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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NOS 65 NGS I .The statistics of residuals and the detection of outliers. Allen J. Pope, May 1976, 133 pp.

NOS 66 NGS 2 Effects of Geoceiver observations upon the classical triangulation network.

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NOS NGS-4

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Robert E. Moose and Soren W. Henriksen, June 1976, 65 pp.

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Adjustment of geodetic field data using a sequential method. Marvin C. Whiting and Allen J. Pope, �Iarch 1976, 11 pp.

Reducing the profile of sparse symmetric matrices. Richard A. Snay, June 1976, 24 pp.

National Geodetic Survey data: Availability, explanation, and application. Joseph F. Dracup, June 1976, 39 pp.

Determination of North American Datum 1983 coordinates of map corners. T. Vincenty, October 1976, 8 pp.

NOAA Technical Report NOS 67 NGS 3

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

Foster Morrison

Geodetic Research and Development Laboratory National Geodetic Survey

Rockville, Md. March 1977

u.s. DEPARTMENT OF COMMERCE Juanita M. Kreps, Secretary National Oceanic and Atmospheric Administration Robert M. White, Administrator National Ocean Survey Allen L Powell, Director

Blank page retained for pagination

Co nte nts

p ... Abstract. . . . . . . . . . . . ........... .................................................... . ........... 1

I. Introduction. .. . . . . ..... . . . . . . . . . ... . . . . . . ...... . . . ...... ...... . ............ . .... ... 1 II. Methods in use.. . ... ... . . . . . . . . . ... . . . . . . ... .. . . . . . . ...... ...... . . . . . ... . . . . . . . . ... 1

Ill. Approximation by Taylor series ... ...... ... . . . . . ;...... ..... .... ............... · 2 IV. The method of singularity. matching... . ... ..... ... ... ......... ...... ......... . 3 V. Programming of the a1gorithms ..... . . . . ... . . . . . . . . �........................... 5

VI. Program testing .. . ... ... .. . ... . . : ................................................... 8 VII. Applications. . . .. . . . . . . . .. . ... . . . . . . ... . . . . . . . . . . . . ... . . .... . . . ..... . . . ... . ...... .... 9

VIII. Constructing the equal.area blocks ... ... .... . . . ... . . . ......... ..... ......... 10 Acknowledgments .... . . . ... . . . . . ,........................................................... 13 References... .. . ....... . . ........ . ..... . . . . ... ..... .... . ... . ..................... . ........ ... . 13 Appendix I. Notation ... .. . . . . . . . . . ..... ...... . . . . . . ... . . . . . . ... ... !........................ 14

Appendix II. Detailed mathematical developments.. . . .. ... ... ... ... ............ 14 1. The Taylor series method... . . . . . . . . . . . . ... . . .... ............... 14 2. The method of singularity· matching . . . ..... . ... ...... . . . ... 17 3. Point mass and numerical cubature algorithms........ . . . . 21 4. Annotated computer listing. . . .. . . . . . . . ... ......... ...... ...... 22

iii

FIGURES

Figure 1. -The block over which �T is evaluated. The origin is in the ... ce!lter . ..... . ... . . . . . . ... ..... . :.............................................. 3

Figure 2. -1640 5° equal·area blocks. Aitoff equal.area projection ... . . . . . ... ;.. 5 Figure 3a. - Equal-area blocks with index numbers. NW quadrant..... . . . . . . . . . .6 Figure 3b. - Equal-area blocks with index numbers. SW quadrant..... . . . . . . ... 6 Figure 3c. -Equal-area blocks with index numbers. NE quadrant... .. ... . . . . . . . 7 Figure 3d. - Equal-area blocks with index numbers. SE quadrant... . . . . .. ...... 7

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

latitude... .. . ... ... . . . . . . ...... . . . . ..... . . ... . . . . . .. . . . . . . . . . ...... . . . ... . . . 8 Figure 5. -The semi-major axis of the orbit of a particle moving in a central.

force field modeled by the surface density algorithm and equation (22l.... .. . ... ... ...... ... .......................................... 9

Figure 6. - The eccentricity of the orbit of the particle ... ... . . . . ... . . ... ... ... ... 9 Figure 7. - The inclination of .the orbit of the particle ......... :.. ...... ...... ...... 9 Figure 8. - Idealized set of equal-area blocks at· pole.... . . .. . . . . . . . . . . . . ...... . . . . . . 11

TABLES

Table l. - Zhongolovitch 10° equal-area blocks for the sphere. . . .. ... . . ... ... .... 11 Ta�le 2. -Zhongolovitch 100 equal·area blocks for the spheroid .... : . .. . . . . . . ... 12 Table 3. - Five·degree �Iocks used for geopotential modeling for the sphere.. 12 Table 4. - Five-degree blocks used for geopotential modeling for the 5pheriod.. 13

jv

Computing the Ceo­Algorithms for potential Using Density Model

a Simple-Layer

Foster Morrison Ceodetic Research and Development Laboratory

National Geodetic Survey National Ocean Survey. NOAA. Rockville. Md.

ABSTRACT. - Several algorithms have been developed for computing th"e 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 {So x 5, blocks on an oblate spheroid is used. The special approximation is used in the sub­satellite region, Taylor series in a surrounding zone. and numerical quadrature in the remaining regions. TIle relative sizes of these zones are readily changed. An auxiliary program can generate aU the parameters for different equal-area block confij!:uratiuns. Different orders may be used in the numerical quadra­ture done in connection with the special apl)roximation. Numerical tests compris­ing 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.

I. INTRODUCTION Chonsing to use a density layer model to represent

a gravitational potential field does not specify an algorithm for the computation. Using an analytic solutifJQ tp a boun�ary value problem. such as spherical or sl)heroidal harmonics. specifies the gen· eral outline of the algorithm. Quite a bit of discre· lion remains 10 Ih� user as 10 whal recursion rela­lions (if anyl 10 apply and what normalizatiun is most efficient: a considerable lileralure has developed throulth 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 Ihe surface is related to the choice of coordinates.· which is another olltinn. Finally. Ihere is a choice of methods uf representing the

density on 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 dis­crete 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 for the combination of gravimetry. and satellite derived data_ which may be dynamical. geometric. or bOlh. might be too cumbersome for computing the ephemeris of a satellite.

