universal model for adjusting observed values

U N I V E R S A L M O D E L F O R A D J U S T I N G O B S E R V E D V A L U E S

LUBOMfR KUB~EK

Institute of the Theory of Measurement, Slovak Acad. Sci., Bratistava*)

S u m m a r y : Fundamental models of the calculus of observations. Regularity and singularity o f models. Universal model. Unbiased estimable and unbiased unestimable functions of the model parameters.

1. INTRODUCTION

The problems of rationalization and automation also have to be considered in the field of geodetic calculations. The use of more accurate and, therefore, also more expensive instruments for observations require a more responsible approach to the processing of the obseryed data, in order to avoid unnecessary depreciation of these data by unsuitable methods of processing. This will require, more frequently than in the past, the use of exact methods of adjustment which are, however, exacting as regards numerical operations. The more frequent use of exact methods of adjustment will also require a unification of the computation algorithms, in order to enable their application to become mechanized.

Information on some of the research results, obtained in this respect over recent years, is presented below.

2. SYMBOLS USED AND FUNDAMENTAL TERMS

A set o f observed values x~ . . . . . x , is cons idered to represent the c ompone n t s

o f a r a n d o m n-dimensional vector ~n,~. A co lumn vector with the componen t s x 1 . . . .

• .., xn is denoted by Xn,1. The mean value o f the vector ~ is a ssumed to read E(~) =

= An,sO~,l. (If the re la t ion between E(~) and 0 is not l inear, it can usual ly be l inear ized

by deve lopment into a Tay lo r series, assuming tha t the app rox ima te values o f the

componen t s of the vector 0 are known. F o r detai ls see, e.g., (4)). The mat r ix A is

known (in present l i tera ture known as the design matr ix) , and the componen t s o f

the s -d imensional vector 0 are unknown values which are to be de te rmined f rom the

values x 1 . . . . . x,. The accuracy of the observat ion , as well as the stoghast ic re la t ions

between the individual elements o f the set o f observed values, are charac ter ized by

the co-var iance mat r ix X = E{[~ - E(~)] [~ - E(~)] '} = o -2 . H. The n x n- type

mat r ix H is a mat r ix o f the weighting coefficients o f the observed values and cr z is

the unit d ispers ion (o- is the unit s t anda rd error) . The mat r ix H is usual ly ob ta ined

in the fo rm H = p - l , where P is the mat r ix d iagona l with elements Pi > 0, i = 1, . . .

. . . . n a long the d iagona l (weights o f the observat ions) , In some cases the u n k n o w n

pa ramete r s 0 t . . . . . 0~ satisfy the system of condi t ions bq, 1 + Bq,sO~, 1 = Oq, 1. The

symbols bq, 1 and B~,~ represent a co lumn q-d imens iona l vector and a q x s - type

*) Address: Dfibravski cesta, 885 27 Bratislava - Patr6nka.

Studia geoph+et geod. 2011976] 103

L. Kubd~ek

Model

Model b=O;B

Model matrix).

matrix with known elements, respectively. (Under non-linear conditions f~(Oa . . . . . Os) = = 0, i = 1 . . . . . q it is usually possible to linearize the conditions, provided the approximate values O~ °), j = 1 . . . . , s, of the components of the vector 0 are known. The r-th component of vector b will then read {b}r~ = f,(O~°),..., 0~°)), and the rk-th element of matrix B will read {B}, k = c3fr(O] °) . . . . . 0~°))/~0k. In this case, instead of the unknowns 0~ . . . . ,0~ we consider the unknowns 80a = 0~ - 0 (°) . . . . . . . . 80~ = 0~ - 0~ °).

In general, therefore, the situation characterized by the following relations is generated in adjusting:

(1.1) E (~ )= A0 ; b + B 0 = 0 ; Z = a 2 . H .

Depending on the structures of the design matrix A, the condition matrix B and the matrix of weighting coefficients H in (1.1) there are 5 fundamental models of adjustment.

1 (direct observation of the scalar parameter (s = 1)): A = i; i ' = 1 ) ; b = 0 ; B = 0 . 2 (indirect observation of the s-dimensional vectorial parameter (s > 1)): ~ 0 .

3 (conditional observation of the vectorial parameter): A = i,,, (identical

Model 4 (direct observation of the n-dimensional subvector of the s-dimensional vectorial parameter with a system of conditions): A = (I .... O,.r); n + r = s.

