PRIMARY ENERGY SUBSTITUTION MODELS*

On The Interaction Between

Energy And Society

c. Marchetti

June 1975 WP-75-88

Working Papers are not intended fordistribution outside of IIASA, andare solely for discussion and infor­mation purposes. The views expressedare those of the author, and do notnecessarily reflect those of IIASA.

PRIMARY ENERGY SUBSITUTION MODELS *

On the Interaction between Energy

and Society

C.Marchetti

Pre f ace

This paper describes an attempt to develop a "synthetic"

mudel of primary energy substitution, using certain rules

which prov8d fruitful in describing the sUbstitution of other

commodities.

This model will be used for forecasting, and for checking

the validity of certain objectives set for R&D in the field of

energy.

*From a lecture, delivered in Moscow, November 1974.

- 2 -

Trends in energy oemand

The first point in forecasting energy demand is obviously

to look at historical trends, over a century at least, and

try to extract the signal out of the white noise and various

medium-scale perturbations that occur along the way. Although

the long-term extrapolation of these trends may require a

more subtle analysis of social and economic trends, they are

good to be kept in mind.

The ones reported in Fig. 1 and Fig. 2 have something special.

They include wood and farm waste. This is necessary to get

a proper basis for extrapolation because part of the growth

of commercial energy sources is due to sUbstitution of wood

and farm waste.

As you see, apart from the big dip, coinciding with the great

recession, "healed" then by 'dorld War II and some "overheating"

coinciding with world war I and preceding the 1930's recession,

the 2% secular trend is followed quite tightly for the world,

even taking into account the compression due to the log display.

In the case of the U.S. we also have a well defined trend with

the bumps in somehOvl different positions. The higher value of

3% does not appear particularly significant as the U.S. popula­

tion has grown roughly 1% faster than the rest of the world

in the period considered (1860-1960).

The second point is to look inside the envelope of total energy

demand for trends in primary fuels demand. I did this exercise

at IIASA, using a methodology completely different from the

"modelling" which is so popular in many places of the world,

and whose contradictory results, when used to forecast over

long ranges, cast many doubts on its reliability.

- 3 -

I started from the somehow iconoclastic hypothesis that the

di.fferent primary energy sources are commodities competing

for a market, like different brands of soap or different pro­

cesses to make steel, so that the rules of the game may after

all be the same. These rules are best described by Fischer

and Pry (1)(2), and can be resumed in saying that the frac­

tional rate at which a new commodity penetrates a ma~ket is

proportional to the fraction of the market not yet covered:

1)

or:

1

F

dFdt

= ex (i-F)

2) In (F/1-F) = at + C

where: F = fraction of market penetrated

ex and C constants, characteristic of the particular

commodity and market.

In Figs. 3, 4 and 5 some cases of market penetration are

reported, showing the extraordinary precision by which those

curves fit the statistical data (which often are not very

precise). All of them refer to a competition between two

products. In the case of energy we have three or four energy

sources competing most of the time so I had to extend a little

the treatment with the extra rule that one of the fractions is

defined as the difference to 1 of the sum of the others. This

fraction follows approximately an equation of type (2) most of

the time, but not always. It finally shows saturation and

change in coefficients.

The fraction dealt with in this way corresponds to the oldest

of the growing ones. The rule can be expressed in the form:

First in - first out.

- 4 -

Fig. 6 shows the plotting of statistical data for the U.S.

in the form In (F/l-F) vs. time.

More than a CGlitury o~ data can be fitted in an almost perfect

way using only two constants, which come out to be twa

dates, for each of the primary energy sources (wood, coal,

oil, gas). The whole destiny of an energy source seems to

be completely predetermined in the first childhood.

As we can see by analyzing the curves and the statistical

data in greater detail, these trends - if we can call them

that way - go unscathed through wars, wild oscillations in

energy prices and depressions. Final total availability of

the primary reserves also seems to have no effect on the rate

of substitution. The only real departures from the curves

are due to strikes In the coal jndustry, but the previous trend

is rapidly resumed and the effects of the strike s9mehow "heal­

ed". On the point of availability it seems that the market

regularly moved away from a certain primary energy source~

long before it was exhausted, at least at world level. The

extrapolation of these trends indicates that the same thing

is likely to happen in the future, e.g. that oil reserves will

never be exhausted because of the timely introduction of other

energy sources.

When I started showing around those curves, people said they

were fascinated, then that the fit was too good to be true,

then that one should find the explanation before accepting

and using them. Nothing to say about the first two points

but the third one is in principle unacceptable: laws work or

don't work, and the only reason to accept a rule as a law is

because all sorts of tests applied to it show that it works.

What most model makers do, starting from elementary relations

and by functional and progressive aggregations going to macro-

- 5 -

scopic variables (e.g. demand) is very similar to what is

done in statistical mechanics in order to "induce" e.g.

thermodynamic laws from mechanistic principles. But thermo­

dynamics is compl~tely autonomous from the interpretation,

in the sense that its "truth" is internal to the set of

macroscopic measurements from which it has been derived.