II. :\IETHODS IN USE Workin!! directly from uptical satellite observa­

lions. "ueh and �Iorrison 11970) derived a density

layer model for the geopotential. In theory. the sur­face 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 den­sity layer model with a surface consisting of the smallest sphere enclosing the Earth. Vinti's model avoids the occurrence of impulses on satellite orbits coming close to any of the point masses. but it is computationally no different"from using the spheri­cal harmonic modelJor 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.

III. APPROXIMATION BY TAYLOR,SERIES

The Cundamental Cormula Cor density layer models is

(I)

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

dcr = i! costJ,JA,dtJ,

and compute (1) as an iterated integral

2tr • tr U = Cf J xlcb. A)": �osfbdAdtb - (2) J.. -tr r

Even if thf: surlace 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 surface only once. i.e., the radi,&;I coordinate of the surface is a single valued function of cb and A.) Even if the surface has 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 topography. An analogous expression could be derived for spheroidal coordinates. Another alternative is to remove r: from the nu�erator of (2) and use density per unit of solid angle: the position of the layer would enter only through, ,*. For the nu merical tests made. only 12) was used.

Given 12). there are various opt inns for ex­pressinj!: the integrand. The simplest. perhaps. is to write

(3)

and expand h into a Taylor series. (See appendix II.) 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 lines of latitude and longitude. The result is of the form

2

(AT), EI J J [ho+h"41+hlA + 111(1&,.,.412 +2h"l.A + huA:!)]dlbdA (5)

if we truncate the series at the second-order terms. The function Uo is a spherical harmonic expansion used for the central force term. oblate ness and lon,­wavelength parts of the potential. Eq. (5), an inte­gral of a polynomial. is evaluated quite easily . . The derivatives are somewhat complicated. but straight­forward.

To obtain the gravity force. the gradient operator is applied

I='VU.

(" a " a' i_ a ) 'iJ = i-+ j-+ K-a". ay. az.

(6) T

Numerical tests have been made at altitudes of 300 and lOOO km. using 15Dx 15D and 5DX5D plltches. Results are satisfactory for the 300-km ahitude 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.-= 0 (l/r*2).

Nevertheless. the method is fa!'t and accurate for altitudes above 1000 km and softens the impulse!l of a purely point mass approximation. , Spherical coordinates are used for the sak.e of

rapid computation. and for the area difl'erential the , spherical approximation

du = r= cos Ib dAdcb + O(e:l)

suffices. Simplifications occur if h is expanded about a point whose latitude is the 8\'era�e of the north and south edlll's of the block and similarly

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

where c6H and A.H are the extent of the block in latitude and longitude.

1 lil'

1 4� -_ ... . ..... _-

1

1

Flla:RE I. -The block over which .l.T is evaluated. The "rildn is in the "center."

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)

r* == V(r - r.)�.

here Ir - r.)� is an analytic function nf the coordi· nales everywhere. bUI r* is not al r*==O. and I/r* is not well approximated by Taylor series in that rettion. The method of singularity·matching I)rovides

al)proximation techniques for computing such things as convergent iml)rOper integrals where exact analytic solutio�s cannot be found. The trouble· some integrand h,x) is factored by an elementary function pCx) with the same type of singularities so that one may write

h'xi == IIIXI PXI. (8) 3

Now px) may be expanded as a Taylor series. The desired results may be obtained provided the expressions x Ifp (x). n == O. . . .• N can be inte· grated analytically. Often this can be done by reo peated application of integration by parts. Computing the potential of a surface layer is

complicated by the (acts 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 in­tegral in the spherical coordinates f/) and A. as was done (or a Taylor series expansion. For the factor with a singularity we choose

where D. E and F are functions of til. For practical applications at satellite altitudes. it is simplest to use the Taylor series expansion for ,.: to com­pute D. E and F:then we have

I(A.) == G",2 (f/) cos f/)= "constant"

which greatly simplifies the results. The "integration with respect to f/) now may be done by a conven­tional nume�cal quadrature. Newton·Cotes formu­las of order 3 through 9 were programmed (Abramo­witz and Stegun 1964). All o( the integrals with respect to A are of the

simple form

J h(A)dA = ( (A±8A.±CA2),IA

J" D+EA+FA2 (9)

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

be made over what has been programmed. For low altitudes. the matching of the singularity in I,(A) and 1/r* should be precise. Gaussian quadrature should be faster. and more accurate than Newton-. Cotes. Some of the integration with respect to f/) also could be done analytically to save time and reduce error. Density variations within a block and the depar­

ture of the apl)roximation p( A) from l/r· can be modeled by expanding

in a Taylor series. No matter what order Taylor series is used. the' in tegra! may be done quite simply IGradshteyn and Ryzhik 1965. p. 80). so

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

f= G'X.r� cos c() (10)

6= (r-r.)�

J

dA, - I "h • (2FA,+E) "YX-V'F sm - vq . (IS)

To have q .. O. it is necessary to have F > O. since D .. 0 for this applicatiun. If F < o. one can use

J dA, I " I (-2FA,-E)

V'X= :.;::j!-sm- v=q (16)

p=r'/: and use the approximat ions

( 1 1) It may happen that F > o. but q < o. Such" may be the case if F is relatively small and E large. In this case one has

f =A+BA,+CA,:

A =1.,+ f.d>+ f .. d>:/2

B= Ii.. + /A.c() C=i/u.