Model 5 (indirect observation of the s-dimensional vectorial parameter with a system of conditions).

Within the scope of the fundamental models, mentioned above, it is possible to determine the adjusted values of the parameters by means of the least-squares method, provided the following conditions (conditions of regularity) are satisfied:

(1.2) Model 1: R(H) = n.

Model 2: R(A) = s (i.e. n > s); R ( H ) = n.

Model 3: R(B) = q (i.e. n > q); R(H) = n.

Model 4: B = (B1, B2), where B 1 is of the q x n-type, B 2 is of the q x

x r-type, R ( B ) = q , R ( B 2 ) = r(i.e, r < q) and n +

+ r > q ) , R ( H ) = n.

Model 5: R(A) = s (i.e. s < n); R(B) = q (i.e. q < s); R(H) = n.

latter model can also be studied under the conditions R ( A B ) . = s ; since the (The \ - - /

object was to give the explicit relation (2.1.e), the assumption given above was employed). Here the symbol R(A) represents the rank of matrix A.

104 Studia geoph, et good. 20 [1976]

Universal Model for Adjusting Observed Values

3. SOLUTION OF R E G U L A R SITUATIONS

The adjusted values of the parameters in regular models are obtained according to the following relations (for details see [3 ] - [6 ] ) :

(2.1.a) Model 1 : 0 = (i 'H-~i) -1 i ' H - l x .

(2.1.b) Model 2 : 0 = (A'M-1A) -1 A'H-lx.

(2.1.c) Model 3 : 0 = [ I - H B ' ( B H B ' ) -1 B] x - H B ' ( B H B ' ) -1 b.

(2.1.d) Model 4 : 0 = 02 = Q21B 1 ] x + _ 021 ] b ,

where {BaH B'I, BO2) = ( Q I , , Q,2" ~ . \B~, GI21 , GI22]

(2.1.e) M o d e l 5 : 0 = { ( A ' H - I A ) -1 - ( A ' H - t A ) -1 B ' [ B ( A ' H - 1 A ) ] -~ .

. B ( A ' H - ' A ) - I } . A ' H - * x - ( A ' H - 1 A ) - 1 .

• B ' [ B ( A ' H - 1 A ) - 1 g ' ] - ' b .

The co-variance matrices for the adjusted values of the parameters determined within the scope pf the regular models, are defined by the following relations:

(2.2.a) Model 1: o'2(0) = o'2. ( iH-1 i ) - , .

(2.2.b) Model 2: Iga = c, 2 . ( A ' H - 1 A ) - I .

(2.2.c) Model 3:12 a = o'2. [H - HB'(BHB') -1 B H ] .

(2.2.d) Model 4: lg(a,} = o ' 2 . { H - HB;QA1B,H, - HB'IQ12~. \02! ~k Q21B1 H , -- Q22 J

(2.2.e) Model 5: £a = a 2 . {(A'I-I-IA) -1 - ( A ' H - ' A ) -1 8 ' .

• [ B ( A ' H - I A ) -~ B ' ] - ' B ( A ' H - ~ A ) - ~ } .

4. S I N G U L A R I T Y OF THE MODEL A N D G E N E R A L I Z E D INVERSION OF THE MATRICES

If the appropriate conditions from (1.2) are not satisfied within the scope of the individual models, the solution of the normal equations cannot be presented in the form (2.t.a) to (2.1.e) because the matrix inversion, indicated there, does not exist. However, from the point of view of numerical operations a similar situation will occur, if the matrices which have to be inverted to solve the normal equations, have determinants close to zero, even though conditions from (1.2) are satisfied.

The procedure for treating these situations was presented by Bjerhammar [1], Rao [9] and Rao and Mitra [8]. Considerable use of the generalized inversion

Studia geoph, et geod. 20 [19713] 105

L. KubdEek

of matrices (the so-called g-inversion) was made in this respect. Apart from solving the problem mentioned above, this theory enables a universal model of adjustment to be established, as will be proved below.

The fundamental concepts of the theory of the g-inversion of matrices are given in the subsequent sections (Ref. [8] and [7]).

There always exists at least one n x m-type matrix A - , for which A A - A = A and which is called the g-inversion of matrix A, to any m x nrtype matrix A. (It should be pointed out that A X A = A can be considered a linear system in which the unknowns are the elements of matrix A - . To write a computer program for solving this system presents no problem).

