NOAA Technical Report NOS 67 NGS 3

Algorithms for Computing the Geopotential Using a Simple-Layer Density Model

Rockville, Md. March 1977

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NOAA Technical Report NOS 67 NGS 3

Algorithms for Computing the Geopotential Using a Simple-Layer Density Model Foster Momson

Geodetic Research and Development Laboratory National Geodetic Survey

Rockville, Md. March 1977

U.S. DEPARTMENT OF COMMERCE Juanlta M. Kreps, Secretary National Oceanic and Atmospheric Administration Robert M. White, Administrator

Allen L Powell, Director National Ocean Survey

Blank page r e t a i n e d for p a g i n a t i o n

C o n t e n t s

Abstract ....................................................................................... I . Introduction ........................................................................

11 . Methods in use ..................................................................... 111 . Approximation by Taylor series ................................................ IV . The method of singularity-matching .......................................... V . Programming of the algorithms .................... 1 ...........................

VI . Program testing ................. i ................................................... VI1 . Applications ......................................................................... VI11 . Constructing the equal-area blocks .......................................... Acknowledgments .............. .t' .......................................................... Appkndix I . Notation ............................................. ........................

Appendix I1 . Detailed mathematical developments .............................. 2 . The method of singularity-matching ........................ 3 . Point mass and numerical cubature algorithms ............ 4 . Annotated computer listing ....................................

References ....................................................................................

1 . The Taylor series method .......................................

.. 1 1 1

. 2 . 3

5 8 9

10 13 13 14 14 14 17 21 22

iii

FIGURES

Figure 1.-The block over which AT is evaluated . The origin is in the

FigurC 2 . - 1640 5" equal-area blocks. Aitoff equal-area projection ........... :. . . Figure 3 a. -Equal-area blocks with index numbers. NW quadrant .............

. "center." .....................................................................

Figure 3b . -Equal-area blocks with index numbers. SW quadranl ............. Figure 3c . -Equal-area blocks with index numbers. NE quadrant .............. Figure 3d . -Equal-area blocks with index numbers . SE quadran t... ............

. Figure 4 .-Subdivision of blocks near pole into blocks equal in longitude and latitude .....................................................................

Figure 5.-The semimajor axis of the orbit of a particle moving in a central . force field modeled by the surface density algorithm and

Figure 6 .-The eccentricity of the orbit of the particle ...........................

Figure 8 .-Idealized set of equal-area blocks at pole ................................

equation (22) ................................................................

Figure 7.-The inclination of the orbit of the particle ...............................

3 . 5 .6 6

7 ' 7

8 .

9 9 9 11

TABLES

Table 1 . -~hongo~ovitch 10" equal-area blocks for the sphere .................... Table 2 . -2hongoiovitch 10" equal-area blocks for the spheroid .................

. Table 3.-Fivedeqee blocks used for geopotential modeling for the sphere .. Table 4 . - Five-degree blocks used for geopotential modeling for the spheriud ..

11

12 12

13

iv

Algorithms potential

for Computing the Geo- Using a Simple-Layer

Density Model Foster Moriison '

. Geodetic Research and Development Laboratory National Geodetic Survey

National Ocean Survey. NOM. Rockville. Md.

ABSTRACT. - Several algorithms have been developed for computing the gravita- tional attraction of a simple-density layer; these are numerical integration. Taylor series. and mixed analytic and numerical integration of a special approximation. A computer program has been written to combine these techniques for computing the higher frequency components 'of the gravitational acceleration of an artificial Earth satellite. A total of 1640 equal-area. constant surface density (5" X 57 blocks on an oblate spheroid is used. The special approximation is used in the sub- satellite region, Taylor series in a surrounding zone. and numerical quadrature in the remaining regions. Tlie relative sizes of these zones are readily changed. An auxiliary program can generate all the parameters for different equal-area block configurations. Different orders may be used in the numerical quadra- ture done in connection with the special approximation. Numerical tests compris- ing integrations of equations of satellite motion and static gravity simulations indicate the simple-density layer model is not only feasible. but highly practical and very easy to use.

a

I. INTRODUCTION

Choosing to use a density layer model to represent a gravitational potential field does not specify an algorithm for the computation. Using an analytic solution tp a boundary value problem. such as spherical or spheroidal harmonics. specifies the gen- eral outline of the algorithm. Quite a bit of discre- tion remains to the user as to what recursion rela- tions (if any) t o apply and what normalization is must efficient: a considerable literature has developed through inquiries into these questions.

Density layer models provide us with many more options. For one thing. we must specify the surface. or boundary. upon which the density is to be defined. The shape of the surface is related to the choice of coordinates.. which is another option. Finally. there is a choice of methods of representing the

density vn the surface. Intricate surfaces and exotic coordinates both would require more' information storage and computation to obtain the gravity vector at a specified point. The choice of analytic or dis- crete representations of the density upon the surface 'will affect the efficiency and speed of one's calculations. also.

To a large extent. the application of the gravity model is a decisive factor in its choice. A model suitable fur the combination of gravimetry. and satellite derived data., which may be dynamical. geometric. or both. might be too cumbersome for computing the ephemeris of a satellite.

11. METHODS IN USE Working directly from optical satellite observa-

tions. liocli and .\lorrison (1970) derived a density

layer model for the popotential. In theory. the sur- face layer was on the Earth's surface. while in fact the mathematical model was equivalent to point masses fixed onto an approximate surface (Morrison 1971). Improved solutions (Koch and Witte 1971). and (Koch 1974) utilizing additional data have retained this type of algorithm.

Vinti (1971), on the other hand, has used a den- sity layer model with a surface consisting of the smallest sphere enclosing the Earth. Vinti's model avoids the occurnnce of impulses on satellite orbits coming close to any of the point masses. but it is computationally no different' from using the spheri- cal harmonic model for the geopotential. Moreover. having the surface layer above the Earth's actual surface yields a model unsuitable for computing mean gravity anomalies at the surface or for doing combination solutions.

111. APPROXIMATION BY TAYLOR,SERIES

The fundamental formula for density layer models is

where r*= r - r., (See appendix I for explanation of undefined terms). If the surface. u. is a sphere. we may use

du = + cosddXdd

and compute (1) as an iterated integral

Even if the &face is not exactly a sphere. (2) may be used provided the surface is starlike with respect to the origin. (For a starlike surface a straight line starting at the origin intersects the surlace only once. i.e.. the radial coordinate of the surface is a single valued function of 4 and A.) Even if the surface lias high frequency ripples and a resulting very large surface area. the total mass of the surface will be very little affected because the surface density is divided by the slope of the lopopaphy. An analogous expression could be derived for spheroidal coordinates. Another alternative is to remove r : from the numerator of (2) and use density per unit of solid angle: the position of the layer would enter only through. r*. For the numerical tests made. only 12) was used.

Given 12). there are various options for ex- pressing the integrand. The simplest. perhaps. is to wile

and expand h into a Taylor series. (See appendix 11.) Since the domain of validity of such a series would be restricted. expansions would have to be done within different regions on the Earth, usually tesserae bounded by tines of latitude and longitude. The result is of the form

..

if we truncate the series at the second-order terms. The function UO is a spherical harmonic expansion used for the central force term. oblateness and long- wavelength parts of the potential. Eq. (5). an inte- gral of a polynomial, is evaluated quite easily. The derivatives are somewhat complicated. but strkght- forward.

To obtain the gravity force. the gradient operator is applied

n=VU. (6)

Numerical tests have been made at altitudes of 300 and 1000 km. using 15" X 15" and 5"X 5' patches. Results are satisfactory for the 300-lim altitude only when the smaller patches are used. This is due to the fact that the Taylor series has many properties of the point mass. Specifically. these properties are

h u e 0 (l/r*a), etc.

Nevertheless. the method is fapt and accurate for altitudes above loo0 km and softens the impulses of a purely point mass approximation. ' Spherical coordinates are used for the sake of

rapid computation. and for the area differential the

du = rz cos & dAd4 + O(e2)

, spherical approximation

.

suffices. Simplifications occur if h is expanded about a point whose latitude is the average of the north and south edges of the block and similarly

2

for the longitude. (See fig. 1.) When h is expanded to the second order and integrated. the odd order terms drop out

.. r 1

where Qn and An are the extent of the block in latitude and longitude.

A9

I Fi(;cRE I.-The block over which AT is evaluated. The orifin is

in the "center."

\

Now J1.r) may be expanded as a Taylor series. The desired results may be obtained provided the expressions x " p ( x ) . n=O. . . .. N can be inte- grated analytically. Often this can be done by re- peated application of integration by parts.

Computing the potential of a surface layer is complicated by the facts that (1) is a double integral and. in most cases. r* is close to but not exactly zero. We begin by converting (1) to an iterated in- tegral in the spherical coordinates 4 and A. as was done for a Taylor series expansion. For the factor with a singularity we choose

where D. E and F are functions of 4. For practical applications at satellite altitudes. it is simplest to use the Taylor series expansion for r*z to com- pute D, E and F: then we have

f ( A ) -CX r * ( 4 ) cos 4="constant"

which greatly simplifies the results. The 'integration with respect to 4 now may be done by a conven- tional numerical quadrature. Newton-Cotes formu- las of order 3 through 9 were programmed (Abramo- witz and Stegun 1964).

All of the integrals with respect IO A are of the simple form

.

(9)

IV. THE METHOD OF SINGULARITY- MATCHING

The limitations of the Taylor series approxi- mations to U are caused by the presence of the square root function in the denominator of (2)

here fr - r.,): i! an analytic function nf the coordi- nates everywhere. but r* is not at r*=O. and I/r* is not well approximated by Taylor series in that region.

Tlie method of singularity-matching provides approximation techniques for computing such things

which can be expressed in elementary function for various values of D, E and F, e.&. see Peirce and Foster (1956). It is more efficient IO branch to different coding than to compute (9) as a function of a complex variable.

Various improvements and generalizations can be made over what has been programmed. For . low altitudes. the matching of the singularity in plh) and l/r* should be precise. Gaussian quadrature should be faster. and more accurate than Newton- Cotes. Some of the integration with respect to Q also could be done analytically to save time and reduce error.

Density variations within a block and the depar- ture of the approximation p i A ) from l/r* can be modeled by expanding

as convergent improper integrals wliere exact analytic solutions cannot be found. Tlie trouble- some integrand A / x j is factored by an elementary "'= p(A) r * function p ( x ) with the same type of singularities . so that one may write

c x r c 0 s d#

in a Taylor series. No matter what order Taylor series is used. the integral may be done quite simply (Cradshteyn and Ryzhik 1965. p. 80). SO h l x t = p r x t J ? x t . (8)

3

any level of approximation can be achieved. Even surfaces of complex shape could be modeled if it were really necessary.

If we define

and use the approximations

we can perform the integration with respect to A in closed form. If a higher approximation to g is required. the integrals will be elliptic and a degree of simplicity will disappear. An alternative to the elliptic integrals is to subdivide the blocks. Doing one of the iterated integrals 'analytically removes the problem of appruximating l/r* by a Taylor series. The'integration with respect t o precedes smoothly by numerical quadratures. Some part of the quadrature with respect to 6 could be done analytically.

Numerical tests of the potential function have proved vhry promising at the 300 km altitude even with 15' patches. These results are far more prom- ising than those obtained with the direct numerical integration algorithm (Koch 1971).