"s=D +"EA, + FA,:

F=tllu

(12)

(13)

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 alternat ive to the elliptic integrals is to subdivide the blllcks. Doing one of the iterated integrals "analytically removes the problem of approximating l/r* by a Taylor series. The" integrati lln with respect tn cb precedes smollthly by numerical quadratures" S lime part of the quadrature with respect to c() could be done analytically" Numerical tests Ill" the plltential function have

proved very prumising at the 300 km alti tude even �ith IS° patches. These results are Car more prom· ising than thllse obtained with the direct numerical integration algorithm (�och 1971). If F ;i: O. we may use in evaluating (9)

J

A+BA+ CA: dA VD+EA+FA: (14)

= BVX +C [�_2�] VX+-Q J

dA F 2F 4 F: YX'

where X= D+ Ell. + FA: and

Q=.4. _ BE _ (3E:-4DF)C "1.F 8F:

If the quantit y q .. O. q=4 DF-E:. then 4

. J t1= & log (�+"A, �� 2&) · (17)

These well known formulas (Weast and Selby. 1967: Gradshteyn and Ryzhik, 1965) are sufficient for aU 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

J A+Bl\+CA,2 dA,=(..j ) VD+EA

D+EA" ( 18)

x [24 _ 2B(2D-EA) + 2C(8D:-4DEA,+3E:l\:] E 3E: ISE3

Equ!ltion (18) is the application of equations 101. " 102. 103 of Peirce and Foster (1956). When both E and F are very small compared to

D. 114) is computed best by expanding the de· nominator in the binomial series and integratin� term by term. This IJrocedure is identical in the formal sense to the Taylor series approximation. but there are differences in the numerical applica­tion. For one thing. the quantities D. E. and F. required to perform the tests indicating whether (4). 1 15). ( 16). (l7). or (18) should be applied. are used at once. Also. the binomial series ex"pansion might not be used for every value of the latitude in the secon� iterated integral . which is done numerically. The results are

J

. Ill. {

(A +BA+CA:)X-'iIJA=D-I"'. 2A3A -ftA

+ � 811.3 [ C-! BE* +2 AE*:_lAF*] 3 2. 8 2

(19)

+ 8AS [2 CE*:+2AF*:+"2 BE*F* -CF*] 5 4 4 2

+ :8

8A:CF*: +0(8A9) } .

where E*==-E/O. F"=F/D.

Equation 119) is given as a definite integral since aU the even powers of 8), 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 orden of differentiation and the integration implicit for U 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 (I1U) = V I I fg-"Idtbd).

= I dtb { V I fr If I 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.

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 soXSo at the equator and fairly uniform in shape everywhere (aee lip. 2 and 3). Blocka near the poles are somewhat extended in longitude. for example. those at the poles cover 900 in longitude. : For this reason these blocks are subdivided into roughly 50 longitude bands and the' algorithms are applied within each sub block generated. TMs is illustrated -in figure 4. To save computer lime, different algorithms are used depending on the distance from the satellite to the center of the block.

FII;l'RE 2. - 1640 five-II!!'"r!!'!!' !!'lluaJ·ar!!'a bluds ... \iluK l'llual.arn pn.j!!'c,jun.

5

222 ·ail 0 - " • 2

FICt:RE 3a. - Equal.area block. wilh ind�i numb�n. NW quadranl.

FU;l HE :Ih. - Equal·an·. hlud,1i with in.dl'x numhC'rs. SW quadrant.

6

711 ; JlO; 'It: JlI ' JlI

FIGURE 3c .. - Equal·area blocks with index numben. NE quadrant.

FICt.:RE 3d. - Equal·area hlncks with index numillon. SE quadrant.

, 1

, J - -flla:RE 4. - Subdivision of blocks near pole into blocks equal in

Ion�tude and latitude.

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

8=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 or­biting s'atellites out to infinity.

For operational purposes and to facilitate incor­poration 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 'that 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 geopoten­tial and lU'avity. 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 sohllion.

8

Some time is saved by direct application of the Newton-Raphson algorithm instead of the compiler subroutine to compute the square root 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 6rst approximations to this distance are available. except for the case of the fint 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 1000 km and one eacb at lOR. lOOR. lOOOR. 1.OOOOR. and 1.000.OOOR. If the density on an oblate spheroid is chosen so that

(22) 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 compar­isons with a spherical shell of constant density. If we set U = IJ./r. in (1) and attempt to solve the in­tegr!!l equation for ](. we can observe that du = r2 (f/» cos tbdtbdA.. which in tum implies that (22) is

. an approximate solution good to an order of about ·e2• 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 welle. known GEODYN program. which is now operational on the NOAA IBM 360-195 at Suitland. Md.

Program linkage and compatibility hite 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 1 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 I)eriod for which the integration · was done. Since the mean value of a obv.iously is not the same as the initial value. drift occurs in the mean anomaly. In practice. the process of orbital adjustment would rectify this I)roblem. A very large secular or long· period perturbation is present in the inclination. I. Runs made Ion the CDC 6600) wilh an inclina­lion of 59.4° did not exhibii this effect. so it is most

likely the result of resonance caused by the trun· cation errors in the Coree 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 (Morrison 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 Cew 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 computer simulations and a numerical verification oC that study would be desirable.

: • · • • · . ! ...

...

fICL'RE 5. -The semi·major axis of Ihe orbil of a parlicle muyinlC in a central force field modeled by the surface densily algurilhm and rqualilln (22).

...

...

-

· · Ii · : u ::

UN�'---T---r--�--�--�--�--r-�r-�---­o

FII.L:RE 6.- The eccentricity ui the orhit "I" thr partidr.

9

_ ....

--

-

--

o , . '

FIGURE 7. -The inclinalion or &he orbil or the pardcle.

VII. APPLICATIONS

Most of the work in developing the density layer method oC geopotential modeling has been devoted to finding approximation techniques {or computing (1). Inasmuch as this is part oC 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 com· binalions of various kinds of data .

The preseAt program. using 50 blocks. is suitable for using satellite tracking. satellite altimetry. and gravity anomalies which have been averaged over areas of about 50 "squares." The model corre. sponds. more or less'. to a sphericid harmonic ex­pansion to degree and order 36. A study of the frequency response of the model is included in Morrison ( 1976). Since the data quantities and density values can be related through linear integral transforms. eit�er 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 advo­cates of discrete variable representations. whereas the techniques known as least-squares collocation and spherical harmonic sampling functions have used. basicaUy. orthogonal function interpolation. The motivation to develop the discrete appraach has been mostly intuitive. based on the observation lhat 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 yourselC 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 cOBting. The advanta�e of the surface coating over a discrete variable representation of the gravilY is that the upward (,ontinuation transformation is much

simpler. i.e .. (l) 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 inter­polation of deflections of the venical. point anom­alies. 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 (Rapp. 1971). Making the blocks exactly equal in area i& 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 fb on an oblate spheroid is given by Unguendoli (1972):

A ("") .