Now, putting philosophy aside, I played the game of fore­

casting (i.e. of backcasting) within the historical period.

E.g., I took the data for the U.S. from 1930 to 1935 and tried

to forecast oil coverage of the U.S. market up to 1970. As

Fig. 7 shows, the predicted values even for the saturation

period fit the statistical data better than 1%~ which after

all is the minimum error that can be expected froIn this kind

of statistics. This means that the contribution of oil to

the U.S. energy budget, e.g. in 1965, was completely predeter­

mined 30 years before, with the only assumption that a new

primary source of energy (e.g. nuclear) was not going to play

a major role in the meantime. As the history of substitutions

shows, however, the time a new source takes to make some in­

road in the market is very long indeed, about a hundred years

to become dominant starting from scratches, so that also this

assumption appears really unimportant for predictions up to

50 years ahead.

As our game worked so well in the last hundred years, why not

make a try for the next hundred years, just to see what happens?

The results are shown in Figs. 8, 9 and 10, and some quite im­

portant consequences can be drawn from them.

The first one is that substitution has a certain internal

dynamics largely independent from external factors like final

reserves of a certain primary energy source. So the coal share

of the market has star-ted decreasing in the U. S. around World

War I in spite of the fact·that coal reserves were in a sense

infinite.

- 6 -

The second is that substitution proceeds at a very slow pace,

let us say of the order of a hundred years to go from 1% to

50%. The "acceleration of the times" which we all perceive

does not show up in the statistics. Perhaps the increasing

number of changes is giving us that sense of ac:celeration

even if the rate of each individual change stays constant and

low.

This fact rules out the possibility of having fusion or solar

energy covering a sizable fraction of the energy market before

the year 2050 and leaves us with the narrow cholce: go nuclear

or bust. A resurgence of coal appears improbable too, and I

found very nasty reactions on that point from everybody except

from coal people who appeared in a sense relieved from a mission

well above their forces.

The problem, however, of how to consider a SNG plant, a coal

consumer or a primary energy producer, as in fact it is seen

from the market, is still an open question. This leaves some

ambiguity in the interpretation of the curves in the case of

important intertransformation of fuels.

Figs. 11 and 12 show the same curves for the world, with dif­

ferent hypotheses concerning the timing and rate of introduc­

tion of new sources of primary energy.

These curves relate to fractions. To get absolute values, one

has to mUltiply them by the total level of energy consumption.

Fig. 13 gives the result for the world, using a 2% secular rate

of growth, Fig. 14 for the U.S., using two different rates of

growth. In both cases, the amount consumed in 1970 is taken

as a unity.

In case of scarcity it appears that energy saving is much mOre

efficient than substitution.

Phasing out of a source does not necessarily mean reduced pro­

duction in absolute term~.

- 7 -

The following step is to inte~rate this consumption allover

the cycle of a certain primary fuel and compare it with the

re50urces. I did this exercise and found that after all the

world will not be short of oil, if nuclear energy will keep

the present rate of penetration and perhaps even if not, but

that there m~y be problems with natural gas. As everybody

has his own figures for the reserves, I prefer not to raise

a row on this point and leave it to you to make comparisons

and draw COllclusions. After all, the scope of this presenta­

tion is essentially methodological.

Productivity VB. energy

People in the world rightfully try to improve their lot, and

the numeri~al indicator for this is GNP. So the linkage bet­

ween GNP and energy consumption, and the possibility of making

this linkage looser as it nppears now, are of the utmost im­

portance both jn order to better understand and plan the work­

ing of our society and perhaps have a guess on the evolutionary

trends.

Although I will not be able to draw final conclusions, I hope

the next figures will show you that there is much purpose in

the research and tho linkage is not as rigid and indissoluble

as much of the pertinent literature tends to indicate.

History as usual is a good mine for digging and I will start

giving a little hint. Fig. 15 shows Europe in 1890, a very

homogeneous system for technique, cultural and societal organi­

zation. But strangely, Gt~P vs. energy consumption organizes

over two different lines. In the first one you have the nations

who don't have coal mines, in the second the ones who have them.

For the same GNP, the ratio of energy consumption between the

two groups of nations is 4!! Large differences appear also if

you compare widely different systems as Pakistan and Sweden or

nations at different times.

Apart from energy, the other inputs to a productivity function

are raw materials, know-how, capital and societal organization,

and one may expect a certain degree of substitutability between

them. The most convincing analysis in that sense has been made

by H. Millendo~fer and C. Gaspari (3) and I report here some of

the results.

-8-

One of the most obvious indicators of the level of know-how

is literacy and in fact the correlations between GNP and

literacy work well, as shown in Fig. 16.

The very interesting point is, however, that the nations of

the world, bunched into a certain number of parallel lines

essentially five in all, indicating another factor at work

which we may call "efficiency parameter" or "societal effi­

ciency". The different groups are geographically identified

in the following Fig. 17. Societal efficiency seems to cor­

relate strongly with religion.