If F # 0. we may use in evuluating (9)

A+BA+CX: d A VD + E A + FA:

where X = D+ EA + FA: and

BE (3E2-4DF)C 2F 8F2

Q = -4 - - - If the quantity q 3 0. q=4 DF-E'. then

To have q D B 0 for this application. If F < 0. one can use

0. it is necessary to have F > 0. since

It may happen that F > 0. but q 0. Such.may be the case if F is relatively small and E large. In this case one has -

These well known formulas (Weast Snd Selby. 1967: Cradshteyn and Ryzhik; 1%) are sufficient for all the cases that occur in this application. provided F is not too small.

If F is exactly zero. which may happen in an actual computation. (14) simplifies to

U 2B(2D-Eh) + 2C(80'-'-4DEA+3E!P' [H- 3E' 15E3

Equation (18) is the application of equations 101. 102. 103 of Peirce and Foster (1956).

Wlieii both E and F are very small compared to D, (14) is computed best by expanding the d e nominator in the binomial series and integrating term by term. This procedure is identical in the formal sense to the Taylor series approximation. but there are differences in the numerical applica- tion. For one thing. the quantities D. E. and F. required to perform the tests indicating whether (14). f15). (16). (17). or (18) sliould be applied. are used at once. Also. the binoniial series expansion might not be used for every value of the latitude in the second iterated integral. which is done numerically. The results are

I + - 3 6A;CF *' + 0 (6A") . 28

where E*=EID. Fa=FID. 1

Equation (19) is given as a definite integral since all the even powers of 6A have zero coefficients with the limits taken thus: and these limits are the ones used. All the constants of 'integration have been omitted from (14)-(18).

There is a definite advantage in interchanging the orders of differentiation and the integration implicit for V in (6). The gradient operator is applied inside the integration (9): the gradients of D, E. and F in (14) are required. The quantities A, B,' and C do not depend on the point at which the potential or gravity is being evaluated. so they act as constants in the differentiation. Gradients are computed for equations (14) through (18). which in turn are integrated by the same numerical quadra- ture formula used for the potential.

V(AU) = v J + p d & d ~

(20)

When F is small. (14) is plagued by small divisors. Upon taking gradients. the divisors become smaller still. so it is necessary to use (18) or (19) to prevent excessive rounding errors.

= I d$ [ Vjfg-'!s dA).

To optimize the numerical integration method. Koch (1971) and Friihlich and Koch (1974)' have studied different configurations of the evaluation points. Recently. Friihlich (1975) published some algorithms of very high accuracy.

V. PROGRAMMING OF THE ALGORITHMS

To construct a program suitable for application to anticipated high precision satellite altimetry data. the computer programs for these algorithms were changed to use a scheme of 1640 equal-area blocks on an oblate spheroid: these blocks are about S'XS" at the equator and fairly uniform in shape everywhere (see figs. 2 and 3). Blocks near the poles are somewhat extended in longitude. for example. those at the poles cover 90' in longitude. a '

For this reason these blocks are subdivided into roughly 5' longitude bands and the' algorithms are applied withjn each subblock generated. This is illustrated in figure 4. To save computer time, different algorithms are used depending on the distance from the satellite to the center of the block.

FILL'RE 2. - 1640 fivedegree equal-area bluclir. AiluB equal-area pmjectiim.

222.;.53 0 - 11 - 2

FIGURE 38. -Equal.urea blocks with indc; numbcn. XW quadrant.

FIGL H E 311. - Equal-arc-a Iiluclrs with index numlien. SB' quadrant.

6

FIGURE 3c..- Equal-area blockr with index nurnliera. NE quadrant.

FIGURE 3d. - Equal-area Iiloeks with index nurnlwn. SE quadrant.

FIWRE 4.-Subdivision of blockr near pole into blocks equal in Longitude and latitude.

Density is assumed constant within each block in this production version of the program. so that

B - C - 0 (21)

in expression (14). The method using (14) is employed for the closest blocks. Taylor series for a zone surrounding that. and numerical integration for the farthest blocks. No loss of accuracy is suffered if the distances of R/4 and R. (R=Earth radius). are used to define these zones. Accuracy is. main- tained from the lowest possible altitudes for or- biting satellites out to infinity.

For operational purposes and to facilitate incor- poration into large-scale orbital analysis programs. some optimization has been done. The equal-area block system has an eight-fold symmetry which has been used to reduce storage requirements. Certain sines and cosines are handled as complex exponential functions. which can save time and 'memory. Some loops were written out explicitly to sacrifice memory for central processor time. It was found ihat the Taylor series and numerical quadrature algorithms can be computed using the short single-precision word length of the IBM 360' with double-precision accumulation of the geopoten- tial and gravity. The more complicated formulas which arise from applying '(2) require double precision at almost every step. but just for Certain troublesome satellite locations. More detailed study is required on this question. though the use of double precision is a satisfactory if not optimal solution. .

Some time is saved by direct application of the Newton-Raphson algorithm instead of the compiler subroutine to compute the square mot needed to evaluate the distance from the satellite to a given block. Since the blocks whose contributions are evaluated successively are always adjacent. very accurate first approximations to this distance are available. except for the case of the first block. of course. The compiler function is used to initialize the process.

VI. PROGRAM TESTING

Timing and debugging tests were made using 90 satellite positions- half at 300, km. most of the rest at lo00 km and one each at 10R. 100R. 1OOOR. 1.0000R. and 1.000.000R. If the density on an oblate spheroid is chosen so that

the potential will be nearly that due to a point mass equal to the mass of the spheriod and located at its center. This was determined through compar- isons with a spherical shell of constant density. If we set U = g / r # in (1) and attempt to solve the in- tegral equation for x. we can observe that du = rz (4) cos tpd&dA. which in turn implies that (22) is

. an approximate solution good to an order of about .e*. A higher order analytic solution to the problem would be useful for testing, but none has as yet been found either through attempts at solution or in the literature.

The program has been inserted as a force model in two orbit computation programs. The first is a small program on the CDC 6600 that generates only orbital elements. The geopotential modeling program was then converted for use on the IBM 360. optimized. and inserted into the well-. known GEODYN program. which is now operational on the NOAA IBM 360-195 at Suitland, Md.

Program linkage and compatibility h n e been tested by using (22) as the density distribution and integrating a two-body problem perturbed only by the computational errors in the surface density model program. The results in Keplerian elements p. e, and I are given in figures 5. 6, and 7. Both the semi-major axis. a. and the eccentricity. e. exhibit periodic changes only. at least for the 9 hour period for which the integration 'was done. Since the mean value of a obviously is not the same as the initial value. drift occurs in the mean anomaly. In practice. the process of orbital adjustment would rectify this problem. A very large secular or long. period perturbation is present in the inclination. I. Runs made Ion the CDC 6600) with an inclina- tion of 59.5" did not exhibit this effect. SO it is most

.

'

'

8

likely the result of resonance caused by the trun- cation errors in the force model. A force model with zero total density and with no errors equiv- alent to low order harmonics would be much less prone to cause such problems (Momson 1972). This is a very severe test in many ways. The model will be used with a spherical harmonic model of. perhaps. degree and order eight. so it will be used to model the last few parts per million of the gravity field. A thorough analysis of the dynamical effects of these computational errors on an actual satellite orbit remains to be done. A qualitative analytic analysis of the problem has been done by Morrison (1972). More computtr simulations and a numerical verification of that study would be desirable.

FIGURE 5. -The semimajor axis of the orbit of a particle moving in a central force field modeled by the surface density algwithm and rquahn (22).

-i

FILLRE 6.-fhe eccciitricily ui the orhit a d the particle.

4 1 8 1 . a @

FicuRe 7.-The indilution of the orbit of h e putid&

v 1 1 . 1 0 1 0

VII. APPLICATIONS

Most of the work in developing the density layer method of geopotential modeling has been devoted to finding approximation techniques for computing (1). Inasmuch as this is part of the art of computing. as opposed to a 'purely mathematical problem. it is **solved" in a satisfactory way. but time and experience will bring improvements of some sort. More analysis and programming are needed for adjusting the density values from corn- binations of various kinds of data. The present program. using 5" blocks. is suitable

for using satellite tracking. satellite altimetry. and gravity anomalies which have been averaged over areas of about 5" "squares." The model corre- sponds. more or less. to a spherical harmonic ex- pansion to degree and order 36. A study of the frequenc). response of the model is included in Morrison (1976). Since the data quantities and density values can be related through linear integral transforms. either exactly or to a high degree of approximation. the data can be combined in a con- sistent and optimum way by covariance methods. The density layer method has been used by advo- cates of discrete variable representations. whereas the techniques known as least-squares collocation and spherical harmonic sampling functions have used. basically. orthogonal function interpolation. The motivation to develop the discrete appmach has been mostly intuitive. based on the observation that when you have to add a new term to your ex- pansion in eigenfunctions for every new data point. you are making unnecessary work for yourself and your computer. and a discrete variable method is better. It seems best to represent the long-wave portions of the geopotential in spherical harmonics and the short-wave parts with the surface coating. The advantage of the surface coating over a discrete variable representation of the gravity is that the upward continuation transformation is much

9

simpler. i.e.. (1) is simpler than the Stokes' integral and more flexible as far as choosing a reference surface.

Even for 5" blocks. the application of the covari- ance methods for interpolation is not completely straightforward. Williamson and Gaposchkin (1973) have shown that gravity is not stationary. not even for 1" and 5" blocks. Point anomalies should be much less stationary than these means. A problem of interest to some geodesists is the inter- polation of deflections of the vertical. point anom- alies. and gravity gradients. To do this will require a program with a much smaller block size than 5".

VIII. CONSTRUCTING THE EQUAL-AREA BLOCKS .

Equal-area blocks have been a popular means for determining a uniform sampling of data distributed over spherelike domains t Rapp. 1971). Making the blocks exactly equal in area is fairly simple for spheres or oblate spheroids. and eliminates the need to store and reference a lot of weight factors in operations such as numerical integration.

The area of the zone between the equator and latitude Q on an oblate spheroid is given by Unguendoli (1972):

where a= semimajor axis. and e== meridian eccen- tricity. There is no need to resort to. series .to evaluate (23). since the integrand is a rational function if w e use the substitution

x = e siaf . (24) The result is

+ - 1. log l + e s i n d ] 2e 1-esinb (25)

Strictly speaking. (25) is not defined for e=O. but

Lim A (6) ='Lna'sin d~. (26)

which is the correct result for a sphere. For small values of e it is necessary to evaluate the logarithm portion of the expression by series and cancel the divisor e to avoid loss of numerical significance. To establish a system of equal-area blocks. one

needs first to pick a block size. say 5" X 5". Then a number of zones are chosen. ./=18=9O0/5" for one hemisphere (the other hemisphere can be obtained by reflection). One next chooses the num- her of blncks in a zone. starting I'rom

r-(I

where the zones are numbered starting from the equator: ( 4 1 ) = mean latitude in zone i land for 5" any other appropriate value may be substituted): the [I symbolizes truncation to the nearest integer. The area of the zone from 0 to 4j is then

Z n ( i 1 . 8 - 1 ,

TO find &j one substitutes A (4j) from (25) into (28) and_ uses the Newton-Raphson iteration. since the transcendental equation .is not readily solved. For a starting value the spherical approximation may be used.

The derivative needed is quite simple.

a'(1 - e z ) cos4 dAldtj = 2n (1 -e* sin* 4)+ (30.1

When a set of blocks is generated. they may be tested for "squareness" using a criterion suggested by Paul (1973)

length of side along mean parallel the same along a meridian

.Ri =

(31 1 1.