2 ·'(1 .') J.'" cos ede · (23) '41 = 7Ta- -e- (1 ," ,' 1: ) "

II -e- sm- '" -where a= semi·major axis. and e= meridian eccen­tricity. There is no need to reson to· series ·to evaluate (23). since the integrand is a rational function if we ul\e the substitution

%= e sin' � , (24) The result is

[ sin 0 "A (fb) = 7Ta= (l-e�) 1- e� sin� 0

+ .1.10" 1 + e sin 0] (25) 2e .. 1 - e sin cb

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

Lim A (tb) =21Ta� sin fb. (26) ,-u

which is the correct result for a sphere, For small values of e it is necessary to evaluate the logarithm ponion of the expression by series and cancel the divisor e to avoid loss .,f numerical siJ!:nificance.

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 . .I = 18 = 90°/5° for one hemisphere nhe other hemisphere can be obtained by reflection I. One next chooses the nurn· her of blncl ... s in a zone. startin� from

n(i) = [�O cos ( fbi) ]. (27)

where the zones are numbered staning from the equator: (fbi) = mean latitude in zone'i (and for 5° any other appropriate value may be substituted): the [] symbolizes truncation to the nearest integer. The area of the zone from 0 to til} is then

} I neil

A (til ) _' . ..;;,,- .:....1 - A (goo). J .1 I n(,n ,.1 .

(28)

To find fbj one substitutes A (fbj) from (25) into (28) and uses the Newton-Raphson iteration. since the

. transcendental equation 'is not readily solved. For a staning value the spherical approximation may be used.

10

(29) The derivative needed is quite simple.

a� (l ..;. e:l) cos fb dAldtb=2w (1 � ' � ""' ) �' (30) -e- Sift - '41

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

R. = length of side along mean parallel . ' I the same along a meridian (31 ).

If the values· 9f Ri are not satisfactory one may change the values of n(i) in a trial-and-error way to see if improvement is Jlossible. Selling up a rigorous adjustment of the n(i) by some criterion of obtaining a better approximation in (31) 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 groUI) into 12 (llee fi�. 3 and 8) and so on. all

n(J) -4

n(J-l) -12

n(./-k) -4(k+ 1 p-n(j-k+ 1)

=4(2k+ I)

/,;<$. J,

( This is because Ihe zones are nearly concenlric rings al Ihe poles and for Ihose regions

pole follow this rule. ISee figs. 2 and 3. and lables 3 and 4.)

R.1_,· == rr/4

gives a better criterion for "squareness."

The 5° equal. area . block model chosen follows the well· known Zhonogolovitch 10° blocks as closely as was considered practical The zone boundaries near 10°. 200• 30°. and 60° are the same. Table 1. gives the parameters for the Zhongolov,tch blocks on a sphere. table 2 for a spheroid. For blocks of 5° only. the IWO zones nearest either

FuaiRE 8.-Idealized leI or equal·area blocks al pole.

T AS L£ 1. -Z hongolovitch J00 equ.al.area blocks for the sphere

a = 1.0000000000 II -1.0000000000 t' = 0.000000000000 Seelon .. l Area -0.03064968442527

J N PHI-2 PHI8AR DPHI DLAM8DA RJ PHI-2 SPH

• 1 36 10. 114144088 5.057072044 10. 1 14 1 44OH8 10.000000000 0.984865728 10.114144088 2 34 19.966058297 15.040101193 9.851914209 10.588235294 1.037923 106 19.9660.0;&:297 3 32 29.838766211 24.902412254 9.872707913 1 1.2SOOOOOOO 1.033560986 29.838766211 " 30 40.083433915 34.96 1100063 10.244667704 12.000000000 0.95996:235 1 4O.08:W33915 5 25 49.982997 154 45.033215534 9.899563239 14.400000000 1.027967879 49.982997154 6 21 60.260840340 55.121918747 10.27784.1186 17.142857143 0.953783344 60.:260840340 7 IS 70.298j'88126 65.279814233 10.037947786 24.000000000 0.9911854867 70.:298788126 8 9 80.185857012 75.242322569 9.887068R86 4O.000u00000 1.0.10S64385 80.185857012 9 3 90.000000000 85.092928506 9.814142988· 120.000000000 1.045917823 90.000000000

Tnlal :!O5

11

\ TABLE 2.-Zhongolovitch LOa equal-area blocks for the spheroid

G'" 1.0000000000 6- 0.9966470765 ,= .. 0.006694605000 Seeton'" I Area -0.03058119674137

J N PHI-:! PHIBAR

1 36 10.158613279 5.079306640 2 34 20.048575322 15.103594300 3 32 29.949659806 24.999117564 4 30 40.209905973 35.07971tl889 5 25 50.109306146 45.159606059 6 21 60.371219506 55.:Z40262826 7 15 70.380067424 65.375643465 8 9 80.228851341 75.304459382 9 3 90.000000000 85.114425671

Total 205

DPHI

• 10.158613279

. 9.889962043 9.9010B4484

10.26024616; 9.899400li3

10.261913360 10.00B847918

9.848i83917 9.771148659

Dl.A�IBDA

• 10.000000000 10.588235294 11.250000000 12.000000000 14.0100000000 17.142857143 24.000000000 40.000000000

120.000000000

RJ PHI-2 SPH

0.980520745 10.092007853 1.033621594 19.924951454 1.029789802 29.7834599CM 0.957114533 40.020265940 1.025712019 49.919815831 0.952431222 6O.� 0.999117443 70.258024753 1.030310595 80.164278502 1.045929007 90.000000000

The CoUowinlleKend ia Uled Cor all tables in this text:

G'" Semi·major axil 6 ... Semi·minor axil , ... Eccentricity of the spheroid

PHI :! = Latitude of non hem ed(le of block PHIBAR ... Average latitude of block

DPHI .. Extent of block in latitude

DLAMBDA -Extent of black in loncitude RJ -Ri-See (31)

PHI-2 SPH - Spherical latitude for PHI 2 J -Index of zones in one hemilphere

N ... Number of block. in one sector of a zone

TABLE 3. - Five-degree blod's used for geopotential modelinll for the sphere G'" 1.0000000000 6 -1.0000000000 ,1 .. 0.000000000000 Sectnn-4 ."rel .. 0.07662421106316