Inside each of the groups, the productivity function becomes:

y = C m1;'; eb F + O. 8qz s

where

y the GNP per capita in U.S. dollars

C z the zonal constant, or societal efficiency

m the indicator for the material input (per capita

electricity consumption)

b the indicator for the immaterial input (literacy,

or engineers/l0,OOO popul., when this indicator is

saturated)

q mineral resources, expressed in per capita value of

production

Fs is a "stress function" indicating the non-complete

substitutability of the material and immaterial in-X bputs. F s = 1 for m 4 = e and bends somehow the iso-

quanten as it appears in Fig. 18. It is fitted through

one parameter only.

Y4 b I-liP1 (~) - p. + 1. (~) - P"2 eb 2 m /4

p fitted by regression

-9-

The results of the calculations are given in the following

table:

Table 1

calc. obs. calc. obs.

Canada 25~0 2380 Great Britain 1830 1700

Australia 1970 19'10 Switzerland 2150 2310

Belgium 1770 1740 U.S.A. 3870 3670De.nm3.rk 1850 1950 Sweden 2230 2500France 1780 1950 Holland (2250) 1520

W.Germany 1760 1 r(50J

for 1969 - in U.S. dollars per capita

The only real departure is for Holland. One interpretation

being that it really belongs to the "Catholic" group, i.e.

to the second one, with a lower societal efficiency.

The results are graphically displayed in Fig. 18 where it

appears very clearly how different nations have organized

themselves, and how high GNP with low material input, e.g.

energy can be obtained via a high level of engineers, l.e.

of know-how.

It is unfortunate for Japan to have such a low level of so­

cietal efficiency, revealing perhaps the difficulty of adapt­

ing its society to an economic system developed by a protes­

tant society.

One might, in abstract, speculate on the consequences of

trying to adapt western technology to the Japanese society,

the reverse of the option taken a century ago.

-10-

Con c 1 u s ion.:

A new approach in the analysis of the internal dynamics

of primary energy substfrution and of energy use is attempted.

The results are very encouraging and promise a deeper

insight into the subtle links between energy use and society

operation.

-11-

Bibliography

1. J.e. Fisher, R.H.Pry

A simple sUbstitution model of technological change

G.E.Report 70-C-215 (June 1970)

2. R.H.Pry

Forecasting the diffusion of technology

G.E.Report 73-CRD-220 (July 1973)

3. H.Millendorfer, C.Gaspari

Imrnaterielle und materielle Faktoren der Entwicklung

Zeitschrift fur Nationalokonomie 31 (1971), 81-120

6

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economy. Japan appears to be the first to use in­

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of the world into four layers. A fifth one 5.s n~t

included bcc:..:.use tile in;j.icCt~0i.".1;';

indicator in this case is percentage of engineers in

the population.

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Fip-;.lB - Iso-GNP as a function of the two indicators for

material and immaterial inruts. Dash~d lin~ indi-

t · 1 . 1 .. b D d"cates helr ba ancc, l.e. m = e otta _1ne has

been dra\'m for F = 1 and shows the effect of incoi;:­splete substitutability of the production factors.

It is very intere::;iinc to note that tho US and 0:·:GC..:::1

have rouL:hly the sa:nc material index, and the Inuch

higher GNP per capita of the US appear~ to be due

essentially to a hiC;hcr immats:rial production fac:tc\~·.

Appendix

Methods of Calculation

N. Nakicennvic

We will describe here the first attemp~ to implement ~he

idcCls presented in the paper .for determining the market pene-

tration behavior of energy sources on the basis of the historical

data.

First, by definition, the sum of all fractional market shares

must be always equal to 1:

'1

I: F.(t) = 1i=l 1

(1)

where Fi(t) is the fraction of market penetration of the source i

and n is the numb~r of energy sources. Expression (1) must hold

for all t.

As shown in the paper, the logistic functions appear to

give a good description of the market share of a given product.

These logistic functions can be written as:

[

F.(t)"jIn I-F:(t) =

.:) 0(l.t + c.

1 1 (2)

The superscript 0 for parameters (li and c i will indicate that those

parameters are defined on the basis of historical data.

-2·-

(1) can be rewritten:

oexp(a..t +1

l+exp (a.?~1

oc. )1

o+ c.)1

(3 )

Our intention is to project the market penetration trends

of the primary energy sources over the time interval longer than

the time period for which data are available.

This projection in the future leads to situations where one

or more energy sources are penetrating the murket at a higher

rate than other energy sources leaving the market. Thus,

in such situations expressions (1) and (3) would be contra-

dietary. The expression (1) is a statement by definition and

cannot be violated. Therefore, the logistic function in

expression (3) cannot hold for the whole time interval in] Ii

question and for all energy sources. In fact, it will hold

universally only for those energy sources which are leaving

the market from the beginning.

The methoJ used is that the oldest still growing energy

source must decrease, i.e. the expression (3) would not hold for

that, oldest, energy source from that time point on, and will

be equal to the difference between 1 and the sum of all other

energy sources. By oldest we mean the energy source which is

anterior to all other energy 30urces. This chosen energy source

then enters a transition period after which the market share starts

decreasing. vlhen this decrease starts, the originally second oldest

still growing energy source Dust be labeled "oldest".