If the values. of Ri are not satisfactory one may change the values of n ( i ) in a trial-and-error way to see if improvement is possible. Setting up a rigorous adjustment of the n ( i ) by some criterion of obtaining a letter approximation in (311 does not seem worth the effort.

'For the poles there are some complications. The blocks at the polar zone are best divided into four. and the next group into 12 (see figs. 3 and 8) and so on. as

n ( J I - 4

n ( l - 1) = 12

n ( . / - k ) = Q ( k + l ) f - n ( J - k + 1) . . .

=4(2k+ 1)

ka 1.

This is because the zones are nearly concentric rings at the poles and for those regions

pole follow this rule. (See figs. 2 and 3. and tables 3 and 4.)

The 5" equal-area. block model chosen Collows the well-known Zhonogolovitch 10' blocks as closely as was considered practical. The zone boundaries near lo', 20", 30". and 60" are the same. Table 1. gives the parameters for the Zhongolovitch blocks on a sphere. table 2 for a spheroid.

R.r -I . = 1~14

gives a better criterion for '*squareness.** For blocks of 5" only. the two zones nearest either

FI(;URB 8. -Idealized set of equal-area blocks ai pole.

TABLE 1.- Zhongolouitch IO" equal-area blocks for the sphere

J N PHI-2 PHIBAR DPHl DLAhlBDA RJ P H I 4 SPH

1 2 3 4 5 6 7 8 9

Total -

36 34 32 30 25 21 IS 9 3

205 -

0

10.114144088 I Y .Y66058297 29.8387661 I 1 40.083433915 49.982997 15, 60.26o&uX3Jo 70.298:88126 80.1858570 12 90.000000000

0

5.0570'72044 15.wo101193 24.Y02412254 34.961 lo0063 45.03321 55.34 55. I219 18757 65.2798 14233 75.212322569 85.09'2938506

I

10.11414QOHB 9.8519 1 Qo9 9.R72707913

10.2446677W 9.899563139

10.277&23186 10.037947786 9.88706@?86 9.8 I4 I 4B88'

0

10.000000(1 10.588135294 11.2500000UO 12.- 14.- 17.141857lM 24.000000000 40.-

120.000000000

0.984865728 1 .O3r)23106 1.0335609A6 0.959962351 I .027%i0l9 0.953783344 0.99YW867

1 .W5917823 I .awstians

0

10.1 14144088 19.9660!!7 29.83876621 I 40.083433915 49.982997 154 60.26o840340 70.2987881% 80.185857012 90.000000000

1 1

\

TABLE 2. -Zhongolovirch 10" equal-area blocks for the spheroid

a= 1.0000000000 b0.9966470765 r'=O.- Sectors= I Area=O.0305811%74137 ~~ ~~~~~~~~

J N PHI-2 PHIBAR DPHI D I A M BDA RI PHI-2 SPH 0 0 * 0 0

1 36 10.158613279 5.079306640 10.158613279 10.000000000 0.980520745 10.092007653 2 34 20.048575322 IS.1035943W . 9.889962043 1 o . n 1.033621594 19.924951450 3 32 4 3 0 5 2 5 6 21 7 15 8 9

3 9

Total 205 -

29.949659806 24.999117564 9.901084484 11.- 1.029789802 29.183459904 40.209905973 35.079781889 10.260246167 12.000000000 0.9571 14533 40.0#)265940 50.109306146 45.159606059 9.899400173 14.400000000 1.025712019 49.919815831 60.371219506 55.240262826 10.261913W 17.142857143 0.952431222 60.205546409 70.380067424 65.375643465 1O.O08&07918 24.000000000 0.9991 17443 10.258024153 80.228851331 75.304359382 9.848783917 40.000000000 1.a3a310595 80.364218502 90.000000000 .85.114425671 9.771148659 120.000000000 1.045929001 90.000000000

The following legend is u a d for dl tables in this text:

o = Semimajor axis 6 = Semiminor axir e = Eccentricity of the spheroid

PHI 2=Latitude of northern edee of block

DPHI =Extent of block in latitude

DLAMBDA=Euent of block in longitude

PHI-2 SPH= Spherical latitude for PHI 2 RJ=Ri=Sce (31)

J =Index of zones in one hemisphere N = Number of blocks in one sector of a zone PHIBAR = Average latitude of block

.

TABLE 3. - Fivedegree blocks used for geopotential modeling for the sphere

.4rea~O.O7W421106316 a-1.0000000000 b= 1.0000000000 es = 0.000000000000 Seetiws =4

J N PHI+ PHIUAR DPHI DLAUUDA RI PHI42 SPH

0

1 18 5.037335850 2 18 10.114144088 3 17 14.983246004 4 17 19.Y666058297 5 16 24.803794163 6 16 29.838766211 7 15 34.801265705 8 15 40.083433915 9 13 45.017042172

10 13 50.4196410i2 11 10 55.0359Y2386 12 IO 60.260840340 13 7 64.580541509 14 7 69.485799940 15 5 73.=3(u116 16 4 78.662(36714Y 17 3 84.338423235 18 1 90.-

Told=% X 4= 1640

e

2.518667925 7.575739969

32.548695046 17.474632151 22.384Y26230 27.321280187 32.320015958' 37.&3498 IO 42.5502380.13 47.718341622 52.7278 167B 57.64841639 62.370690925 66.983 I70724 71.7132197 13 76.30 I8033 I8 81.joo695192 87.16921 16 17

m

5.037335850 5.076808238 4.849 101 Y I6 4.982812293 4.837i35865 5.034972048 5.9624W494 5.28'21682 10 4.933608258 5.4025- 4.6 1635 131 4 5.224845954 4.219701 169 5.00525853 1 4.455839536 4.72327663 5.675)56086 5.661576765

0

5.000000000 5.000000000 5.m 1 1 7647 5.294117647 5.625000000 5.625000000 6.000000000 6.U0000(10UO 6.923076923 6.923016923 9.000000000 9.000000000

12.85714m7 12.857 152857 18.OlMlWMO 92.500000000 30.- 90.000000000

0.y91629292 0.976275270 1.06131;1857 1.01w2456 I.Oi5117645 0.992560304 . 1.021753384 0.901863137 1.033751358. 0.8621 I n 0 4 0 1.1906i6862 0.Y21752781 1 .5 I301 2628 l.ooO376l IO I .a7816931 1.12U9321i 0.781245113 0.785078675

0

5.037335850 10.114144088 14.9832- 19.966058291 24.803794163 29.83876621 1 34.801265705 40.m433915 45.017OS2172 SO.41%41072 55.055992386, '

60.z6Ow340 64.480501509 69.485199940 . 73.94ww 7R662%7149 84.33w3235 90.000000000

12

TABLE 4. - Fii:e-tlegree Ilocks used for Feopotentinl modeling Jor the spheroid

PHI-2 SPH RI DIAMBD.4 J N PHI-:! PHIUAR DPHl

1 2 3 4 5 6 7 8 9

10 11 12 13 14 IS 16 17 18

18 18 17 17 16 I 6 15 15 13 13 10 10 7 7 5 4 3 1

e

5.059836888 10.1586 13279 15.047475056 20.048575322 24.901674946 i?9.94%59806 34.921630212 40.209905973 45.145343214 50.54559a66. 55.156394527 60.37121M 64.580143945 69.569857941 74.008705762 78.712308656 84.363550821 90.00(1000000

0

2529918344 7.609225084

12.603014168 17.548025189 22.475125134 27.425667376 32.4356o5009 37.565768092 42.6i7624593 47.845467740 52.8509933% 57.7'63807016 62.475681725 67.075000943 71.789281851 76.360507209 8r.535929738 67.181 775410

a

5.oj9836888 5.098776391 4.888861777 5.001100265 4.853099625 5.047984859 4.971970406 5.288;1?5761 4.93W724 1 5.400249052 4.610802261 5.214824979 4.208924439 4.9897 13996 4.538817821 4.7W2894 5.651242164 5.6364491i9

0

5.000000000 5.000000000 5.294117647 5.254117647 5.625000000 5.625000000 6.000000000 6.000000000 6923076923 6.923076923 9.000000000 9.000000000

12.857 142857 12.857142857 18.000000000 22.500000000 30.000000000 90.000000000

0

0.987210982 0.971992256 1.056801567 1.009328610 1.071017871 O.B!Nl68218 1.018502939 0.899333629 1.031256833 0.860386605 1.118755745 0.92Q586272 1.411648910 1.003702695 1.26727191 7 1.12WL?569 0.781180387 0.78508 1504

0

5.0&137369 10.o92001853 14.9512634M 19.9249514% 24.755008330 29.783459904 34.741 194898 40.020265940 44.9529127M 50.3w34602 54.975721014 60.- 64.430619998 69.43646351 73.w649 I693 78.6382051 10 8S.325809758 90.000000000

ACKNOWLEDGMENTS

The computer cartography to generate figure 2 was provided by Robert H. Hanson. Clyde Goad did most of the optimization, of tlie programs. and aided in adapting them to the IBM 360 and GEODYN. Some of the computing time was funded under NASA Purchase Order P-55.674 iC). .

REFERESCES

Aliramiiwitz. 11.. and Sterun. I.. 1065: HondbuuC o/:~iiihernnrical Functions. .Vdonul Bureuu of Standards App!ied :Itathemotics Series. 55. U.S. Government Printing OHice. Washingtiin. D.C., 104i pp.

Friihlich. H.. 1975: Vcrfuliren zur k u n g des Oberfl2clien. integrales f i r das Xlidell der cinfachcn Schic:ht in der Satel- litenseodiisie. Series C: Dirserttrrion. Vul. 207. German C e d e t i c Ciimmissiun. 1lunir.h. .17 1111.

Fri;hlich. H.. and liuch. E. R.. 1974: Intepratiiinsfeliler in den Variations-gleichunren I'iir daa h d e l l der einraclien Schicht in der Satellitcnpudisie. . l l itteilun~en our drm Insr. f. thror. L'eodasie der Uniwrsitiir Bonn ICimniilnirution of the Ciim. mittee hlr Thenrcticrl (;eodesy. University uf Ilunnl. Sn. 25. 33 PP.

Gradshtcyn. T. S.. and Hyshih. T. 11.. 1965: Ttrl,le of Inrerralr. Series. und Pruductr. Academic. i 'nss . S r w Ylwk. itranslated fmm Russian by Alan .letTrey). 1 0 6 iip.

Koch. K. R.. 1974: Earth'* (iravity Field and Statiiin CiJiirdinut,rs from Dtqililcr Data. Satellite Triangulation. and ( k r v i t y Anumalirr. :$0.4.4 rrrhnicnl Hrporr 505 62. Srtiutiul Oceun Survey. Natiiinal Oceanic and .4tmusybrric .4dminirtration. Rockville. Jld.. 29 pp.

Kuch. K. R.. 1971: EITON of Quadrature Connected with the Simple layer Model of the Geupntential. .VOAA Technicd Memornndum NOS 11. National Ocean Survey. National Oceanic and Atmospheric Adminictration. Roekvilk. Md.. 18 pi,.

Kticti. li. R.. and Witte. U.. 1971: Earth's gravity field repre- sented by a simple l a y r Initriitial frnm Diilipler tracking of Satrllites. 1. Crophys. Rrs.. 56. pp. 8 4 7 W i 9 .