N PHH PHIUAR DPHI DU:\IUDA RJ PHI--l SPH

1 18 5.037335850 2.51866i925 5.037335850 5.000000000 0.991629292 5.037335850 2 18 10.114144088 7.575;39969 5.076808238 5.000000000 0.976274270 10.114144088 3 17 14.983246004 12.548695046 4.869101916 5.294117647 U1613148Si 14.983246004 4 17 19.966058297 17.47465:!151 4.982812293 5.29411;647 1.013442458 19.966058297 5 16 24.803794163 22.384926230 4.837i35865 5.625000000 1.075117645 24.803794163 6 16 29.838;66211 27.321280187 5.034972048 5.625000000 O. 9925603� . 29.838766211 7 15 34.801265;05 32.320015958· 4.962499494 6.000000000 1.021 iS3384 34.801265705 8 15 40.083433915 37.442349810 5.282168210 6.000000000 0.90186.1137 40.083433915 9 13 45.017042172 42.550238043 4.933608258 6.923076923 1.033751458· 45.017042172

10 13 50.419641072 47.718341622 5.402598900 6.9'13076923 0.862118040 50.419641072 11 10 55.035992386 52.727816729 4.616351314 9.000000000 1.180676862 55.035992386 . 12 10 6O.2608403.w 57.64841636.1 5.22484;954 9.000000000 0.9'lI752781 6O.26CI84034O 13 7 64,480541509 62.370690925 4.219701169 12.857142857 1.413012628 64.480541509 14 7 69.485799940 66.983170724 5.005258431 12.857142857 1.004376110 69.485799940 15 5 73.9406.19-186 71.713219713 4.454839546 18.000000000 1.267816931 73.94063M86 16 4 78.662967149 76.301803318 4.72:!327663 22.500000000 J.121t..!9321 i 78.662967149 17 3 84.338423235 81.500695 1 9'l 5.67S456086 30.000000000 0.781245113 84.338423235 18 I 90.000000000 87.169211617 5.661576i65 9!).000000000 0.785078675 90.000000000

Total'" 205)( 4"'1640

12

TABLE 4. - Fil:e·de/!:ree blocks wed for geopotential motlelin� for the spheroid

0 .... 1.0000000000 b'" 0.9966470765 e� - 0.00669460S000 Secturs-4 .",rea" 0.076452991853-13

N PHI-2 PHIIJAR DPHI DU�f8DA RI I'HI-2 SPH

• • 1 18 5.059836888 2.5ml8444 5.059836888 5.000000000 0.987210982 5.026137369 2 18 10.158613279 7.609225084 5.098776391 5.000000000 0.971992256 10.092007853 .1 17 ·15.047-175056 12.603014168 4.88886In7 5.294117647 1.OS6801567 14.95116J466 4 17 20.048575322 17.548025189 5.001100265 5.2MII7&47 1.009328610 19.9249510154 5 16 24.901674946 22.475125134 4.853099625 5.625000000 1.071017871 24.7S5008330 6 16 29.949659806 27.425667376 5.047984859 5.625000000 0.989068218 29.783459904 7 15 34.921630212 32.43S645009 4.971970406 6.000000000 1.018502939 34.741194898 8 15 40.209905973 37.565 76809:l 5.288275761 6.000000000 0.899333629 4O.O"lO2659-&O 9 13 45.145343214 42.677624593 4.935437241 6.923076923 1.031256833 44.952912706

10 13 50.545592266 . 47.845467740 5.400249052 6.923076923 0.860386605 5O.3S6634602 11 10 55.156394527 52.850993396 4.61080"2261 9.000000000 1.178755745 54.975721014 12 10 60.371219506 57.763807016 5.214824979 9.000000000 0.920586272 60.205546609 13 7 64.580143945 62.475681725 4.208924439 12.857142857 1.411668910 64.430619998 14 7 69.56985 79-& 1 67.075000943 4.989713996 12.857142857 1.003702695 69.443646351 IS 5 74.008705762 71.789281851 4.438847821 18.000000000 . 1.267271917 73.906491693 16 4 78.712308656 76.360507209 4.70360'l894 22.SOOOOOOOO 1.128O"�'569 78.638205110 17 3 84.363550821 81.53;929738 5.651242164 30.000000000 0.781180387 8U25809778 18 I 90.000000000 87.18177�10 5.636449179 90.000000000 0.785081504 90.000000000

TUlal .. :ZOS x 4 ... 1640

ACKNOWLEDGMENTS

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

REt'ERENCES

Ahramuwitz. �1 .. ilDd Slellun.l.. 1965: H.m./buuk alMII/hematica/ Func/ions • . VII/ional Bureau o/Standards AI.plitd :\lalhemll/ics Series. 55. U.S. Govemmenl Prinling Office. Washinglnn. D.C .• 100i pp.

Friihlich. H.. 1975: Verfahren lur l.iiaung del' Oberniichen· intelU'ales fiir das Mlldell Iler einfachen Schidu in der Satel· lilengeod.lie. .'ierie5 l.': Oiuerlllli.m. Vul. 207. (;erman Geudetic Cummilsiun. :\Iunidl. ·17 1'1'.

• "ri;ldich. H .. and Io.:uch. 10.:. R .. 1974: Intellratinnsfehler in den Varialions·g1eichunllen fiir da. :\Indell der einfachen Schicht in der Satellitenjteudiisie . . lliuf'i/an(!en trllS tit'''' Inst. /. tht'or. (;eodasie ./er Unillf!rsitii/ Bonn ICllmmllnil'Utilln nf Ihe Cnm· millee fnr Thenrelical (;e"desy. l!nivenily uf lIunnl. �n. 25. 33 PI'.

(;radshteyn. T. S .. and Ryzhi .... T. :\1 . . 1965: T.d,le ol/n/l!llra/s. Serie •• und ProdllCIS . . ",cadenlil· Press. �I'w Y"rk. ,Irantilaled from Rus"ian by .o\Jan .Jf'lfreYI. 1086 I'll.

Koch. K. R .. 1974: Earth'b (;ravity Field and 5tali"n C,,"rdinalf'lI from DIII'Jller Data. Satf'lIile Trianllulatiun. an.1 l;ravllY Anumalil's. :VD.4.4 Tt'C'hniC'tr/ Rr"ort :'\OS 62. :'\aliunal Ocean Survey. !'ialinnal Oceanic and .",Imu�pheric .",dminiblralion. Rockville. :\Id .. 29 pp.