All the solutions x of the consistent system Ax = y (i.e. a system which has at least one solution for a given vector y) are obtained from the relation x = A - y , where A - all possible g-inversions of matrix A are substituted for A - .

If N,, , is a positive semi-definitive matrix (i.e. x ' N x > 0 for any n-dimensional vector x), the symbol tlxlI, represents the value x/(x 'Nx) which is called the N-seminorm of the vector x.

The matrix A~', which is the g-inversion of the matrix A and which has the pro- perty that JIA~-y[IN < IIA-yHN for any value of y for which the system Ax = y is consistent and for any g-inversion A - of the matrix A, A - + A~-, is called the minimum N-seminorm g-inversion of matrix A and is denoted by the symbol A~-,(N).

Assume the matrix Nm,,, t o b e positive and semi-definitive. The matrix A~, for which it is true that the inequality I IAA;y - YlIM ~ [l A x - - YlIM holds for any m-dimensional vector y and any n-dimensional vector x, is called the g-inversion of matrix A with regard to the least-squares method and is denoted by the symbol

A~M).

The symbols Ai-(M),m(N) and A + M,N are used to denote the g-inversion of matrix A +

which is of the A~-(M)-type and for which IIA. NYllN _--< IlAr,.>YllN for any m-dimen- +

sional vector and any matrix A~M ) + AM, N.

It should be mentioned that the matrix A2(N), as well as the matrices A/(M) and A~, N can be determined with the aid of the g-inversion of certain matrix expressions, as will be shown in Section 5 and 6.

It is now possible to demonstrate on Model 2 to what extent the adjusted parameters can be determined if the regularity conditions are not satisfied.

Assume that f (O) = p'0 is the functions of the parameters, the adjusted values of which is to be determined. We shall consider a certain functionf(x) of the observed values x to be this value ; f (x ) = L'x. If E(L'~) = p'O for all values of 0, f (x) = L'x

is then called the unbiased estimate of the function f(O). Since E([(~)) = E(L'~) = = L'E(~) = L'AO, the function f (O) = p'O can be subject to an unbiased estimate, provided L'AO = p'O for all values of 0, i.e. provided the system A'L = p is consistent. (The concept of estimability, which was unknown in the theory of the calculus of observations, was introduced in 1944 by R.C. Bose (2)). Since the dispersion

1 0 6 Studia geoph, et geod. 20 [1976]


o_2(f(~)) = a2. L'HL = a2!ILIl~ (for details see (4)), L is determined from the consistent system A'L = p to render ltLII H minimum, i.e. we shall use the minimum H-seminorm of the g-inversion of matrix A'

(3.1) L = (A')~,(H)lb.

If the system A'L = p is consistent for pi, where p'~ = (01, 02 . . . . ,0~_ ~, 1 i, 0i+ 1 . . . . ..., 0~), the unbiased estimate of the i-th parameter 0i can be obtained from the observed data x 1 . . . . . x,. I f the system A'L = p is no consistent for p~, the unbiased estimate for the parameter 0~ cannot be obtained from the observed data and we speak of an unbiased-unestimable parameter 0~. Similarly we speak of an unestimable function of the parameters f(O) if f (0) = P'oO, and for this Po the system A'L = p is inconsistent.

The above indicates the difficulties which are encountered in computing the adjusted values of indirectly observed parameters in the singular model.

It should also be pointed out that if the conditions (1.2) are satisfied for Model 2, with regard to (3.1) and (2.1.b) L'x = p '[A')~,(m] 'x = p ' ( A ' H - I A ) - 1 A ' H - ~ x . In this case the vector [A'),~(H~]' x is determined uniquely and can be denoted by the symbol 0, however, in the singular case this vector is not determined uniquely, because it depends on the selection of the g-inversion (A')~,(H). Leter on we shall see, however, that this ambiguity does not hamper the determination of the adjusted values of the values of the estimatable functions and, therefore, also in this case the notation 0 = [A'),~(H)]' x will be used.