Koch. K. R.. and 1lairriwn. F.. 19iO: A Siniiilc layer m d e l of the geoptential frum a roniliinution #if satellite and gravity data. ./. Ceophys. Res., i5. lip. 14KL149P.

h r r i s u n . F.. 19i1: Density layer models fair the gru~wtential. Bulletin Ciotleiique. KO. 101.319-328.

hliirrisun. F.. 1972: I'ruliupatioii id errors in ohi t9 ciimpitted fmm density layer models. b r 1:se nf Artificicrl Satellites for Ceodrsy. AGL' .Ifonopraph 15. cd. by S. a. Hcnrikscn ct d. .American C.eupliyrirol L'niiin. Waaliiiigtun. D.C.

Wirririiii. F.. 19i6: Alguritlimr bir cnimpeting the gaqiotential using a simple-layer density model. 1. Cmphyr. Res.. 81. pp. 4933-4936.

I'aul. 11. K.. 1Y73: On cuniputatiiin # I f equal urea blucks. Bulletin Ghtlirique. No. 107. l ip. 73-83.

Peirce. 18. 0.. and Fiister. N.. 1956: A Shorr Table of Inrcrmls. Ginn and Ce.. Biisti~n. 18') l ip.

Kalip. R. H.. 19il: Equal-ureu blocks. Bulletin CCodciiqur. XI. YY. 1111. 113-125.

Lngueiidiili. 11.. 192: Divisiilii ilc la surfucc terrestn en blucs il'rirr c'gale. Uullrtin Ciorfisique. So. IN. PI-23.

Vinti. .I.. 1971: Relwrsetitation 111' tlie Earth's gravitrtiwml ptrnt ia l . Celrrtiol Jlerharics. 4. *Si.

Urast. R. C.. ani1 Srlliy. d. 11. i e l i t a i r s ~ . 1967: Handbook of Tdhs fnr .lIothrnriitirs. third etlilion. The Chrrniral NulDkr G.. Clevrlmd. Oliiu. luio 1111.

Uillianisun. 11. R.. atid (;aptsrhhin. E. 51.. 1973: Estimaic of gravity animalier. in IYi3 Smithsoninn Stundard Eonh rlll). S A 0 Special Heport 353. rditcd by E. 11. Caposchkin. Camliridgr. Jlarr.. 1'1.3-228.

13

APPENDIX I. NOTATION Most notations in this paper are standard: symbols

undefined in the text are:

gravitational constant position vector on the Earth position of a point where potential is E

puted '

components of r. anomalous qavitational potential gravitational potential dummy variables ' .

longitude latitude (spherical coordinate) ,

density surface area

!om-

APPENDIX 11. DETAILED MATHEMATICAL DEVELOPMENTS

This section is comprised of a detailed presen- tation of all formulas used. In some cases. options not used in the programs are descrihed so that. this material will be suitable for a'number of programs tailored for specific purposes.

1. THE TAYLOR SERIES METHOD A. Development of the Potential

T = x AT

- -A* - - A I

where

andr*= 11 r* 11 The integrand is abbreviated to

r * = r - r r

XrS cos 4 f- r*

and

(1.1)

(1.2)

The vector rI is the satellite position. r the coordinates of the increment cfu on the reference surface.

r= r(cos 4 cos A. cos 4 sin A. sin 4) (1.1 1 )

ar* r+ ar a4 r* a+ -=-.-

ar* r* ar ah r* ah -=- .-

. .

ar atb - = r (-sin 4 cos A, -sin 4 sin A, cos 4)

ar + - (cos + cos A. cos &sin A. sin b ) 7 84.

ah * =r(-cos4sinA.cosgcosA.O)I

ar E ah

+ -

a + r a+. a+: ra+z. -=-++4--

ar . i- 2 - (- sin 4 cos A. - sin 4 sin A . cos 4) a+ a+

a4aA -=r(sin&sin h.-sin+cosA,O)7

11.12)

(1.13)

(1.14)

(1.15)

(1.16)

(1.17)

(1.18)

(1.19)

(1.20)

ar a& + - (-cos 4 sin A. COS 6 COS A. 0)

Most terms with a W h r are omitted.

Taylor's theorem: Substitute (1.2) into (1.1) and expand by

.

All the terms containing odd powers of (6 -b" )

or ( A - ha) integrate out to zero. which includes all the third-order terms. Hence. we can find

+O(.Atb4, AA4)]* (1.23)

B. The Expressions for the Gravity Vector

The formula

is fundamental. Substitute (1.23) and (1.1) into (1.24)

Ag=OT (1.24)

Ag=V AT (1.25)

( 1.25a)

(1.26)

The only factor of (1.2) that is not constant with respect to 0 is Ill*. and V operates on (1.5) and (1.7) similarly. Actually. (1.6) is not needed and (1.3) and (1.4) are only steps to derive (1.5) and (1.i).

Gradients of derivatives of r*

(1.28)

+ 2 + ( 5 ) ] (1.29)

+ 2 E V ( x ) ] a r* (1.30)

We may fact,or out seven distinct furictions of r* from /id and ik. It is convenient for program coding to write these 8s elements of 4 X 4 matrices @and .I.

C D , , -- l/r* (1.31 a)

Then we can write

. . ( r ~ . v f ~ ) T = @ a

(fAA.v f A A )I= * \ p.

with a and f l being Cvectors:

(1.31b)

(1.31~)

f1.31d)

( 1.324

(1.32b)

( 1.32~)

( 1.32d)

(1.33)

(1.34)

pi = cos d [Jr

/3:i = a3 (1.36~)

p 4 = a4 ( 1.W)

The remaining rows of @ and .I are obtained by applying the operator V to the fint row of each

( 1.37a)

( 1.38a)

( 1.37b)

f 1.38b)

( 1.37~)

( 1.38~)

(1.37d)

f1.38d)

C. Special Procedure for the Cosine Factor in f 1.2) for Large Block Sizes.

F.or larger block sizes one may wish to use an approximation higher than second.order to cos d in I 1.2). I 1 .36~)

16

Let f= H cos 4 (1.39)

(1.40)

The expansion in Taylor series is now done as

f COS 4 { H a + (4 - 40) Hd + ( A - Ao) Hn + - [(d - 40)' H& $. 2Hln (4 - 4ttNA 40)

1 2

+ (A - A o ) ~ Hnn] + . . . } (1.41)

Integrating the zero-order term

= 4AhHn cos 60 sin A6 (1,42)

Comparing this to (1.23) we can observe f i t = Hat cos 60 and sin A4 is fairly small. (less than 7ih9, we can use ,

Since

(1.43) A@ sin Ad e Ad-- 6

was assumed in (1.22). we can add the remaining terms of 0.43) to get a correction to (1.23)

None of the fcrctors in the correction depends tin

the satellite distance r*: they depend only on the block size used.

Another correction of the same order as (1.44) may be obtained from the consideration of the Grst- order terms of (1.41). The term containing I A - AdHA integrates out to zero, but l / d f ~ ~ ~ - d t t ~ cos r j ~ does not. Hence. we may write

The integrals in (1.45) are standard forms and simplify to

AT1=2AAHdsin&1(A4cos A+-sin Ad) - (1.46)

The trigonometric functions of A4 in (1.46) can be expanded in Taylor series: the terms in A$ drop out:

A T , = - i dhA4JHd sin 40

Since the first term of (1.47) is already included in (1.23). the correction is

All other corrections would be of even higher order and have been neglected since the- will not improve the accuracy of the computation.

D. Corrections to the Gravity Vector Applying the qperator V t i ( 1.44) yields .

For (1.48) the result is

=AhHd = - xr2(921. %. 4h1)~ . (1.521

2. THE METHOD OF SINCUL.4RITY- MATCHlNG

A. Computing the Potential Instead of (1.2) w e use

where f is now defined as

12.1)

17

fAr =-sin 4 [ 2 q 2 + r f i~ To compute the integral (2.1) expand f and F in Taylor series about u$ll. All) ah a A

(2.13)

(2.14) + 2gd~A4AA + & d A 2 1 + . . .

Substituting (2.4) and (2.5) into (2.1) leads to

(2.5)

(2.15) ar* g * = 2 P - a4 ..

(2.71)

(2.8)

1 f* =cos Q [ 2 q $+r? ix a h (2.11)

and

(2.16)

(2.17)

(2.'18)

(2.19)

where (1.17) is applied. The derivatives of r* and r"are obtained from 11.12). 11.13).11.14). t1.15). (L16) and (1.20).

The computational strategy will be to integrate (2.6) analytically with respect to A. This wiU elimi- nate the improper condition 'from the integral. The integration with respect to 4 then can be done numerically in a completely straightforward man- ner. The possibility of doing some or all of the integration with respect to Q by analytic means will: be treated elsewhere. Special procedures for very low altitudes also will be treated elsewhere. These formulas will be suitable for satellite altitudes. typically 200 km or more.

We now define

(2.20) x (Di + EiAA + FiAA')-"'dh

18

where the subscript i indicates A, B. etc. are evaluated at sequentially spaced intervals of 4

&,,-hb=41 c 4: e 43 . . . -c 6 , - I 4 M

= 4'1 + A4

4/- I -4 i 641 so that (2.6) can be computed from (2.19) by numerical quadrature formulas. For simplicity the subscripts will be omitted in the detailed formulas that follow.

Let us adopt the abbreviation

X = FAA? + EAA + D (2.21)

and d s o drop the A symbols. so we can write

X- FA: + EA + D

(AI')!=/ ( A + BA*+ CA2)X- IndA. (2.22)

-411 this does is move the working origin to the "center** of each equal-area block. Now ,we can apply standard forms to obtain

(Peirce and Foster 1956. eq. 174 and 177). By collecting terms we obtain

(2.24) with

The expression J ,Y-Ipi dA can be considered to define a function with two poles: one at each mot of,.Y. Hence. for all values of D. E. F it can be expressed as a single elementary function of a complex variable. at least when we exclude negative

vdues of the square root. Since we have the special case that D, E, F are all real and that the result is real and the integral is taken only on the real line. it is more efficient computationally to express J X-'" dA as different real functions of a real variable and use the one designated by appropriate functions of D . E, F. Using the appropriate complex function would about double the amount of corn pu tat ion.

The auxiliary functions needed are F and

qz4DF- E'. The three useful basic forms are

F > O . q > O ;

F c 0, q <O. and .

F > 0.

(2.26)

(2.27)

(1.28)

(2.29)

Form (2.29) is very handy for evaluating definite integrals. since the difference of logarithms is the logarithm of the quotient of the arguments

(2.30) log x - log y = log -. X

Y' hence

(2.31)

where K ( A ) is a function of A. Similar. convenient forms can be obtained for (2.27) and (2.28) by applying appropriate identities.

The inverse hyperbolic sine may be expressed in logarithms as

so (2.31) may be applied to evaluate a definite integral of the form (2.27). Definite integrals of the form (2.28) may be evaluated with the identity

sin-lx-sin-ly=sin-l [ z d i 7 - y d F 7 ]

A& (2.39) ar a+ VD=-2r*-2-&-- 84 a4

In some cases F-0. or F may be so small that (2.21). (2.28). and (2.29) may cause numerical problems. If E is not similarly small the ap- proxiillat ion

(2.34)

may be used in place of (2.6). The analog of (2.23) is. then (Peirce and Foster. 1956. eq. 100. 101. 102):

a+ a u .