222·le. 0 • " • 3

13

Kuch. K. R .• 1 971: ElII'Ors of Quadrature Connected with the Simple layer Model of Ihe Geupolenlial. NOAA Tf!chnical Memor"nd"m NOS II. Natiunal Oceall Survey. Salional Oceanic and Atmospheric .",dminisaration. Rockville. Md .. III PI"

Io.:nch. 10.:. R .. and Wille. 8 .. 19i1: Eanh's gravilY field repre­sented by a .imlde layer I,otf'ntial frnm DlIl'Jller trackin� of Sat .. lliles. J. Cl'ophys. Rt's .• i6. pp. Mil-8+;9.

"och .... R .. and :\llIrri�nn. r .. 19iO: .", 5imille layer muclel ur the llellllOlential from a l'unlhinaliun nr satellile and jEfIlvity data . ./. Ceophys. Rf's., j5.1111. 1483-1-'92.

�Iurrisun. t· .. 19i1: Densily la�'er moof'ls fur Ihe lIeDllUtelllial. Bulietin Gio'/f!siquf!. Nu. 101.319-328.

MurriBun. r .. 1972: I'ru\lallalilln nf errors in Ilrhil� l'nmpllted from density layer mudell!. Tht' Use 01 Artijicill/ S"tellites/or (;eoc/esy. A(;U ,"onollral'" IS. ed. by S. \l'. Henriksen 1"1 al. American GeuJlhy .. il'D1 ['ninn. Wa.hin!dUII. D.C.

�Iurrislln. t" .. 1976: Alllurilhms ru, cumlJluinll the lIetll'otl!'nlial using a simple·layer denliily model. J. Ceopl\)·s. Re ... 81. pp. 4CJ33-·'�36.

I'aul. :\1. 11: .. ICJ73: On CUlllllutatiun .. r elluailirea bluckl. 8"l/eli. C;'o,/psique. Nil. 107. I'I', i3-64.

Peirce. II. 0 .. and fUller. H .. 1956: A .r;l\ort Tahir o/I,.,epal •• l;inn an,l C .... 8u�\rJII. 181] I'l'.

RIII'II. R. H .. 19i1: ['lual'lIreu blllcki. Bull"i,. Giodi.iqur. �". CJIJ. 1",.113-125.

l'nl'uellduli. :\1.. 19i2: Divisi .. n lie la .urfllce terrellre en blul'. .I'airl" Pllale. Bulirtin (;;o.li.iquf!. �II. 104.221-:.129.

Villii. .1.. 11J7 1: Rell"'�elllalilln .. ,. Ihe Eunh's jlI1Ivi''''iuII.1 l'''ll'ntial. Crlrsrio/ :"«ho'lic •• -'. 348-36i.

"·rasi. R. C .. llnel S"lhy. S. :\1. letlilllnol. 1967: Handboo£· 0/ TII"'r. /ot' .llnlht'n/lIl;(',. Ihird e.lili .. II. The Chl"lIIil'ul Hul.iH'r e .... C:1",·"land. Ohiu. 1050 1'1"

Willianl5un. :\1. R .. alld (;ilIM.s,"hl..in. E. �I .. 1973: E.timall!' or jlra .. ily anllm.lips. ;n 19;3 Smi,h,on;"n S,anda,d Eonh ,1111. SAO Special Repon 353. I'diled by E. :\1. Gapolehkin. Camhridll". :\Ia .... IY3-:t.!8.

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

undefined in the" text are:

C gravitational constant r position vector on the Earth r. position of a point where potential is com-

puted x • • y . . z. components of r. T anomalous gravitational potential U "

gravitational· potential x, t dummy variables A longitude til latitude (spherical coordinate) " density a surface area

APPENDIX II. DETAILED MATHEl\lATICAL DEVELOPMENTS

This section is comprised of a detailed presen­tation of all formulas used. In some cases. options not used in the prolD'ams 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

fJ.�= J

olr1> folA ,,(tho AJ: cos <b dAdtl> j

- 4. - 4 A

where r* == r - r.

and r*:: II r* I I The integrand is abbreviated to

and

f== "rl cos til r*

Ei. _ _ � xr� cos th a A aA r*�

+ cos th [ r� � ... :! � ] r'"

a A 1')( il A

( 1 . 1 )

( 1 .2)

(1.3)

f 1 .4)

I�

" (1 .5)

(1 .7)

r* == r - r. (l.8)

r* == 11"*11

== ( .. * ... *)11:1 (1.9)

... == (x •• y •• Z.)T (1. 10)

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

.. == rCCOIi th ens A. cos <b sin A. sin <b) T ( 1 . 1 1 )

0'· r* or -= _ . -aq, r· aq, (1. 12) or ( � - 11.0) integrate out to zero. which includes all the third-order terms. Hence. we can find

ar· r· or - = - - -all. r* all. 4T::: 4""411. [ 4�0+ � a2/ 4t/1:l + � !:.l .:1,V (1.13) . 'fI �. 3 aq, 2 3 a1l.2

� == r. (-sin t/> ·cos A, -sin t/> sin A, cos t/» T

+ :�.(cos f#, cos �, cos q,sin �, sin f#,) 'f

:: == ,(-cos t/> sin A, cos � cos 11., 0) T

- == - r- + r· · - - -a2r· 1 [ ., a2r (ar*)2] at/) 2 ,* ot/12 at/)

- == - - + r* , - - -a2r* I [(ar)� i Pr (ar.)�] aA.� r· aA. aA.� a>.

a�r I' iI�r af#,� = - r+4 -;' at/)� ·.

+ 2 :� (- sin q, cos A . ...... sin eb sin A, cos t/» T

..!:!.. == resin eb sin A, - sin eb cos 11.,0) T arl)a>..

+ :; (-cos eb sin >.. . cos t/1 cos �, 0) T

. (- 1 0 0 ) a " r � == 0 - l O r aA- 0 0 0

Most terms with a�r/ilA2 are omitted.