-3-

We assume that there are n energy sources competing on the

market so that any energy is denoted by ig[l,nl ~ 1 • We want

to consider a time period longer than the historical period for

which data are avilable:

At the beginning we choose, as described above, the oldest

still growing energy source j by:

j={ iii = [l,n] s/~, t=tINI , a~ ~ 0, u~_l< O} (q)

As already described above, for all other energy sources

that are leaving the market, i.e. a~ < 0, expression (3) always

holds. For the oldest still growing energy source j, we define

the market penetration by:

1**. exp(ajt + Cj)

t...l+eXP(a~t+c:)J J

for t INI

for t 0

bJ

for t 0

eJ

< t < t bi (ascending phase)

< t < t ej (transition phase) ~5)

< t < t pIN (descending phase)

In the following we will explain the quantities introduced

by (5). First we define tbo and t 0 as these are instrumentalJ eJ

in defining the transition phase:

-4-

t. J'D

o 0exp(o:.t+ci---~ ..

o 0l+exp(o:.t+c.

J J

+ 2:: F. (t»l} (6)• -I' 1l],J -

and:

t .={t!t:s[tb, ,t"'L.•"JeJ J' i'

,.,~,~ +~~ :: ..

c or (7 )'

wherr~ c is a constu.n i:. •

t}. is, therefore, Lhe beginnin':1 of the transition pel.-iod for)J

ener~y source j. It denotes the time point where the rate of

market penetration of energy source j starts to fall. t . iseJ

the (~lld of this t:ra.nsi tion period, when the ma:;:-ket penetration

of energy source j follows the logi.stic function again (see

Fig.l) .

(8)

*and (' by:j

(~t

*Finally we define a.J

F j (t)

d [ 1n (I~:-f.':-nT) J- J*a.. =

J

anu:

*- a.tJ

at time pojnt t = t .eJ

(9 )

Beyond t . the energy source market penetration follows then theel

*logistic function 1n the descending mode, i.e. a j < O.

AJ 1 of th~~ quanti ties on the right hand side of expression (5)

are no\-] defined, and there fore also the raark€t penetration

Dehavior of the energy source j.

-5 ...

The energy source j thus defined, however, is only the

first energy source that must leave the market due to defini-

tion in expression (1). As the time goes on we might still

encounter the problem that we must reduce the current, oldest

still growing energy source. Accordingly, we go through the

process source by source. Generally, we therefore have:

where:

r 0 if 0 < 0 for tnn~t~tFIN. a. 0'. .( J. J

(t) = ! 0 if 0 > 0 for tINI~t<tbj0.. 0'.. 0'..J l J. J

I

! ...if 0

~ <t<ta; a· 0 for t\ J J ej - - FIN

and similarly:

0 if 0 < 0 for tINI:s.t'::tpIN! c j OJ

(t)= ) 0 if 0 > 0 for tINI:s.t<tbjc. c. 0'.. -JJ

J J

I...

0'.'I c· if 0'.. > 0 for t <t<t\.. J J ej - - FIN

(10)

(11 )

Finally expression (5) becomes:

for t < t . or t > t .bJ eJ

(5 *)

·t < t .eJ

-6~

To summarize, it follows from the relations presented

above that a. (t) and c. (t) are piecewise defined for t.b.ose1 1

energy source which penetrate the market and later must

leave it. They are' not defined over the time interval [tbif teil.

During this interval equ.ation (1) is .employed b..., go\;e7~'J.ll:he be-

havior of the energy source in transition. Those energy sources

which have the descending market penetration from the beginning

of the time period ana.lyzed, have a.(t) and c.(t) defined over1 J_

the whole time period. The definition for this is: ai(tINI ) =

oa. < o.1

We developed a computer code for expressions (~) through

(11) • The initial values of parameters aC? and. c? v?"ere determined1 J.

on the basis of historical data. The program Q then. generated

the values of F i (t) for any chosen interval of time bet,,7een

t > 0 and t < 00

In order to express consumption of different energy energy

sources in absolute terms, a base year t = 1970 is given as inputo

and a constant growth rate (r) for total energy consumption is

given. The consumption in the base year for each SOl.u:ce has

been taken as unity for this source.

The source code for this program has been written in

Fortran language. It has been implemented on the PDPll/45 with

DOS version VO'] opera·ting sys·tern and later ~1i t.h max operating

system. The graphical software used in the program is a version

of Calcomp plotting library specifically modified for PDPll/45.

Attached is the flow chart of this program.

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---------..- ~cA"O (5, (, 0) --,:fA x. _ _ . READ (5 ~ 60) _HIt~ _

READ (5,60) IREFY______ ___ _ __ Rf:: ~ D ( 5 p 5 I2J ) _R ~ .