VF=--. (2.41)

The derivatives of r are given in (1.14). (1.15). (1.19). (1.20). and (1.21).

The gradient of (2.24) is given by

Both EAA and FAA: may be small compared to D. so that the binomial series is the most accurate method to use.

(2.36)

Computing ( A T ) i is done by multiplying and integrating power series in the usual way

+=AA:I 9 cy- BE + ---I 3AE' AF 3 [ 2 0 D: 2 0

+ - 3 ;3A' - CFZ i 0 (JAW)]. (2.37) 28 DZ The use of binomial series might be feasible over an entire block. but in some cases E and F will be small only for a limited band in latitude and the expansion of AT would not be accurate if done in terms of d as well as A.

B. Gravity From the Singularity-Matching Method

Let us determine the gradients of the factors in (2.24):

The firsi part of (2.42) is written symbolically ' because for. the case of constant density blocks on an oblate spheroid B 4 = 0 moreover, for that case V Q=O. Then (2.42) reduces to

If (2.27) or (2.28) is used. it is convenient to define

Y = (2FA + E ) ( q l - I p : (2.43)

and obtain

V q = 4 ( F V D + D V F ) -2EVE. (2.48)

If (2.29) is used. define

2 = X 1'2 + A F ~ ! ? + 2EF- I / t (2.49 1

20

and obtain.

with

Where B = C = 0, (2.35) simplifies to

(AT), = a and V ( A T ) ; - 2AVE - E:' m

A . E +- ( h V E ) ( D + E h ) - l : " .

For B=C=O (2.37) simplifies to

(2.51)

(2.52)

, (2.53)

(2.54)

( 2 . 5 )

V W = $ Ah:! 6A V ( E / D ) - $ V ( F / D ) ] [ " 3 AF 10 D +-AA"V(FID) . (2.58)

V (EID) ( D O E -EVD)IDz (2.59)

V ( F / D ) = ( D V F - F V D ) / D : . (2.60)

3. POINT MASS AND NUMERICAL CUHATURE ALGORITHMS

The simplest possible numerical cubature scheme was used by Koch and Momson (1970) and Koch and Witte (1971). The constant density blocks were divided into lour sub-blocks and the distance r* from the sub-block to the satellite was used: each sub-block was weighted by its area Obviously. this is the same as using four point masses. Any conventional numerical cubature woald corre- spond to some array of point masses.

One sets up the array of sampling points in the 4. A coordinate system and applies the numerical cubaiure formula to

to obtain

T = 2 AT,,. I

(3.4)

where u;r are the cubature formula weights. For the formula actually used. N = 1. and tuI

A&dA cos 41= area of the block or sub-block. Hence, 13.2)simplifies to

21

P

15

20

PRDGR1.M . DIlOUR CDC 6bOo f I N U3.0-324 OPI.1 8 ~ 1 Z b 1 7 6 I~.oS.18. ?AGE

PROGRAM Df IOU#( INPU1,OUIPUIl

1 3 w

5

IO

IS

20

25

30

35

40

45

50

55

SUEROU?IHf 1 1 f S l

102 106

c 65 c . . .

c . . . c . . .

70

75 105

c c . . .

80

65

90

95

100

lo?

C D C 6000 8 1 N V3.0-326 OP1.l 03 /26 l?b lS.05.18.

I f (KOUY? . G T . 5 2 ) KOUHl a 0 co 10 10 E N D

5

SUBROUlIYE Sf IUP CDC bb00 f 1 W V 3 . 0 - 3 2 4 O P l = l Q312617b 13.05.18. PA6E 1

10

I5

20

25

30

35

40

b 5

50

33

C . . . N.B. I H f CDC 6600 f O R l R A W I V ALLOYS OWE 10 LOAD CORMOW BY D A l A c... Pn isPn SPntn ic iL LA^. of Y. BORDER of w . nfrtispnEnt%ocKs

0111 PHI ,mi I 0 . 0 8 7 7 2 2 b 4 5 ? 3 7 7 , 0 . l 7 b l 3 8 7 b 5 l b ~ 5 , ~ . 2 b ~ V ~ 8 7 ? 4 7 8 5 8 , O . 3 4 ~ 7 5 b 0 0 b l 3 b 5 . 2 0.43205~401b823,Q.51~8lVb37Vl2V,0.b0b34823bVV2V,0.bV8485408l328. 3 0.7845763350315,0.87888V073V73~,0.~5V5073402V~b,l.O5078501525bl. 4 1.12452bb580253,1.212020273388b01.28VV~lbl8bl0b,1.3?24V55V70138.

D A I h I ) / 1 0.~Vbb55ZbVbb7b 0.VVb72027b3778,Q.VVb83bVVVb55b.0.VVbV??Qlb3b58. 2 Q.VV715V428V735~0.VV73b784b7lVV,~,VV7b0b77?l352,0.VV78?b3Z7Z4VV. 3 0.VV81b4218b1S7,0.VV8bbb24l?47b,O.VV87b~2423835,0.VVV04ll?bb387. . 4 0.VVV2V334l85b3,~.V~V5l3b8b7237,O.VVVbV7505b35V.~.VVV84lb3bbb8l. 5 0.VVVV41b8V7b78,0.VVVVV352I22501

5 1 ~ 4 7 1 7 b 3 0 2 5 0 2 3 1 . ~ ~ 5 7 0 7 V b 3 2 b 7 V 4 V I ~ c... I) SPHEROID RADIUS A 1 C E N l E R O f EACH BLOCK.

C... DR D f P I V A l I V f Of I) Y R l LAl . DAlA

1- 0.0033 1730bb 1 15,0.003 I 8 9 2 5 0 ~ l l 1.0.002Vb3VQ~bVOb. 0.002b838li3208, 2 0.Q02322V7bb3b0,0.00lVl072?V2~2.0.O0l43b3l52502,0.O00V~0550blV6. . 3 0.0Q032b5?57432.- .000211001)LLO.- .~~~8bb0V7b843.- .00143QbQ00230~

bo

65

70

75

80

85

90

v5

100

10s

110

suenoui iut s t TUP CDC bb00 f 1 N V l . 0 - 3 2 4 OPT.) 0312617b 15.05.18.

c c... c c c . . .

C...

C c. . .

1 .99b.lOb4, 11~2,1ZO4,1Z~b, 1348,I420.14F8,155b, 1bZO. l b w 2 , 1744,1804.1908,2012.2092.2172,2228,2284,2344~2408,24~0,2552/

I N f L Q U l l l O N NUMBERS I N 1HE M I T H E M A T I C A L DESCRIPIION A R E 6 l V E N IN COLUMNS 72.. . ..BO

I M E T 4 GREENWICH HOUR AWGLI. S E I 10 0.0 IN THIS T E S 1 . 1 H f l A = 0.0 SINTH = 0.0 C O S I M 1.0 . IMU 8 1.0 IWlGl IN161 = 7 I Y l L * l N T G l - 1 l l Y l L = fLOA1~( IN IL ) D V R = 1.0 P I = fOURPII4.0 TUOPI = fOURPIIZ.0

NUMBCR OF P O l i T S IN NCYTON-COIES IUTEGlATlON.

COMPUTE IHE AREA f01 ANY SUB-BLOCK IN ALL 56 ZONES. DO b 2 I V = 1, 3b

CACt 2

. .

I I 5

1 2 0

5UBAOUIINf 5E IUP CDC bbOO f U VI .0 -524 OP1.l 05126176 15.05.18. PA6E 1

125

I c L A n i I D I L Z . 2 ) = - f c ~ ~ n i l ~ 1 ~ . 2 ) 60 CONIINUE

l H l S 5 f C I l O N LOAD5 IHE D E N S l I V VALUES IO l O O E 1 APPROI l lAlELV A CINIRAL IORCE f l f L D BY A COAIING ON AN OBLATE SPHEROID.

c . . . c

c . . . 7 5

SUBROUIINE D I I D I CDC bb00 f I N V1.0-124 OPl=l 0112617b. 13.OS.10.

SUlROUl INE D 110 I c C . . . D I I D I COIPUlfS 1Hf POIENIIAL AND AIIRACIIOW DUE IO A SURfACE C D f W S l l V ON AN O B L A l f SPHfROID. In€ DfWS11V IS PbRAIEI fRIZED DV 1Hf C USf O f 1640 fOUAL A R C h BLOCKS. E A C H hBOU1 5 OfCRC(: ON A SIDE. c--- 5

20

2s

10

1s

10

40

4 5

so

SI

c...

C C... C C

C O f f f ICILlfIS IOR NEWON-COlES NUlltRlCbL QUADRAlUII D A l A ALPHAIl.O84~O.O, 1 .O. I .O 1.0.3.0.0.2.0. I .o, 2.0, Se0.0.3.0.

I 1.0..9.0 1.0.0.6.0, 14.0 ~4.0.12.0.2~0.0.4S~0. 2 95.0 17i.O 210.0 2.0.0 580.0 41.0,Zlb.O 2?.0,1Sb.0.0.O8l40

b 19Sb.O: 2 1 S S Z .O: -17 I 2 .a, 4 1884 .O. -9000.. I 4 I?) -0. 5 ?S? l1 .O. 14 lbb9.0,9?20.0.1?409b.O, S2002.0.89100.01 . 3 m?:o z5oio.o 9i~1.0.~0021.6.0.0. i72~o.~,

DAlA~AL01~1.2B = 0.0) . (ALBl(2.2) = 0 . 0 ) DA1A(DRDLII~5)=0.0). (DZRDLZ(l)=O.O). ( f (4J.O.OJ D A I A (DZIDPL(1)=O.OJ

1.0.

C...

0101 RS

, b,? 0 9 10

1.2.1 4. s

PbCt

7 5

80

85

90

95

100

I05

110

c . . .

c . . . c . . .

t . . .

c . . .

c . . .

CDC bd00 l I N V3.0-124 OPl.1 0312bi?b 13.05.18. P U f

(1.11) (1.10 (1.8) (1.8)

(1.10 (1.14)

2

cc, 0

SUBROUlIYf D l I D I COC bb00 f I N V 3 . 0 - 3 2 4 'OP1.1 ' 03126176 '13.05.10. PAC1

115

120

l I 5

130

135

140

1 0

150

155 '

1 bo

lb5

c

c c c . . . c

c

c

c...

c

RS

II)

11.14) (1.151 (1.151 (1.151

1.19) (1.191 t 1.19) (1 .21) 11.21) (1.14) (2.151 11.121 (2.16) ( 2 . 16)

-,I 1.13)

12.191 (2.171

(1.20) c1.20) (1.20)

(1.171 (2.18) (2.0) (2.9) 12.10) (2.101 (2.10)

t2.1A) (2.7D) (2 .71 1 12.71)

3

SUBROUlIYl D l l D I CDC bb00 111 V 3 . 0 - 3 2 4 OP1.l 0312b17b 13.05.18. PAC( 4

(2.61) 12.41)

- . 170 A l l = A e S t i t l ) )

l f ~ ~ I 1 . 6 1 . 0 . 0 0 1 - A D l ~ GO 10 443 A l l = A 8 S I f l l ) ) lflAfl.G1.l0000.0~Afl.AYD.Afl.Gl.O.Ol*ADl~ GO IO 0 9

r 175

180

185

I90

200

c . . .

c c.. . 1 439 c

1001 f

205

2 10 c

215

220

I I .