5. T H E U N I V E R S A L M O D E L

The purpose of this section is to demonstrate that all other models, regular or singular, can be expressed with the aid of Model 2, if we omit the conditions of regularity of this model. We shall first demonstrate it on Model 3. Here, E(~,,I) = = 0,,,1; b~,l + gq,,O~,l = 0. The vector b may be considered a special case of a random vector with the mean value equal to b and with a zero covariance matrix. Therefore, the conditions b + B0 = 0 can be expressed as E ( - b ) = B0; IE b = 0. If we now form the vector (~', - b ' ) ' from the vectors ~ and b, Model 3 can be expressed like Model 2,

where the vector ~ is substituted by the vector ( _ ~), the design matrix A by the

matrix ( ~ ) , and the covariance matrix a2H by the matrix E(_g) .

Stuclia geoph, et geod. 20 [19761 107

L. Kubf6ek

Similarly, for Model 4

and for Model 5:

. ~ 0-2 (L (:: 0) g = ~ 0 -2

From the above it is clear that any fundamental model can be expressed as Model 2, provided the vector, ~, the design matrix A and the covariance matrix o-2H of Model 2 is expressed with the aid of vector ~ and b, the design matrix and the matrix of conditions, and the covariance matrix 0-2H and the zero matrices of the model being expressed, respectively.

Model 2, taken in this way, is called the universal model. Note. The assumption that the system of conditional equations b + B0 = 0 is

consistent, is also valid within the scope of the universal model.

6. T H E O R Y OF T H E U N I V E R S A L M O D E L

Within the scope of the universal model E(¢) = A0; X¢ = o-2H is the adjusted value of the estimable function of the parameters f(0) = p'O, given by the function of observed values f(x) = L;×, where

(5.1) A'L o = p and IIL011H __< IILIIH

for any solution L of the consistent system A'L = p, for which L =~ L o. This L o is defined by the relation /-o = (A'),~(H)P. In (8) it was proved that the matrix (H + + AA') - A[A'(H + AA') - A ] - can be taken for (A')~(H), so that

(5.2) f(x) = p'{(H + AA')- A[A'(H + AA')- A ] - } ' x .

The dispersion of this adjusted value is defined as (for details see (7))

(5.3) a2(f(¢)) = 0 -2 p '{[A'(H + AA' ) - A] - - I} p .

The vector

(5.4) {(H + AA')- A[A'(H + AA')- A]-}'x

will be denoted by the symbol O, and the matrix

(5.5) 0-2. {[A'(H + AA')- A]- - I}

by the symbol I~.

108 Studia geoph, et geod. 20 [1976]


The vector 0 from (5.4) and the matrix N~ from (5.5) behave like the vector of the adjusted parameters and the co-variance matrix of the adjusted parameters for all unbiased estimatable functions of the parameter 0. (A few comments on the unbiased- unestimatable functions of the parameters are in Section 6).

If the conditions of regularity (1.2) are satisfied and in (5.4) and (5.5) we consider the structure of the plan matrix A and the co-variance matrix a2H, formed by Model 3 it holds that (compare (2.1.c) and (5.4)):

[I - H B ' ( R H B ' ) -1 B ] x - H B ' ( B H B ' ) -1 b =

([(": :)+ (',,)~" "~]-('){~" "~ L,,o,r'" :) + (,) ~,. ,,'~]- (,,,)}-)'. ( : )

It also holds true (compare (2.2.c) and (5,5)) that

~ . [H - HW(BHW)-~ BH] =

"~ ([~" "~ [(:: :)+ (:)~" "1-(')}- - ' ) If the matrices A and H are generated in (5.4) and (5.5) by Models 4 and 5, res-

pectively, the estimates from (2,1.d) and (2.1.e) and the co-variance matrices (2.2.d) and (2.2.e) are expressed analogously to Model 3 provided the conditions of regularity are satisfied. (For details see [7]).

It can also be proved (cf. [9] and [7]) that if any 9-inversion of the matrix

.: ~ o ~)

H , = ~ o~) - (~:: ~ : ) (5.6)

the vector

(~.7) =C~x, or O=C3x

has the same properties as the vector ~l from (5.4) and the matrix

(5 .8 ) ~z~ = ~2C~

has the same properties as the matrix E~ from (5.5). If, for example, we determine

H,O i'\- o, oLB) ....... ;'-t-

\,, B , o/

s tud ia geoph, et geod. 20 [i976] 1 0 9

L. Kub55ek

/ Y \ for the regular Model 3, we arrive at C ~ ( . _ ~ b ) = [ I - H B ' ( B H B ' ) - I B ] x -

- H B ' ( B H B ' ) - ~ b , and C 4 = H - HB' (BHB' ) -1 BH. The matrices C1, C 2, C a and C4 for Models 4 and 5 are determined in the same

way. As regards Model 4,

and for Model 5

! , , . o \ -

(--BV~--6.--iS-/ o, Bi o , o /

( t C1 C2

C3 [ - C 4 \ /

/ . , o i A ] - = I t ° ' ° i B ! .... 1 . . . . . .