RE:.AD (5,50) RGiRi:: AD (5 I 5 ~n ~~_~OB

READ (5,(,0) NP' -_____________. RI:. AD (5 I %i , CLI) L E ( I 1J~_LL.~10~) _

DO 20 I~l~N

RE. A, D ( S , , :3 0 , (!. Y.p ~_q_, Jl1..J_~1.!c...4.:..!)~ __CON-UNUEWRITl;: (b,~~~,

-.-.--------.- -WR I TE (b, 1-( U) NP\<IR 1 TE (b, lHU _DO 1 1~1,N

P.(:.AD (5,Uj~) l{1, Flwin '(I: (b, 1(£0) :c i~- Ff----- ---.------ ---- .-.---- -- .--------- -------------.---- -READ (5,100) X2, F2

-- -.. __..._._-~---_. '~~r~lrE (b,14~) ·X2'-F2"

Yl::AL(JG(r:l)Y2:ALOG CF 2) --- -- . - -- --_._--

CAL:.. FIJNt:CY1,Y2,Xl,X2,A(I),BCI»CONTINU~ - - .' ---- ,. _... - .. - ..------

WRITE (0,40)FO~f~AT (urjFORMAT (FS.3)

.. _-- _.__ ..- -_.._---_._- -- - ..•... -FORMAT (14)F() R'-1 AT ( Ee lJ 1)FOrlM~'f (- ~~LOG(F/(l";F»=AI!rT"C·)'---- -- --FOR~1AT (10A2)FOR tU T (F 5 • ~ , F5-.3 )--­FORMAT ('fP)FORMAT (4A2,' TOTAL AMUUNT CONSUMED :~,E14.1)

FO~r1AT (lU2)FORMAT C1H ,3(10 X,E14.7)FORMAl C10X,'CONSUMED BEfORE ',14,' ~',E14.7)

FO~MAT (10~,'CONSUMtD AFTER ',14,' =·,E14.7)FORMAT (l~~,'PLOT # ',12)FO~MA' (lU~,'TOTAL CONSUMPTION IN ',14,' ISFORMAT (le~,PG~OWTH RATE (RG) BEFORE ',14,'FORMAT (l~X,iG~OWTH RA1E (RG) AFTER ',14,'FORMAT (lH ,1~~,F6.0,6(10x,Fb.4»

FOR~AT (' R=F*(l+RG)~*(',I~,'-T)')

FQR~AT (lH ,I4,10X,2(10X,E14.7»XMAX=FLOAT(MAX) --lCHIN=FLDAT U1IN)

- --- ---)( 1 "1 I N=)( H HJ ~. 5 0 •X3 "1 I N=j( MI tJ .. 7 " !II

M1=(r~ Ax- MIN) 11·0 ------VMAX=.tt:.+3YHI N=• :i. E&> 1 -- - - -- -- -----------

YMAY,L:ALOG(YMA~)

VMlhIL=ALOGCYMIN)'-- ----- ----,--.---

YL=ALOGC.S*YMIN)_ .._----------- ._-_._-,- -_.- - _._- - .. - .. -.---...- - .._. --~--

XM="lRG211:~G

_____L ..._, _0_•••• _-.• _. __~_.~,_.__ •• _

----------- .-.

.-----------_._--

._------------_.__ . --

---'--_.- '-- _. -_ ....- .._---------..._- ._,

-----_. ,. __.- ._. --_._-_._----_._._._-_.-

--~.- ._._-----_.-

12

8

T

2S3"

3

CON'fINLJECALL Pi L.;OCAL L f" L 0 '1' ( 1 • 3 , p J .l )

CAL L SL;: '._ F ( 1 • , , 0 , ~ ~ , 0 • )CALL f-I't.OT (1.- 3. pD. j----- ---.-.---.-------.--- ...

CllL L. Yl.uGt. C" MIN, Hi AX, .01, /1,5., fa., XMIN, .02, SCy_t . _CALL fGRlG(C/XHIN,YMI~L,10.,M1) I08 LI I:"11·iJ.;r,,,~;:i'.~u. ._._____ . .. . ._~ i :: ~ LUll Y \ £ ) " i ::., "C:\ L L fo L ~: .: [( l XI , '( L, • i 2. , • 15 , 0 • )W~ 1TEl. / ~ b D) - 1 - -....-- _.------- ..:....:;...-----

CUN I' J. :'~utCI~L.L. F C,1 "r( CO'~-~ tJ, vr-n NL-; ;i2-~ f5~-0 D)~-- --------.-------.

l~ In: y~: :. l v 1 v)) .V!! t~J. __ _ _.. ._ _ . .Dr. :.: 'r' ~1 .:: i J()CJ I :i::_ i~.

1l- ~ lJ l:: .-, .~ , j •

8l' I. u: /' L. f) b (/j i~ :C;'Il.,l. i- C,I,CI:~ (;{ jll 1'1 ~--BEi:; .-12 ;~-1 !:r,-0~-)--- _.-...--- -._-- - -----------.. --tJR!iE li, lU.1 lll~ . .__C0 I~ Y! iJ lJ;-':y 1 T 1 :: t.:. UI, (',' ,1£\)~ ,~j" )

t uL L r: (.1" ~.ii l i j t1'iN~· 1"1 "(T;-~2, ~T5-;0.)IF (:lt1<,GY"U.) GlJ TO 30wid 'fl: i i .. U~n· -- ----------(, 0 '\ () ~; ~\