CllON l iSSUMfS f = 0. I6

12.54)

CDC 6600 I1N V 3 . 0 - 3 2 4 O P l = l 0 5 1 2 6 1 7 6 lS.OS.10. PACf

SUBROUIIYI DllDI CDC 6600 llw U3.0-324 011.1 0312b17b 13.05.10, pact b

280

285

290

295

300

305

3 10

3 1 5

120

S25

110

BARGZ 8 S O R 1 4 1 . 0 ARG2..2l DASH = ALOG~(ARG1 8ARG111lAR62 BARGZI I fZ.32)

f A R G 2 = l.OIBARG2

G R A S H l ~ , 1 B l = fAIG2~61RClZ.IB)

GO 1 0 ' 4 4 7

If~ABS~~RGll.LI.l.O.AYD.A8S~ARG2l.Lf.l.O~ GO IO 449 '

l l l F ~ l ~ . G ~ . O . O l GO 10 4 5 2

PRIM1 1002, 00. D l l ) , I l l ) . F ( 1 ) GO 10 4 7 1

I1 = Z . O * ( R l f * R I l I I O * D € L L A I I l t(l) A 2 Z.O*IRlF~RlZ - fll)mD€LLAII) Ill) lfS1 = lllAZ Il~ltSl.6l.O.OJ GO 10 451 m i n i 1003, 11.12

1003 FDR11A1(31. 4HARG6 , ZLZO.12) GO IO 4 5 3

DASH 3 A L O G ( T t S 1 I DO 454 18 = 1. 3 IBP = 18 * 1

GAR6Cl1,IBl = 2.0*lGRRlVl 1 8 ) . R l l R1T*CRlIl.IB) * F ' l D L ) * C l l O P l GAR6GlZ,lBl = 2.0*lCRR1Il l B ) * R 1 2 RlF*GRIlZ.IB) - FIDL)*tllBP) G R A S H 1 1 , l B ) * C A R C G l 1 . 1 B l 1 A l 6RASHI2,lB) * GARGCl2,IB)IAZ 50 10 447

B A R 6 1 * S Q R l l l . 0 - ARCl**21 BAR52 * SOR111.0 - ARC2.*2) XWUR A R G l * B A R 6 2 - A R C Z * B A R C l DLYO = IRGl*ARCZ O A R C l m B A R 6 2 DASH = AIAYZ~XYUI,DtYOI c2.3S) . 60 10 448

F f F ( l l 1 * O R I F * D A S H 12.24) DO 444 10 * 1. 3

1 CRORIFlIBl*DASH O R l F * ~ C R A S l l ~ l , l B ~ - C R b S H ( ~ , I O ) ) 12.45)

4 4 8 C O N ~ l Y U t 11161 = 1.OIBARGl

DO 1440 IB = 1, s GRASWll.lB) 8 l A R C l * C A R G l 1 . 1 B 1

1448 COYllYUf

c. . . 446 COYIIWUI

453 CO*IIWUL 1 0 0 2 FOR~Al(61.6fZO.lOJ

452 COYlIWUI

c... 4 5 1 COWllYUL

. ~ I D L = f(iBP)-otiLan

454 COYIIYUE

4 4 9 COIIIYUf c...

C . . . 447 tOrllYUt

CRADI(II.1Ol 12.44)

444 COYllYUI

419 COYll~Ul C

12.31)

3 4 0

3 4 5 (1.16) (1.18)

IU~IOUIINI DIIDI . CDC 6600 #IN V 3 . 0 - 3 2 4 0Pl.1 01126176 l1.OS.18. PAGE

C. . . N I Y I O N - C O I I S NUMERICAL OUADRAIURI. DD 670 i e = I . 4 DO 476 NI = 1. I N I A C 1 INlNl * I N l 6 1 V - NI

I35 I I C N ( l B 1 = lfCN(18) t AlPHllNI.INlGlI~~fDI~Nl,l~I t ~ D f l l N l M l , l 8 ~ ~ 1 * f A K I O R

60 IO 999 476 C O N l l N U I

c C . . . I W I l A l l O R S I R I I I MI IHOD. C

601 CONIINUE D l Z f C = 6111 - DRSDf l . *Z D1Z fC2 = 6 Z Z A - DRIDLM.*2 DRSZDf = 0 5 I I * D R Z f C DRSZDL = R 5 1 l ~ D R Z f C Z D f l P H l = 0111) DlDP = D Z l ( 1 ) DPHIZ * D I l ( 1 ) 2 Y O l f * 12.0 * (0.1 - O . O Z 3 8 0 9 ~ 2 3 8 I ~ D P H l Z ~ ~ ~ O P H l 2 . ~ 2 DLDPI = DLDPI3.O

I50

c... COMPUIL f . c n

17s

380

185

360'

16s

170

7

suenouiiwE DIIDI CDC 6600 1 1 1 V3.0-324 O P l r l 0312b176 13.05.18.

400

405

4 10

415

420

425

4 30

435

4bO

(1 .51A l (1.510) (1.3211 f 1.320)

460 C

DO 410 J J = 1. 2

fCOIP(11.JJl 0.0 DO 410 KK a 1- 4

DO 410 I 1 = 1, 4

SDZ = 0.2*SINPHI.DPH12 DO 470 IVORV = 1, 4 f conp( i vonv ,o =

1 fCOIP(lVORV,11 I D 2 fPHI(IV0RV) 470 coniinur

C

(1.23) C 1.ZSA)

PA6t 8

445

450

LUBIOUIINf DIlDl

799 CONll*Ut

999 CONIIYUI c r

4 7 5 c a c . . . na

rr I f II II I f I f R f I N

C D C bbOO tlW V3.0-324 011.1 O 3 I Z b l ? 6 13.05.18. ?ACE P

w -1

l.o'4?000 0.000000 282.000000 .V553855701E*00 -.18VV349292f *OO .8V35121503t *OO .1574bObb3Of-O4 .V134?64585€*00

-.27571246?01-01 .27064VO?lOI-03 -.1211867LSS1-02 -.157~b06630€-04 -.1241b5b575€-02

1.047000 1.000000 282.000000 .V553824V50E+00 -.18V862VVOVE*OO .8V~1430484t+00 -.lb05287805t-01 .Vl324lS645€*00 . . .V5510V81?6E*00 -.18Vb353V33E*OO .8021643812C*OO -.15V20bV253E-O1 .V12234801@€*00

-.27215736b2E-03 .. .227597644OL-03 d.V78b672731E-01 .132185515bL-03 -.100b?b2b01€-02

.V551OV817bf*OO -.18V66428OlE*00 .8921002826E*OO -0. .V122348OlV€*00

16S(DG) = .12418f-02 DffLECIIDN .00127173? DEG. l l H E = .4630 SLC.

ABSIDG) .10134E-02 DEfLEC110N = .00?2915bV DIG.' I l H E .4b20 SIC.

1.047000 3.000000 282.000000 .V55382VO51+00. -.18Vb645347E~OO .8V22345418E*OO -.48DbO?b307L-01 .Vl1435?943€~00 .V5510V81?bt*00 -. 18040435lef to0 .8VlO774lb3f 400 -.47742b8068E~01 .Vl22148OlVE *OO

-.27307?8b~?f-01 .2601828V53E-03 -.1157125552t-02 .3180823Vb3E-03 -.120OVV?374€-02 A B S ( D G ) = .1227Vt-02 DEfLECllON = .016055118 DEC. I I H E = .bblO SEC.

1.047000 5.000000 282.000000 .V553?VVV4lE*00 -.18VZ4153bbE*OO .8V01V38100E*00 -.?V?6?b7724E-01 .V15575S3b5€*00 .V5510Vd37bI*00 -.188V425502I~OO .888VO481OVf*OO -.?95Ob501?2E-O1 .Vl22348OlV€*OO

-.27015147VOE-03 .29898b1185E-O3 -.1288VVVlVOI-02 .2611755lbVL-03 -.134073b585€-02 A6S(DG) = .15487E-02 DEfLEtllDN = .009214510 DES. l l H E = .4710 S I C .

1 .Ob7000 7.000000 282.000000 .V55370522bL*00 -. 1885007708Er00 .88b773?083I*OO -. 1 1 134874B?E*00 .Vlf4Olf83~€*OO .V55lOV817bL+00 -.1882505511t+00 .885b4V2118E~OO -.llll7345b3E*00 .91223b8019€*00

-.ZbO684V782E-01 .25921V44OZE-03 -.11244Vb44VE-02 .1752V238?61-03 -.llb65814b1€-02 AiBSlDG) = .1167?E-02 DffLICllON = .DO2432695 DIG. T I H E .4630 SEC.

1.047000 10.000000 282.000000 .V553bl234bE~OO -.1870V8Vb64(~00 .8800540?48E*OO -.158V16b314f*OO .V11bbV6558€*00 .V551OV8376E~OO -.18b7828535E*OO .878?4423b5t~OO -.15840?01101*00 .9122~~8019€*00

'.251SVb782Vf-01 .3161128b70E-03 -.1300838253E-02 .5087204862f-03 -.141485~851€-02

1.047000 11.000000 282.000000 .V55335620bE~00 -.18502015V3E*00 .87040112V1E+00 -.205VV24226f*00 .V133820b33€*00 . V I 5 1OV837bE rOO -. 1848031V6Vf 400 .8bV430b844E 400 -. 2052081805E *OO .9122348@19€ *OO -.2257829270€-01 .225V621V84t-03 -.V?04447528E-03 .7842420937L-03 -.11472b1357€-02

1.047000 20.000000 282.000000 .V552?bOV25E*OO -.1785044555E*00 .8391098?83f *OO -.~110b07?28E*00 .VlS682bSQO€*OO .V5510V8376E*00 -.1782261244f*00 .83848?QVllE*00 -.1120026?771*00 .Vl2Z3480lVE*OO

-.lb62548208f-03 .2783310858€-03 -.112188?011E-02 .10580V50VOL-02 -.1447b5?041€-02

16S(DG) = .14401E-02 DEfLfClIDN = .01bV08847 DEC. l l H E = .4?60 S I C .

ABS(DG) = .126801-02 DEfLfClIDN = .033897945 DES. l l H E . = ,4700 SEC.

A B S t D G ) :15b711-02 DffLEClIDN = .037b52564 DtC. llHI = .bo50 SEC. .

1.047000 25.000000 282.000000 .V5521880841*00 -.1721154338E*OO .8OV54283bVf*OO -.3868174942E400 .Vl357OS233€*00 .V551OV8376€+00 -.1718V4214OE*00 .~086V86V~?L*00 -.38552?0863E*OO .912234801V€400

-.108V?07988E-03 .22121V?87~E-03 -.8441421b2Vf-O3 .12V040788VE-02 -.1315?2116V€-02 ABS(DG) = .15578E-02 DffLfClION = .050308338 DEC. I I H E = .so00 SEC.

1.047000 30.000000 282 .OOOOOO ,9551582216E to0 -. lb437556726 600 .?734bO854b6 *00 - .457490VOV?f 400 . V 1~5419822L 4 0 0 .V551OV8376f*OO -.1642540848E*00 .772754712?€*00 -.45b1174010t*00 .V12214801V€*OO

-.48383VV8OOC-O4 .12148248VZE-03 -.?Ob1418V4Vf-O3 .1373508704€-02 -.130?180224€-02

1.047000 35.000000 282.000000 .V5505V55Obt~00 -.15543Ob?8VE*OO .71124?4790E*00 -.524704V054€*00 .Vl3343?S20€400 .V551OV837bf*OO -.1553638828f*OO .71OVZVb008E*00 -.S2321638b8L*OO .V12234801VE*OO .502870053VL-04 .b67V61bV83f-O4 -.3178782245C-03 .14b15185VOE-02 -.110bV5003VL-02

ABItDC) = .15492E-02 DfiLfC11ON .05218035V DIG. I l l € .53?0 S I C .