(1.14)

(1.15)

( 1 . 16)

( L IS)

( 1 . 19)

( 1 .20)

( 1 .21 )

Substitute ( 1 .2) into ( I . l ) and expand by Taylor's theorem:

a.:.1 2 ] } + (/A� ( A � �o) + . . . d�dcb. ( 1 .22 )

All the terms containing odd powers of (cb - ebo)

15

+O(.4t/)4, 411.4)} (1 .23)

8. The Expressions for the Gravity Vector

The formula A«== fJT ( 1.24)

is fundamental. Substitute 0.23) and ( 1 . 1 ) into ( 1 .24)

(1 .25)

== � [4 V /0+ � V / .. t1t/>1 + � V /u4A2 ] t1t/>4A 0 .25a)

.. * 'V /0 == )Cr2 cos t/) -;; , ( 1 .26)

The only factor of ( 1.2) that is not constant with respect to 'V is 1/r*, and 'V operates on U .S) and ( 1 .7) similarly. Actually. 0 .6) is not needed and ( 1 .3) and ( 1 .4) are only steps to derive ( 1 .5) and ( 1 .i ) . Gradients of derivatives of ,*

'V - = - - - + - -( a ,* ) l a r .. * a r* a t/) ,. a t/) r*� a f/J ( 1 .27)

'V - = - - - + - --( a r. ) 1 0 1' ''* a r'" a A r· a A r·� a A ( 1 .2S)

'V -- = - -- - - -( a 2r* ) r* ( a 2r• ) 1 [ (/ 2r a t/)2 ,.2 a t/)� r· a t/)2

( 1 .29)

+ 2 :�. 'V ( �� ) ] ( 1 .30)

We may factor out seven distinct functions of ,. from I •• and JA�. It is convenient for l'rolD'am coding to write these as eJements of 4 X 4 matrices .and .\.

<1> . . == 1/r· ( 1 .31a)

Then we can write

All == � II I ar· A .. == - -I. ,.*2 ilA

with « and IJ being 4-vectors:

( 1 .31 b)

(l.3Ic)

n.31d)

11.32a)

' (l .32b)

(l .32c)

( I.32d)

(1 .33)

. ih . 4 ' .J.. ar _� h .... (II - - 28; r� �m t/I - )(r 8m Of' ilt/I - )(,- "'os �

. [ or aX o�x ( La

rt/l ):! +

cos t/) 4r-- .... r2 - + 2)( acb adl ac6�

1 1.35a)

(I: == 2)(r� sin q, - 2 cos c6 [ r�1! + 2rx :� ] 0.3511)

O'!I = - Xr= COl <II

1 1 .35d)

+ 2)( ( �) :J . • (lAo

1 1 .3681

16

n.36b)

U.36c)

(l.36d)

The remaining rows of • and .\ are obtained by applying the operator 'il to the first row of each

(l.37b)

n.38b)

U.37c)

(1 .3Be)

(l.37dl

2 ( or* ) ( or'" ) + ;;; ar 'il li . 1 l.38d)

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

F 9r larl%er block sizes one may wish to use an approximation higher than second, order to cos dJ in 1 1 • .2).

Let The integrals in 1 1.45) are standard forms and /=H cos f/J 0.39) simplify to

H = � r*

The expansion in Taylor series is now done as

Integrating the zero·order term

f f H .. cos f/Jdt/KlA. = 24A.H" L:: cos ct>dt/J

(1.40)

= 4�AHn cos fbn sin ilfb · (1.42) Comparing this to (1.23) we can observe /n =

Hn cos ct>o and sin 11t/J is fairly small lless than 7�12°), we can use

0.43)

Since 0.43)

was assumed in 0.22). we can add the remaining . terms of Y.43) to get a correction to (1 .23)

3 (4T) = �ct>��Afn ( 1 -:1cb�

) • 0.44) I 30 42

None uf the factors in the correction depends nn the satellite distance r*: they depend only on the block size used. Another correction of the same order as n .44)

may be obtained from the consideration of the first­order terms IIf 1 1 .41). The term containinl! 1 11. ­Ani H A intejl:rates out to zero, but l1J1 .. 1 cb -cbtl) cos ct> does not. Hence. we may write

- dill J

r!l . ldI ens cbdcb } . .� - l.t ( 1 .45)

I i

4T. =24ili. sinf/>o(4f/J cos 4f/J- sin �ct» . (1.46)

The trigonometric functions of 4ct> in 11.46) can be expanded in Taylor series: the terms in If/J drop out:

4T. == -� 4A.4f/J3H. sin f/Jo

X [ 1 - l� �ct>� + 0 (4ct> .. )] (1.47) Since the first term of (1.47) is alread, included in (1.23). the correction is

1 32 (4n = 15 4.A.4t/J1 n. til sin f/Jo. (1.48)

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

D. Corrections to the Gravity Vector Applying the �peralor 'il t� 1 1.44) yields

I1cbll:1A ( fa.cb� ) 'il £I; • (4T) ] == '"""3i) I - 42 'il /." 1 1 .49)

For (1 .48) the result is

2. THE METHOD OF SINGULARITY· MATCHING '.

A. Computinp; the Potential Instead of 1 1 .2) we use

�T== C J f i dAdcb

where I is now defined as /== ;t(r� cos ct>

and

(2.1)

(2.2)

12.3)

.'

To compute the integral (2.1) expand f and Il in Taylor series about 1(/)11. All)

1-/o +/.4(/) +/A4.A + � V .. .1q,2

+ 'lJ.A�AA + IAA4.A21 + . . . (2.4)

. 1 1 - 10 + 1�+BA4A + i (g.u.A4l:t

+ 2.r.tA�4.A + lAA4.A�1 + . . . (2.5)

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

with I A -/0 + 1.4(/) + i I

.. Atb2.

B - fA + f.A�(/). 1 C - i/lA•

1 D - lo +I.4.q, + i, .. �(/)2. E = IlA + fl.Ab.(/).

1 F - iSu,

(2.7a)

(2.7b)

(2.7c)

(2.7d)

(2.7e)

(2.;0

III = ](r2 cos 4>. (2.8)

; a" .I. 2 ar .I. ., . ... (2 9) J. = 8; r2 cos ... + r 8fb )( cos ... . - ]( r- sm "". .

(2.10)

+ 2" ( ar )�] - sin tb [2r21!.

-4", ar ]-/11 a(/) atb a(/) •

IA - cos tb [ :lrx !!:. +r2 !x ] aA at..