C;Ui~'{ 1 !WEWRX',L:: ~l,2dq I1~EFVCON)'li\IUt: .... ------..----.------ ----------------

YF O{11"G( .. U\J) GO 'f0 ]'1_._--- '-- ._-_._-- ------- -_.-VHi Gd Vi1 ~ HD0 ,_: '3 :,: ,; ., .. " ,I'(I~:i b ~ yil :1. G::·, iJCJ- .-- - -----.------..-------- --- --- -....--. . -...,.. --- ._---. ---.---.-

VI' rGL :; J\ L Ll G( VI! :a: G)CALL FPLor cL iUHN,"VHtG-~-------·---·'_.'--.-C".l L H'L ur (2 t )( MA)( , VHI GL )

- - -------- - _.- - . --- ----caton !l'IUct.:ON'('i(IJU~

ell LL Fpun ( 1, )( MA~ , VHI Gr-'-- - --.-. --._..T1 r d ;; II L UC ( Y;1.1 ~ ~ 1-.:> • )CALL Fe H Aft ()( f1 I rJ ,'( I TH, -';T2-, -~ 15;0;r- - -~. -----.- .- --WRITE Cl,90) (TITLE(Kl),Kl~1,4)H l:: 1-1 .. G'r w(1,,) GO T0 ~ 3 - -- -_.- -_.-.. _. - --- -- .- --...---. .-.-----------.-.-

'11:.; 0 ~

F~UL) ..J.;l

---,------- - SL OPE;; 1D,); .- --------

DO 2 i ;';ILIJ vI1A)(51111o:0, -;c :: FLO !\ ',' ( nII=i u i1:iiJ'dXF (.\ (J) <. ,.'( " (J t)' J lJ J .;. 1

.. --._--_._------_._.. -_.-.~--_... -~- -----_._-----~~--

DO :5 i!..:; 1 , iJy(t1 ) ;: A (/") (: x.... 0 (~1)

F ( t1) .: ' 'J" l ! • I 11 D +E l( P ( v(M) ) ) ,- --- -- -,- . ----. -.S LJ r i -: ~HI !·1 .; r (: j JCONTINUE - - -- - -------,-------------- ---- ----

IF CSU;-1.EQ~l.) GO TO b~ (J) ~ 1 " ~ sU f1 ... f ( J )- - ------------ -------- -_.- ....--.- ._- - -.-----'

If (F(~).LT.1.E":"~)_ G_O._~~ 4 . . ._

......-.. ' -- - _._-- ----_.._.......-_._. - ...._....._._.._. "-'" ..._---_.' - ...-_., .----

.__ .._----

--_ .._-----

------_._---

.._-_..__._-------

-'---- ..- ------.-

--------_.- ..__ ._-_....._- '.'-"'---'--"----.

------ _... --- _. -- ._.. - .. _._- ------- ---"

---------_.._----

----------- -----------------

14

27

21

22

11

10

-- --- '·1 J

. ABCJ)::AREA8/REF(J)~""'" -.--_.--_........ _... ­

AA(J):APEAA/REFeJ)IF (TL (J, MA x.. r~I~J+ 1) ·;iE. 0'. )-tLeJ, MAX-MIN+1) =YtlIN "---'-"'-T~(J;MAX-MIN+l):ALDGeTL(J,MAX.MIN+l)." CONTINUE. _.. _- ..0 --" -.~---' .".- ---- •• - .•• - •••,.-=---- .-..--.---.-------.--.-

WR I TE (6, Q0).. _DO 10 K=l,NCAL L r PLOT (... 2 , X~ IN ,~_(~_,JJJ. ~' . .._.... ...__K5~50 .DOl l I;: /11 ~ , M~XII=IDMIN+lIF (11. EQ • ~ 5) .. G.O l~_ll _'_ _'_.GO TO 22CONT1NUE ----_. __._-- ---------'--1<5:.:K5·~SO

tONTINU£,.---.." --_.- - X:FLOAT(I-)

CAL ~ FPLOT C0 , )( , (j (It, I 1) )CONTINUE - .-_.-._.---- --.----~=------------._--- .-------.-------

CALL fPLDT(l,X~AX,G(K,II»)C[) N T!til Ur: --.' --- .- -. -~ t·ll N 1 :: ~ f11. IJ -' 2 U"

-·----0---.--- -_..- CALL FPLOT (1 ~ )CH',,-X; YKrNL)CALL YLOGA(V~I~,YMAX,.01,4,5,,0.,XMIN,.02,SCY)

.. ----- ... ---- -.. ell L L FCtiA R (It MI til 1, VMI NL:-~.12;· .. 15';-0-;)---'· .