ABS(D6) = .15040f-02 D E f L E C l l O N .Ob3??522b DEC. l l H E .5910 S I C .

I

I .047000

1.047000

1 .047000

1.047000

1.047000

1.04 7000

1.047000

1.067000

1.047000

1 .Ob7000

1.04 7000

pn I

40.000000

42.500000

50.000000

55.000000

bo. 000000

b5.000000

70.000000

75.000000

80.000000

85.000000

89.500000

LlMBDA U U I

282.000000 .V5bV4201V?f*00 -.lb51VVbb3II*00 .V55lOV837bf*00 -.lb52912b78f*OO .lb?817V317f-03 -.Vlb247b483t-Ob

282.000000 .V5b8897742f*OO -.13V7081b481*00 .V5510V83?bt*OO -.OV8351753E*00 .22OOb345OOf-O~ -.12?0305147f-03

282.000000 .V54738507Vf*OO -.1217bV3b8Vf*OO .V55lOV837bf*00 -.12lV138bV21*00.

. .37132V77?bf-O3 -.1443002V?3f-03 A B S ( D G ) .88375L-03 DtlLIClIOW

28?.000000 .V5Cb314808f*00 -.10853b0028f*OO .V5510V837bf*OO -.10878bVb1Vf*00 . .b?835b8742f-03 -.2508bVO41bf-O3

ABSCDG) = .12423C-02 DflLfCllOM

282.000000 .05b5182730f*OO -.945V7b4085€-01 .V551OV817bi*OO -.948321bOO5f-O1 .5715bb604bI-03 -.214bVVlV411-03

282.000000 .V5b4440205L*OO -.?V8b3587&8€-01 .V55IOV837bI*00 -.8015558837f-01 .bb5817164bf-O3 -.~12000888Of-03

282.000000 .V543b8627?t*00 -.bbSV879024L-01 .VS510V837bI*00 -.bb8bVOO42bf-Ol .7418OV8V38f-O3 -.27021402bOf-03

262.000000 .V541051lVlf*00 -.488bZ183bVf-O1 .V5510V837bf*OO -.4VO887278bI-O1 .8Ob5185bO3f-O3 -.22b54b3782f-O3

ABSCDC) = .10078f-02 DIfLLClIOM

ABS(DG) = .V3468C;03 DffLfClION

ABS(Dt1 = .11410f-02 DffLLClION

ABS(D6J .1b783€-02 D L f L E C l l O W

ABS(DC) = .18704L-02 DIfLfCllON

ABStDC) .21131f-02 DLfLfClION

-.3275bO75bO€-Ol -. 3203415b61L-01 282.000000

AOStDG) = . Z C O V I E - 0 2 DIfLICllOM

282.000000 .V5b23S235bL*00 -.lb4111I147I-O1 .95510983?b1*00 -.lb53033120f-01 .8746020554f -03 -. llVlV9737St -03

282.000000. .V542215221I*00 -.lbb22188Ob1-02 .955lOV837bf*OO -.lb55112073t-02 .88831555V8f-03 .710b7328bbI-O5

ABStDG) = .21471E-02 DffLtCIION

ABS(DC) = .24248E-02 DffLECl10M

uv u t 6

. b 8 306 S 5 1. f * 00 - .58 7267 1690I *OO . 0 121 53 5 52bI e00

.b835416733f*OO -.58637322781*00 .912234801VI*OO

.45b151Ob42f-O3 .89394121231-03 -.218750435bI-03

.bS72b2ClllI*OO -.61b9V33bObI*OO .91224b1950E*00

.6578?277bZf *OO -.6lb29b89921*00 .V122348019f e00

.blO?V24832f-O3 .6V6bb11372€-03 -.l13V3027ZO€-Ob

.5?2870?25bI *OO -. bV935bVV4VI *OO .9122050290I e00

.573559565VI*OO -.6988124008f*OO. .V12234801VE*00

.b~O8bO3438i-O3 .54bSVbObVbI-O3 .2V?72V7521L-O4

. S lObO32Ol3I e 0 0 - .747464b917f e00 .V117010~19E*00

.5 1 18024 IbbI *OO -. 7b725VOO2V~~OO .91223480191*00 rllV021312bt-02 .205b8883bbf-03 .5337100253t-03

8 .061779801 D I G . VINE = .bl?O S I C .

= .058700702 D I G . IIMI = .b2VO S I C .

= .055475835 DIG. ll1I = .bt10 S I C .

.O?Ob785bb D I G . 1111 = .bWO SfC.

. bC505905b6I e00 -. 7897769920I e00 .911bb8b5lbI*00

.b411501414L*OO -.7V001651271*00 .91223b8019t*00

.lOVOl8b7?6L-02 -.2415206V4Vf-OS .7bb150532bt-03

. Jl5bb 3 10bbL +OO -.825V97577bI e00 .910V087432I e00

.14592900b~L-02 -.767V270500t~03 .lS2bO5871bf-O2

.3038V41b25L*00 ~.8558Vl?V93L*OO .91053b4b4bE*00

.305184670bC*OO -.8572203ll8€*00 .912234801VI*OO

.1288508125L-02 -.132851252bI-02 .lbV83373b6E-02

.2298648795f *00 -.87934873 l?t*OO .9102085?98I e00

.23094430711*00 -.8811511548E*OO .9122348019E*OO

.1O?Vb27bb2f-O2 -.1802b23lb9L-02 .202b2721b0€-02

.1540847V51I*00 -.8V61330819E*00 .9OV873517bE*OO . lS4Vb4318OL*00 -.8983?59055I*OO .91223b8019L*00

.861522937Ot-O3 -.22428235VZI-02 .23bl48b34bI-O2

.7?18605b89f -01 -.VOb?004925E~00 .91012~V00bI*00

.777bV093901-01 -.V087b3473lt*00 .*122~b801~E+Oo

.583OJVO167I-OS -.20b29805981-02 .2lObVO13SbI-O2

.77V9724483I-O2 -.9097?53012I*OO .9098102585E*OO

.7?866900931-02 -.V122000168I*00 .?lZZSb8OlOI*00 -.lZO34389bVt-Ob -.Z42b?b0585E-02 .ZbZ~SbI~ObI-02

= .OS3117521 DLC. 1111 = .bVbO S I C .

.3?71023Vb5f *OO -.82b76550bbf *00 .-9122348019E*00

= .064655576 DEG. 11111 = . I340 SEC.

.0~.?25575Z DIG. l I I E ' = .7750 S I C .

= .OS7697291 O f C . ll1L = .Ob90 SIC.

= .050015763 DEC. ll1f = 1.0420 S I C .

= .024002464 DE6. Il1t = 1.1oso S I C .

= .0022bbb25 DLC. IllI = 1.102g SIC.

I )

I . 04 1000

1.300000

2.100000

1.047000

1.047000

1.047000

1.047000

1.047000

1.047000

1.047000

1.0&7000

PM I LAneoA U U I

VO.000000 282.000000 .V542214bZlf*OO -.12152Vl5b3t-l! .V55109837bf*00 i-.b4lZV3OVb9I-l4 .88837550?7f -03 .573VV82112t - 14

ABSCDG) = .231V4t-02 DlfLfCllOW

YO.000000 282.000000 .7b8718b783f*OO -.bO3805318Vt-l4 .7bV2307bV2f*OO -.41SV711b24L-l4 .5 120VOVZ4b€-O3 .18783415b4t-l4

ABSIOG) : .10V?lt -02 O f f L E C l l O W

VO.000000 282.000000 .47b13VbOOlf*OO -.200831130bf-16

.5087bO47b8t-04 .4142268063C-l5 t .47blV047b2t*OO -.15V40845OOf-14

LBSCDG) = .b13VOt-O4 OtlLlC110N

-5.000000 282.000000 .V5537352851*00 -.18VlV20585E~OO .V551OV8376f*OO -.188V425502€*00

-.2b3bV08899I-O3 .24950824b8t-O! 1 8 5 1 O G ) . = .112191-02 , DffLlCllON

- 15.000000 282.000000 . V 5 5 3 0 6 5 3 V 4 l 000 -. 183380bV08t*00 .955109837bE*OO -.183201b2b5€*00

-.lVb7017515t-03 .17VOb43404t-03 ABSCDC) = .V3720f-03 OtfLlClIOW

-35.000000 212.000000 .V55031S708E*00 -.1552V11152E*00 .955109837bf*OO -.1553b38828€*00 .784bb7V771t-04 -.727675b0431-04

-60 .OOOOOO 282 .OOOOOO .PI45 172421f to0 - .V453V 19950E -01 .V55109837bt*00 -.Vb83214005E-01 .5925951015t-O3 -.ZVZ94075071-03

A8SIOG) = .1Sbb4t-02 DlfLtCllON

-75.000000 282.000000 .95429Vlb70f*OO -.4887031010t-01 . .VSSlOV837bf*OO -.49088?278bE-01

.810670b502€-03 -.218bl77b53E-0! A B S ( D G ) = .213751-02 DLfLLCllON

-~O.OOOOOo 282.000000 . V ~ 4 2 Z l b b 2 1 t ~ 0 0 - . lO2O?l IZ93€-1~ .V5510V837bf*00 -.b41203OVbVt-l4 .8883755077t-03 .37V4181Vb5€-lb

A B S ( 0 G l = .82088t-03 DffLtClIOW

A d S ( D G ) = .23194f-02 D f f L ~ C l I Q N

bO.000000 2b7.000000 .9545381120t*00 .2381?1825bL-01 .955lOO837bt*00 .2387134014E-01 .571525blV4t-03 .5415777bVSt-Ob

, 1BStDG) * . .115771-02 . DtfLtClIOW

bO.000000 285.000000 .054S385308t*00 -.1177?504bSE*00 .95S1098376t*OO -. 1180518702E*OO .5713067V22€ -03 -'.27b8237bWt-O!

AEScDG) = .1158lt-O2 D t f L f C l I O N

U I uz C

-.2Ob8808Vlbt-O4 -.90VV1549VVt*00 .V09915S0016*00 ..301704b813f-13 -.9122348010t*00 .9122348OlVE*OO

.ZOb8808Vllt-Ob -.231V5020461-02 .231930181lE-O2

- . 1 ~ 8 ~ b O V l 8 4 t - O 5 -.5VObl88~48E*OO .SVOb188S48E*OO .195bVV04551-13 -.SV1715V7b3i*OO .591?159?b31*00 .158~~OVZObt-O5 -.10971215231-02 .10971215211-02

= .001302bV3 D t C . 11111 = .8580 SEC.

= .000153bOb 016. r in t = . w o stc . .4V13bO85071-1b -.22bbV5V797t*00 .ZZbbVS0?9?1*00 .7~VV577V16t- l4 -.22b75?3696I*OO .226757f6VbE*00 .258SVb9428t-l4 -.bl3899027?f-Ob .bl38VV02??t-01

= .OOOOOOOOO D U . l i n t = . i7bo s t c .