Since the value of ~2 is usually unknown, it is subject to an unbiased estimate in the universal model (cf. [9] or [7]),

(5.9) ~2 = v'H-v[R(H, A) - R(A)] -1 ,

where v = A~ - x (the vector 0 is from (5.4) or (5.7)). For the regular case within the scope of the individual models the following holds (7):

Model 1: R(H, i) - R(i) = n - 1 (number of surplus observations);

Model 2: R(H, A) - R(A) = n - s (number of surplus observations);

M°delB: R[H' O' I ) - R ( I B ) = n + q - n = O,

Model4: R ( H , O , I O , O, B1, O ) - R ( I , B z gl , O ) = n + q - ( n + r ) = q - r ; B 2

Model5: R(H, O, O,

In [9] the following relation is given (proof thereof is in [7]):

(5.10) R(H, A) - R(A) = Tr(HC1),

(where Tr(HCl) is the trace, i.e. the sum of the diagonal elements of the matrix He1) , which makes the calculation of the values of R(H, A) - R(A) considerably simpler if in adjusting we use the matrices Ci, C 2, C 3 and C 4. It also holds that ([9, 7])

( 5 . 1 1 ) v ' H - v = x'Clx,

which can serve as a check.

110 Stadia geoph, et geod. 20 [1976]


Equations (5.6), (5.7), (5.8), (5.10) and (5.11) are used for adjusting within the scope of the universal model if the method of matrices Ca, I22, C 3 and C+ is employed. Otherwise Eqs (5.2), (5.3) and (5.9) are used• Any of the live situations can be solved by a single universal computational algorithm.

At this point it should be emphasized that the functions of the parameters, the adjusted values of which are to be determined according to the relations given above, must be estimable (i.e. the system A'L -- p must be consistent for a given design matrix A and vector p). The case of the unestimable functions of the parameter is treated further on.

7. U N B I A S E D U N E S T I M A B L E F U N C T I O N S O F T H E P A R A M E T E R

I f the function of the parameter f(O) = p;O does not guarantee the consistency of the system A'L = p for p = P0, it is impossible to determine the function of observed values f (x) = L'x for which E(L'~) = p'o 0 for all values of 0. In this case it is natural to try and select the vector L so that the difference L'AO - p '0 is minimum in absolute value. However, since this difference depends on 0, which is unknown, the minimum condition may be substituted by various other conditions impossed on vector L (for details see (7)). For the sake of simplicity we shall only discuss one of the possible conditions.

The quantity b(O, L) = L'AO - p'O = (A'L - p)'O at a given value of L will vary with the direction and magnitude of the vector 0. Assuming the magnitude of vector 0 to be constant, b(O, L) will be maximum in absolute value when the direction of vector 0 will coincide with that of vector A'L - p. If we minimize the vector A'L - p by adopting a suitable value of the vector L, i.e. the quantity [(A'L - p ) ' . • (A'L - p)]1/2 = IIA, L _ PII,, we shall obtain the minimum of the maxima b(O, L). This estimate o f f ( x ) = L'x is called minimax.

It can be proved (8) that this particular L is defined as

(6.1) L = (A')7(,) p ,

where ! is an identical matrix.