Ft1 1N:: V:-tI N/ ( 1 ..... Vt\ IN'·-·--------· ..wkITI: (7.50) FMt~

DO 17 I ~ 1 ,9,2_..- --.-----. --- vVA:: FLO AT 0 )/Hf..-·

Y~A~ALOG(YVA/C1.~YVA)

CAL L F CHARC XMIN f; y[1--;-;-1'2-;-;15·, 0 • )I-mITE U,50) YVA

.. ·..--·---11-·-----· 'CONTItJUE--' -'------------~------._----

CALl.. FCHAR (XHIN1, YHAXL, .12, .1::..;5=...:,~0=...;.:..:)~ _FM h X=yMAl( I (1 ~+VMAXl' ---WRITE (7,50) FM6X

..- --' - CALL f CHAR CXMINi;Tlrr;-;12;-;1~-0·-.~)~ ----.-------------l~RITE 11,110)It M0 =)( M1N+ 150 =...:~-------------

CALL FPLOTC1,XMAX,YMINL)-------.---".- XM="l-~---

CAL~ SCALFC1.,t.,0.,0.). -..--- ... -' '-C ALL F PL 0 T ( 1 , r~' ,-~:r;) .-=...::..--------

GO TO it?. CON'(rr.ru~-00 14 1(~t,N

---.----.--. CALL' FPCOTT;;;2,XMlN;Il:'{l{IT)1<5=50

-_.-_.~- - _.~._-"."." DO 15 I:aMIN,"M"Ar

I'i::I-MIt~+i~. ....-.. -.. --.....-. rF ( I I .. E. I). K5 r-G<f-T O--~"'3

GO TO 24··· ..-·---·23...------ CONT INOE------·------------- -------

K5::1<':).;.S0CONTiNUElC=FLOAT(I)

- - ..- ..---- CALL FPLOT e0,x~TL-(tr;Tnl

15 CONTINUE- ----.-- ...- - CALL FPLOTlf~··)(MA-X,TL-(l<__;lnT·-----·--------------::

CONTINUE

-

--

C

I

!

II

I

I

II,

---------

-----~-----_.----------------

--- ----- _._ .._---_.

- ._.._---_.----------- -- ------ --_ .._---- ------

5

3&

&

28

. - -9 ­

2

r--"- -.... --------.. -

~._ .. .-:.. ~.,_~ " __ ...~. __......... __ ..__. .. . __ .__ . _. __ .~ _ ...- -..:. ._._,",,----L.

Y( J ) :: AUil; (F (J ) I (1 , • Ftj}ff--­SLOPEN::Y(J)"YII F ( SL nPEN .. t: Q .. 0 .. ) G0--- TO 3(,SOIFF:SlOPE;SLOPENER T1:: 1 • - t: RR011- ----- -

ERT2~1."'ERROR

IF l SD I Fr .. GE, ERT 1~ AN[f;-SD-ffF~-LE;ERTi. AND ~SI.OPEN~GT • 0~ '--YC-OGM CJ)CONT1NUEl F (SLOPE .. Lor ~-e ~-~ AND ~ FTJ fll-LE--.-YLOGi'fcJn -- GO--"O'-s- -------:-----

;. XF lS [) H~_F. ~ (j~J!._ER H, A~D, SO I FF pLE, ERT2, AND ISL.Q_~_E ,J.. T, 0.) GO TO 5YI=V(J)

' - rOl:: F (,J) _5L.l.IF E.; ;:H.OPEN ---------------~-----

GO III ()-~-4--' ~ _.- - CON ~f ~: hJ lJ \::'-'" -_._-

F (J):.: 1. ,,1::"')--- -v (,1) ;~ i\ ~~ Olj (F (jY/t l-;';'F(Jf) )

'( r :.; ., (" J.----------.- --.-- FX::f (.j) ---------------------- --------

L: LJ i'l .,. '1 ~IU E-... ----- .. ---- -.. - tA:.t_ FUI~C(Y(jj~Y1~X~-X·l.,Atjl~-·Bl~n)

J~.Jd_...._.- -_._-_._-- .. -- ._---------------------..:---------------Vlt:U"FIGU oSLOfJ~"--t u~,~,---

CON'fY rWi:DO C) K;;(~IJ---------------------- ..

G (K i XI , :oJ V(\()-x F (Y lin ~ L,jl' ~y I,; AXU--G ~~-;I 1) =YlfIXI:

n: I: " (I-( j u;. T " V1-11 NL) G(K , 11) II YMI NL._--. H' O"I.;T

uIREfY) RG:RG1----- =------

XH~(l"~RG)~'(XuIREFV''- -.. ----- - -- - ! Ii Ci J()" i ih.: FY, ·RI:~oeIO-cFnO-.-

)( t, ;; f' ll~ ) wlP1. r F (" 11 .. 1.. Y. Ytfl N)-nr=l''R'IN

"( L (\( , 1 ! ) ~ Xt1COI.,jTrNU~ -- ---~-----------~-oo----. -------..

corHXNU~

II =f1f·:.c .. lDO 2'1 J~t~N

----.- .-.---'-' - !\fd::A::I:1-oJ-oo--------~---------~------~--------

AflEA~~~u

AII EAII ~ [il-::;-------

Ai? EA;i S~ 0 L'_.,._._._-- ----

12':5.;l'o0 1 I) I ~ rn Np :n

.---.oo ------.---- :.: 1 a;:i ... ili. i~·;.l·----------------------.-.-------- ---

~M~(TL(~,!i)~YL(J,Il+l))/2,-- -- -- :iF CiliJ,Il+l)"LT.