.88VVb7SZb7t*00 .7V7741lbZVt-O1 .V133b5358lL*00

.888VO48lOVt*OO .795Ob50172t-O1 .V122348OlVE*OO -.lOb2715862t-02 -.Zb7b1456711-03 -.111055b200E-02 = .010854275 D t C . rinf = . ~ 7 4 0 SIC.

.8b2541774lt*OO .23b7588020t*OO .913050b6b8f*OO

.8b18VS88801*00 .23b1037403f*00 .Vl223b8OlVE*OO -.b4588bOP52t-O3 -.b55061bS24t-03 -.8158b484S7f-03 = . 028V5bOZZ D t t . 11111 = .LO10 SEC.

.?30627b8bOf to0 .5239962bV 1ttOO .V lZI lbb894t*OO

.7~092Vb008l*OO .52323638681*00 . .V1223b80101*00

.301V147822t-03 -.7598b?3188t-03 -.1818874807f-O3

.444827?443t*OO .78V231732%*00 .V108?bSbl l t*00

.13223V7lO3t-02 .78b78OZlb4t-OI .15582bOb72t-02

.OS0271152 Dt6. l i n t = .bo70 s t c .

.44b1501414t*OO .7V0018512?t*00 .VlZZ34801V1*00

= .04V04112Z Df6. l l l l t = .7010 S I C .

.8792146522t *OO -18 1 1S 1 1 S481 4 0 0

.229925bV70t*OO

. lOl8b 101V3t -02 .18bbSOZb23i-Oi 2072~80~53~-02

.20b88088VVf-Ob .VOVV15~V9Vt*OO .V009155001t*00

.101704b813t-l3 .9122348010t*00 .V122348019E*00 -.ZObI80889bE-Ob .2119302Ob4t-02 .231V301808E-02 = .001302b93 DCC. 1lHf = . I 7 7 0 SEC.

.SS43b441b?1*00 -.78V?bfZIQtt*OO .~114~VSlbVE*OO

.455bV2307Vt*OO -.7V00185127t*00 .V122348019t*OO

.11278V1242i-O2 -.2552V46414t-O~ .78528500311-03

.43V48OVVb2t*00 - .78V?b~b4lLt*OO . 9 ~ 1 4 ~ ? b 0 8 ~ E * O O

.bbO5?55??bt*OO -.7900185127E*OO .V1223b8OlVE*OO

.109b5811Vb1-02 -.2578712990i-03 .78?3V15318f-03

.230944S0711*00

= .03203bVbl Dt6. l l l lt = .8890 SEC.

= .051CbVb24 D f C . l l l l t = , .7180 S t ( .

= .0533b3?03 Dt6. IlMt = ..?ZOO s t c .

1.157000

1.157000

1.157000

1.157000

1.1~7000

1.157000

1.157000

I . 157000

l.157000

l.l5?OOO

pw I

00.000000

90.000000

89.9b8750

89.937500

89.875000

Ob. 000000

70.000000

55.000000

4 2 . bOO000

30.000000

1s .oooooo

LIRBDA U U I U 1 ut 6

208.000000 .95453778511 *OO. -. lC0595Cb391 *OO, .4127274578f *00 -.78V?b2,170bitOO .Vll4S18218i~00 .9551098376f*OO -.lCOQ480283f*00 .4317934Zb4f*00 -.7900185lZ?fr00 .Vl22348OIVf*00 .57205229231-01 - .352566b238f-03 . lOb59b8552f -02 -.2Sb3420592i-03 .?82V8Ol?l2l-O3

A B S I D G I .115171-02 D E l L f C l l O W a .05306bb31 DfC. 11111 = .?OS0 s i c .

282.000000 .8b3bZO?84lf e00 -.832b362b?lf-14 -.571b8179021-05 -.74544?7874ftOO .745447?875f e00 .804304235lf*OO -.52514QbO74f-l4 .2470b3559bf-l3 -.74?0218108f.*00 .?b?0218108L*00 .88345103251*03 .107b8b459?f - 1 4 .57 168 179276-OS -. 15740233SSi-02 .15740233311-02

A B S ~ D G J = .15#40f-02 D L f L I C I I O W = .000439400 D f C . 1 1 N i = .7bbO S f C .

282.000000 .8b3b20?8781*00 -.843b74Qbb4i-04 .391201OOb5I -03 -.?454C??132i*00 .745b4?820bi*OO .8b430423511*00 -.84?109S155f-04 .3V85336932f-03 -.?4?021bVV?it00 .74702181081~00 .b83447253SE-O1 -.J4345580?8f-Ob .7332686?15f-05 -. 157398b518f-02 .lS?3V9020lf-02

I B 5 l D G l = .15740(-02 D E l L f C I I O W = .00049921b DfC. l l l l f = .Ob70 S I C .

282.000000 . 8b3b20192 11*00 - .1b8?$4Q9 1 I f -03 .?E8 1 188805f -03 -. 7 ~ S 4 4 7 4 Z O S i *OO .?bS44?8Sb Ii e00 .8b4304235lf*00 -.lb92218?79f-03 .?970b?2b791-03 -.7470213bb4f*00 .74?0218108t*00 .b834429732f-O3 -.6868862235f-Ob .894818?397f-OS -.15?394597Of-O2 .15?3054?02f-02

282.000000 .8b3b2080221*00 -.33?4b98983f-O3 .158l95433lf-02 -.?4S44bl?03i*OO .?bSC479S42i*OO .0bL304235lf*OO -.338843S542f-03 .1594133587i-02 -.7470200130f*00 .?4?0218108i*00 .b834329097f-03 -.1373655888f-05 .121?9ZSbbOf-O4 -.157~85373Vf-02 .l5?38?bSSSi-O2

282.000000 .8b3bZSb330E*00 -. 1019205 15VL-01 .50766829851-01 -.?43b8lSV4bf *00 .?4S4902789f *OO .8b43042351f*GO -.108341~b5?f-O1 .5097088739f-01 -.74520210321*00 .?470218108f*OO .b?8bO212791-03 -.421449?773f-O4 .20405753l6f-01 -.lSZO?O8b55f-02 .I53l5SI853f-02

A B S ( D G 1 * .15740f-02 DElLiCllON = .0005SV27b D f C . llRf = .9510 S f C .

1 9 S t D G ) : .157391-02 D E F L f C l I O N * .OOOb79857 D I G . t i n f = . v n o s t c .

A B S ( D 6 I = .15349f-02 D L f L f C l l O N = .007822539 DfS.. i i n i = . a 0 0 ILC.

282.000000 .863726422?6 *DO -.52V40242blf-O1 .8b430&2351f *OO -.5312071073f-01 .5778123bb?f-03 -.1804601148f-03

I B S ( D C ) = .138131-02 D E f L C C r l O N

282.000000 ' .8b39184090f * 0 -.888?b7?2Obf-O1 .8b4304235lt*80 -.890847?b29f -01 .38S82bO44Of -03 - .20800623 I81 -03

IBS(D6J .10421f-02 DLILLCIJON

282.000000 .8b411?96811 a00 -. 1141815718f 400 -8643042 3 3 1 f (00 - - 1 IC 3266OOVf 400 . IEbZbb?Vb5f-b3 -. 14SOZPO757f -03

282.000000 .8b42950633f *OO -.13 475IO48i*OO

.9171793b43fa05 -.512Sb42834f-O4

l iBS1DG) .7477bf-03 DEfLfCllOW

.ab~10~zs5tt *oo -. i3tsobs613t *oo A 9 S ( D C ) .57b181-03 D C F L f C l I O N

.24VObO8405f*OO -.7008VV0323f*OO .?4S?lb?lVOL*00

.2499132V51f*00 -.?019?088S2f*OO ..7470218108i*00

.852454bO?SL-03 -.1071850904f-02 .1305OVI?V5f-O2 = . o M ? ~ ~ ~ s s ots. . r i m = . w o s i c . . . .4I8131~1024i*OO -.bllb3b??V2i*OO .?4b211Vfllt*00 ~41Vl10V208f*00 -.blIVZ44b34t*OO .74702l8IO8i*OO .V79818398bf-03 -.287bb4Z4Zlf-03 .8OV8??bV?3f-O3

.53718?802bt*00 -.505V2Sb242i*00 .?4b705b?ZOE*00

.5378b43690E*00 -.SOSb4lllZbi*OO .?b10218108f*OO

.b?b5bb3bZlf-O3 .2845ll6435f-OS .3lb138804bf-O3

= .050331429 DES. l l l l f = . S W O s i c .

P .OS198Sb40 016. llnf = .5S20 S i c .

.b326488299€*00 -.l?LObS2LSbE*OO .7471b12S?81*00

~8~.000000 .8b4b22lV'l3f *OO -. l5OlObl488f *00 .?Ob2041 14VL*00 -. lV3702V018t*00 .74?~37171U*OO .8b4304235~f*OO -.1500223522i*OO .70579Vb752i*OO - .W33b34? I7 i *OO .~b?02181081*00

-.11705b21596-03 .83?0bbZZOlE-O4 -.404430b8V6f-O3 .44VSZOO384t-Ol -.5153bbSb18€-OS

e 0.

1.157000 5.000000 ;82.000000 .86bb66bO93f*OO -.15b8b7498~f*OO .7285280310f+00 -.bS2721b42Of-O1 .747b572064f*OO .8bb30b2351f*00 -.I5b7235489f*00 .7279170b70f*00 ~-.b5107240?71-01 .74?0218108t*OO

-.16217kl88If-O3 .1239b99bb51-03 -.6109bb0207f-03 .1bb9234299f-03 -.b35393b04lE-03

1.15~000 0.000000 282.000000 .8Ckb?135bOf*OO -.155bbb393Of*OO .7313304?47t+00 .3153818584f-05 .?476b78558E*00

-.lb?1198113€-03 .1298252273t-03 -.b328827733f-03 -.3153818584f-05 -.bbbO449b9bf-03

1.07000 -5.000000 282.000000 .8bbb655987€+00* -.15b8C5095bf*OO .728517b891f*OO .b5277595751-01 .747bLb9087C+OO .86b30b235lf+00 -.15b7235b89f+OO .727917Ob7Of*OO .b510?2b07?E-O1 .7470218108E*00 -.1613b356b9€-03 .1215bbb787C-03 -.b0042213161-03 -.1?035497b?f-03 -.b2509?9062E-O3

.1OOOOOf*O~ b5.000000 282.000000 .1000004b13I-02 -.87867bb4QbL-O7 .4133848bZZt-06 -.90631178bbC-Ob .100000~L12E-05 .1000000000f-02 -.8780727737E-07 .4133830387f-Ob -.901307?870f-Ob .1000000d00E-05 -. b4 I3349Sb?(-O8 .38?5885018t-!Z -. 1823bbOb991- 1 I .39993b4557f-11 -. C41249931Of -11

bBSID6l = .64b861-03 D f r L f C l l O N , .008b38b38 Of&. llUf = . .4lbO S f C .

.8643042351f*00 -.15531b5b78f+OO .7306975919f+00 -0. .~b702181081 *Ob

1 B S I Q 6 1 = .6b607€-03 D f f L f C l I O N .OOOb27121 DES. 11UE = .4180 S f C .

b B S f D 6 J = .635851-01 D E f L f C l l O N = .008925601 OEC. l I U f = .4140 S f C .

M S f D G ) = .bblZSf-ll D f r L f C l I DN = .000000038 D f G . IlWf = -1190 SEC.

0