .1779543E+02GAMMA w .553E-04

CHI SQUARE = 1.27110

ITERATION NO. 6 NO. OF FUNCTION CALLS = 45

DETERMINANT = . 1162E+02 MIN. PIVOT .2462E-01 ANGLE IN SCALED COORD. =85.01 DEGREES

TEST POINT PARAMETER VALUES

7327E+01 .1210E+02 .5996E+01 .8047E+01 .9807E+01 .6888E+01

DETERMINANT . 1145E-02 MIN. PIVOT .5560E-01 ANGLE IN SCALED COORD. =76.06 DEGREES

TEST POINT PARAMETER VALUES

.760E+01 1276E+02 .4827E+01 8855E+01 9599E+01 6886E+01

TEST POINT SUM OF SQUARES a . 1706E+02

PARAMETER VALUES VIA REGRESSION

(* DENOTES FIXED VALUE) LAYER RESISTIVITY THICKNESSI .218136E+04 .701156E+04

2 .346579E+06 147567E+05

3 .124818E+03

TO(1)m .215S1E-01

RUN TIME = OOHRS 21MIN 16. 114000SEC

GAMMA =m.527E-04 SUM OF SQUARES AFTER REGRESSION = .1706059E+02

CHI SQUARE = 1.21861

ITERATION NO. 7 NO. OF FUNCTION CALLS w 53

DETERMINANT .1644E+03 MIN. PIVOT .1725E-01 ANGLE IN SCALED COORD. =85.80 DEGREES

TEST POINT PARAMETER VALUES

7766E+01 .1276E+02 .4865E+01 .8169E+01 .9459E+01 .6889E+01

TEST POINT SLIM OF SQUARES = .5092E+02

0

DETERMINANT = 1069E-02 MIN. PIVOT = 3533E-01 ANGLE IN SCALED COORD. =82.35 DEGREES

TEST-POINT PARAMETER VALUES7484E+01 1335E+02 5903E+01 8488E+01 .9260E+01 6887E+01

TEST POINT SUM OF SQUARES = 1250E+02

PARAMETER VALUES VIA REGRESSION(4 DENOTES FIXED VALUE) LAYER RESISTIVITY THICKNESS

1 177998E+04 485467E+042 .625894E+06 .105039E+053 366052E+03

TO(1)= 20442E-01

RUN TIME = OOHRS 24MIN 55.3610005EC

GAMMA = 516E-04

CHI SQUARE = 89255ITERATION STOPS

REDUCED CHI SQUARE LESS THAN

STANDARD DEVIATION OF RESIDUALS =

FINAL PARAMETER VALUES(* DENOTES FIXED VALUE)

UPPER CONFIDENCE LIMITS (BASED ONBAKOND: FER 10 @ 020545

( DENOTES FIXED VALUE)

LOWER CONFIDENCE LIMITS (BASED ONBAKGNU: FER 10 020545

(* DENOTES FIXED VALUE)

SUM OF SQUARES AFTER REGRESSION = .1249572E+02

1.000

FINAL PARAMETER VALUES AND ASSOCIATED STATISTICS

.9447E+00, 14 DEGREES OF FREEDOM

LAYER RESISTIVITY1 .177998E+04

2 .625894E+06

3 366052E+03

TO(1)= 20442E-01

LINEAR HYPOTHESDIS)

LAYER RESISTIVITY1 .142269E+05

2 .170141E+393 250390E+04

TO()= .11875E-01

LINEAR HYPOTHESIS)

LAYER RESISTIVITY1 .222699E+032 170141E+393 -53514E+02

TO(1)= 28934E-01

THICKNESS.485467E+04.105039E+05

THICKNESS.797997E+05.323890E+05

THICKNESS.295337E+03.340648E+04

CORRELATION MATRIX OF THE PARAMETERS AND ASSOCIATED INFORMATION

PARAMETER PARAMETER

NUMBER TYPE

1 Ri1

2 R 2

3 R 3

4 DlI

5 D 2

6 TO I

NORMALIZING ELEMENTS

PARAMETER NUMBER

ELEMENT VALUE

CORRELATION WITH PARAMETER -1 2 3 4 5 6

1. 00

-. 14

-.57

.95

-. 66

-. 99

1. 00

.34

. 17

-. 60

. 08

1.00

-.49 1.00

.02 -.83 1.00

.52 -.96 .71 1.00

1 2 3 4 5 6.110E+01 .238E+03 .102E+01 148E+01 .596E+00 . 461E-02

EIGENVALUES OF MOMENT MATRIX

1 2 3 4 5 6

.149E+08 . 195E+04 .971E+02 .246E+01 138E+00 -. 158E+00

OBSERVED AND FITTED DATA VALUES

NUMBER1

2

3

4

5

6

7

8

9

10

*****************MAGNITUDE********************

FREQUENCY OBSERVED S.D. COMPUTED (OBS-COMP)/SD

.500000 29.199993 .730000 29.484276 -.36943

1.500000 82.599976 2.065000 83.670959 -.51864

2.500000132.999969 3.324999 133.418915 -. 12600

3.968200201.999969 5.049999 198.105530 .77118

5.500000261.999939 6.549999 254.539001 1.13907

6.982109302.999939 7.574999 298.679871 .57031

8.986090353.999939 8.849998 343.481506 1.18852

11.947599373.999939 18.699997 383.424438 -.50398

15.960798359.999939 35.999992 400.428223 -1.12301

20.930199322.999939 64.5999?1 384.022827 -.94463

OBSERVED S.D. COMPUTED (OBS-COMP)/SD

79.099976 1.500000 79.814667 -.47646

63.899986 1.500000 64.714737 -. 54316

51.799988 1.500000 50.378677 .94754

29.899994 1.500000 30.048962 -.09931

9.469997 1.500000 9.657454 -.12497

-10.199999 1.500000 -9.337234 -.57518

-36.199997 1.500000 -33.990028 -1.47331

-71.999985 2.500000 -68.558792 -1.37648

-112.000000 5.000000 -112.490204 .09804

-158.999969 10.000000 -163.129883 41299

CHI-SQUARE = .89255

ITERATION STOPS

REDUCED CHI SQUARE LESS THAN 1.000

PLOT MAGNITUDE - - OBSERVED=O, THEORETICAL=+

+0

+ 0

+ 0

+ 0

~0 +

PLOT PHASE - - OBSERVED=Oo THEORETICAL=+0+

46

414

B9. Acknowledgements

This work was supported by the following grants: National

Science Foundation GA 37171, DES75-04879, EAR 75-04879 AOl; Office

of Naval Research contract NOOO14-67-A-0128-0026; and University of

Wisconsin Industrial Research grant 160706; C. S. Clay, principal

investigator.

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