u0i) 'XMae-.-------·------ --- .. -

IF (TL(J.ll).LE.~.) TLCJ,II).YMIN'fLo (,J ,:CO ~~A!.OG CTL(J p 11»------ -.--.---- ----ARE !I, '. AII t::: ,l v :: '1if ({ uGE. IilEFV) -Gtf-l'O?S--------- ------ .-.----.-------- .. -----------AR EAB:; A/?E AU+)( 11GOT 0 r~ ,) "-_ ... -- - ---- - - .--_ .._---- ._. - -- --_. - -_ .. -..CONTitJlt':AR£AAI.:.IlI£flA+~<t'1-·-----·--- - --~--- ----------- --

L:ONTINUt:C:ONl YNUt: ------------------------- - .. -'-" .. - -_. --------.---..--

ARE(J)=ARcA/REFe J )

_. - ._- _.. _. ---- - ---~ -. ... - _.- - - ._- . ~ ...'- -, .. '-' - .._---- .._-~.- --._._--.. -----_.. --_.- -_.- --.-....-.:-'-~

.._.._--- -_._----

,.. - .. - _.- -

i&

------ .._-----

Ct.LL Sr:ALrCL,t.,0.,0.) ------.. - ... -- .-...

CALL FPLOT(1,1.,-3~)CALL P1130 ----.--.------------ -------

CALL FPLOT(1,~.,3.)

CAL L S t.: /. U: ( 1 0 , 1 • , e. , 0-~-'---­H101~lo

Hl02:; ~ ~

CA~ L Fe t1A R ( XMOl, (:.1" , , 12, • 15,0. )w'" 1TE (I. 1d ':) J H~ EY V - .. - --.-----Ct. LL Fe l-lii 11 ( ).. M0 i ... II j , " 12, • 15,0. )

.-.---.. -.------ - \iJR 1 Tt: (7, 19{1' IR;;'F V, f-lG2 - . - -----'''"''---.---

CALL ftH~R(~MU1'-.~'o12,.S5,0.)vlRt'fr: 1I~1(5) !ilE:F'(·,RG·l--·---- -=-----DO 10 J q , iJHJG:;fLOtIT CJ) - - -'--- ---...--------.---------- ...-----.-. - --..--------

hlGl:;11:L\.j""q3HIG2=H(l1l'-',,;JCAL L FPLOT ( ... 2 ~ II t'\\.) 2 9 II I G)-.._._------- ..._----'- - -.- - -.-.-'----------- ..C1\ LL F PLa l' ( 1 , x t"U C! .. III G)Ci\ LL Fe IH ~ ( )( MU1, H ! G, • 12, • 15, " • )WH IrE (7,120) (l YPE (,1, K3) , K3z: 1~-4 j, ARE [J) --- ---.-----.---- --.---- _.. -

l: Ill. L F L 11 M< ( X" 0 A , HI {j 1 , • 1 i~ , • 15 , 0 • )..-------.-- ---- WRr 'f J: ll, 15 tJ j .. I t1 E l~ V; AU( Jr---.'---"-..:.-""""""'-~----------

c t~ LL F eli ,; R (X j., 0 1 I Ii 1G2 , p 12 , • 15 , " • )..... - .-.. --_. - . - Iran n: (l 01 (;, 10 ) III t. FV, AA(J )-----.---=.-=------

CON'f I NljiH 1G;.; FLO u" (i'J'd '- ---

CALL FtHA~(XM02,HIG~.24,.3,0.). ~J 1\ I TE" ( ( , 1 I ~ r·NP-----=--...:..--=---------

CALL 3C~LFll~~lQ,~~~0.'

CALL rPI.UT'u,::i.;-·w:);r-CAL I.. ~ PL 0"1' (2 , 3 • 1 , .. :s v )

_.-_.~- ---.----- _.- .... t: ;'~ LL r Pl... 0)- ( 1 , .J 0 1 '-'~:1 0 )------

S)'OP~ND-

.__._._._------

------------_._._..-

--------_...-------

--- .-.. ----_ ....

_., .- .....:.- __.._.... ----_._- .----_. _. ---_._. ------_._._--------

-------- --- ._._......__.__ ... -- .._--_.- -_.. _---

_.L.•....-....-_ -.. ...._

S1I BR0 UTpJ E fUN C tY 1 , Y2 , )( i , X2, A ~ Br-A:: (Y 1- Y2) I (Xl-L~ 2J__ . _B;Y2-A~X2

RETURNEND - -.------

--_.\,.-.- -- -_.- -_ .. ---- .- _.- '--~- ...

----------------- -- ------- -_ ....._--_ .. -_.

------------------------- .------- .-.--

_. -- I

- .... --'-"--"-'-- - ._._." -- ----_.._----_.__._._--

... _..._-_._- ._-_._--_.- _..._---_._----------_..------

- _.. - --- _...--_._-----_.- -'--' ._---_._---------

._.- ._------_._------------- --------_..- -'- --' - -- - _..._.

._---_. -----_.__...- --_ ....

._ ..-- _._-------,

- ---------------- -- ------------