Since the dispersion of the deviated estimate f (x) = L'x = p'(A') /( , )x of the unbiased unestimable function f(O) = p'O is az(f(~)) = a2L'btL = aZItLI} 2, a special matrix (A')/(w) must be adopted to minimize this dispersion, in this case with the

' - = (A '~+ is (A '~+ It can be proved that the matrix ~ )t,H help of the matrix (A)z(,),,,(n) ~ mH" a special case of the matrix (A')~,(H). Therefore, if in the case of estimable functions of the parameter we adopt the matrix (A')/(,),,,(H) instead of the matrix (A')m(m (e.g., from Eq. (5.2) we arrive at the adjustment of the estimable functions of the parameter, as well as at the minimax estimates of the unbiased unestimable functions of the parameter. The adjustment of the estimable functions of the parameter will be

Studia geoph, et geod. 20 [1976] 1 | 1

L. Kubdt?ek

identical with the original adjustment according to (5.2) and (2.1) in the case of regular models. In [8] it is proved that the matrix

(6.2) ,{Dq+,,.v : (V + DD')- D[DD'(V + DD')- D]- D

can be used instead of the matrix (D q+ \ ) l ,V"

Assume that the matrix V, the matrix D, the vectors t /and y have the follownig meaning in the individual models:

(6.3) Model l : V = H , D = i , g = ~, y = X ;

Model 2: V = H, D = A , r/ = ~, y = x ;

Model3: V = ( M , O ) , D = ( / ) , q = ( ~ ) , y = ( ; ) ; O 7 - - - -

(Oo) (o ) (:) Model 4: V = H, D = !, t/ = , y = , O , B z B 2 - -

Model5: V = ( H , O ) , D : ( B A ) , q = ( b~), y : ( b )" O 7 - - - -

As regards the estimate of any function of the parameter f(O) = p'O it then holds that

t + (6.4) f(×) = p'[(D')~v]' y , a2(f(~)) = cr 2 . p'[(D')~v]' V(D )l,v P ,

D/D q+ = v ' V - v [ R ( V , D ) - R(D)] -z v = ~ j , , v Y - Y , a'z

Equations (6.4), (6.3) and (6.2) cover all the situations in all the five models, whether regular or singular, whether cases of unbiased estimable or cases of unbiased unestimable functions of the parameter.

If the computer contains a procedure for determining the g-inversion of a given matrix, we are able to determine in turn (V + DD'), [DD'(V + DD')- D]- and, finally, (6.2) and V-. The inputs into the computer, i.e. the matrices V and D, and y are determined by solving the problem according to (6.3). The computation using (6.4) then becomes elementary.

8. CONCLUSION

The purpose of the paper was to point out some of the results of investigating the methods of the calculus of observations, which may be useful in unifying and mechanizing the adjusting procedures for all types of situations occurring in practice. In general the solution is defined by Eqs (6.2), (6.3) and (6.4). If we wish to limit the adjustment to estimable functions of the parameter, it is sufficient to use Eqs (5.6) to (5.11).

Received 1.8. 1975 Reviewer: F. Charamza

][12 Studia g e o p h , et geod. 20 [1978]

Universal Model for Adjustin9 Observed Values

References

[1] A. B j e r h a m m a r : Theory of Errors and Generalized Matrix Inverses. Elsevier, Amsterdam 1973.

[2] R. C. B o s e: The Fundamental Theorem of Linear Estimation. Proc. 31 st. Indian Sci. Congress 1944, 2 (Abstract).

[3] L. Kub~ieek: On the General Problem of Adjustment of Measured Values. Matem. eas., 19 (1969), 270.

[4] L. K u b f i e e k , A. P~izman: Guide to Statistics in Measurement. Acta metronomica, I.)TM SAV, 8 (1972), zogit 2, 1.

[51 L. K u b f i e e k : Odhad nepriamo meran~ch parametrov so syst6mom podmienok. Zbornlk vedec, v:?sk, pr~ic 1)TM SAV (venovan6 70. v~roeiu narodenin akad. E. Kneppu), Brati- slava 1973.

[6] L. K u b f i e e k , S. ~;ujan: Z~.kladn6 poznatky gtatistickej te6rie odhadu (skriptum). Vydav. Univ. Komensk6ho, Bratislava 1975.

[7] L. K u b R e e k , G. W i m m e r , J. V o l a u f o w i : Matematick6 met6dy te6rie odhadov v metro- nomike. Z/tver. spr. eiastkovej filohy SPZV III-7-1/1, I )TM SAV, Bratislava 1975 (not published).

[8] C. R. R a o , S. K. M i t r a : Generalized Inverse of Matrices and Its Applications. J. Wiley, New York 1971.

[9] C. R. R a o : Unified Theory of Linear Estimation. Sankhya, 33 (197I), 371.

Studia geoph, et geod. 20 [19761 113

universal model for adjusting observed values

Documents