[ (ar ):: . � � !x 1AA. = cos (/) 2]( FA + 2rx aJt.2

+ 4r aA all.

(2. 1 1)

�] +

r ilAcl • (2.12)

18

(2.14)

(2.15)

2r* if,. IA - ar .

(2.16)

(2.17)

(2.18)

and

(2.19)

where ( 1 . 17) is appUed. The . derivalives of r* and r"are obtained Crom 11 . 12). 1 1.13). 1 1.14). 1 1. 15). (:kI6) and (1.20).

The computational strategy wiD be to integrate (2.6) analyticaJly with respect 10 A. This �iO elimi­nate the improper condition 'from the integral. The integration with respect to (/) then can be done numerically in a completely straightforward man­ner. The possibility of doing some or aU of .the integration with resl)ect 10 tb by analytic means will: be treated elsewhere. Special procedures for .ery low altitudes also will be treated elaewhere. TheBe formulas will be suitable for satellite altitudes. typically 200 km or more.

We now define

(2.20)

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

.p, • I - .pi = 3.p, so that (2.6) can be computed from (2. 19) by numerical quadrature formulas. For siniplicity the subscripts will be omitted in the detaile� formulas that follow.

Let us adopt the abbreviation

X = FAA'l + EAA + D (2.21 )

and ·also drop the � symbols. so w� can write

x == FA� + Ell. + D

(AT) ! = f (A + BA '+ CA�)X - IlldA. (2.22)

All this does is move the working origin to the "center" of each equal·area block. Now we can apply standard forms to obtain

(AT) = A J �+ B J �+CJ� I Vi vx Vi

=A f �+ B [ VX -!.. J � ] vX . F 2F vX (2 .. 23)

[ ( A 3E ) · . . + C 2F - 4F� vX

+ 3E2 - 4DF J

dA ] 8F� vX

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

(.1T) · =--+ C - - - - vX + Q -BvX [ A 3 E ] J

dA , F 2F 4 F: vX

with

Q = A _ BE (3E� - 4DF)C 2F + 8F-; .

(2.24)

(2.25)

The expression f .'(-I!: dA can be considered to define a function with two poles: one at each root of .'(. 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 ex('lude negative

19

values 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 f X -I/� 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 computation.

The auxiliary functions needed are F and

q = 4DF - E2. (2.26)

The three useful basic Corms are

(2.27)

F > O, q > O:

f�=_I- . _ 1 (2FA+E) VX V_ F

sm v=q ' (2.28)

F < 0, q < O. and

(2.29)

F > O.

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

hence

% log % - Iog y = log -: y

log [IC{A) ] ��l == log [ IC ����)] .

(2.30)

(2.31 )

where IC (A) is a function oC A. Similar. conven·ient forms can be obtained for (2.27) and (2.28) by applying appropriate identities.

The inverse hyperbolic sine may be expressed in logarithms as

sinh- I % = log (% + V%� + 1 ) . (2.32)

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

sin - I % - sin - I y= sin - I [z Vl - y� - y Vl -z=]

== tan - I • [% Vl -y� - y Vl - %2} .xy+ Vl -z� V l -Y::

(:l.33)

In some cases F = O. or F may be so small that 12.27). 12.28). and 12.29) may cause numerical problems. If E is not similarly small the ap· proximation

�T ... f f II + Bi1A + C�A2 dA.dtb

VD + E.lA (2.34)

may be used in place of (2.6). The analog of (2.23) is. then (Peirce and Foster. 1956. eq. 100. 10l . 102):

(�T) ' = VD EA [24 _ 2B (2D - EA ) , + E. 3£2

2C (8D� - 4DEA. + 3E2 A� )] I

'U' . + 15£:1 - .u. (2.35)

Both £�A and F�A� may be small compared to D. so that the binomial series is the most accurate method to use.

(D + EA. + FA.2 ) - 11. = D- I'. [1 - EA + FA� 2D .

� (EA. + FA�) 2 _ 1. (£A.+FA�) :' ]

+ 8 0 16 0 + . . . . (2.36)

Computing (i1T) ; is done by multiplying: and integrating power series in the usual way

III (A + BA. + CA,2) X_ I/. dA. = D- I/. {24�A - �l .) [ BE 3AE2 AF] + 3 lA:I c :- 2D + D2 - 2D

+ �A� [� CE2 + � AF2 + � BEF _ CF] 5 4 D2 4 D2 2 D2 D

+ - �A1 - + O (�AII ) 3 CF2 } 28 D2 . . (2.37)

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 :l.T would not be accurate if done in terms e,f cb as well as A.

B. Gravity From the Singularity·Matching Method

Let us determine the Itradients of the factors in 12.24):

'VQ = - .f... 'VD - (2 EC -.!!..)'VE

2F -1- F2 '!.F . _ (BE - CD _ 3CE�)'7F ( 2.38 ) :!.F� �F"

20

'V F = _ _ a;.,.I.;;.,"_ a 11.:& (2.41 )

The derivatives of .. are given in (1.14). (tiS). (1.19). (1 .20). and ( 1 .21).

The gradient of (2.24) is given by

'V (flAT) . = 'V { B-..IX + C [. �'""'! .! ] VJ.} • F 2F 4 F:

+ ('VQ) f dA. + Q'V f dA . (2.42) -..IX -..IX

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

J

dA. 'V (;1T) ; = Q 'V vX' (2.42A)

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

Y = (2FA + E) I q l - 1/2 (2.43)

and obtain

'V f dA = - ! F-:':� ( 'VF) sinh- I Y -..IX 2

or + [F( l + y2 ) ] - li2 'V Y. (2.44)

'V f � = � (-F) - :1!2 ( 'VF) sin -1 Y

+ [-F ( I - Y�) ] - I/� 'V Y.. (2.45) with

'V Y = I q l - ' :2 (2A. 'V F. + 'V E)

1 - 2 (:!.FA .... E) l q l - :\I�'V l q l (2.46)

'V Iq l .= 1:1

'V q (2.47 )

'V q =4(F'V D + D'V F) - 'lE'V E. (2.48)

If 12.29) is used. define

Z =X 1 '2 + A.F I:2 + 2£F- I/2 (2.49 )

and obtain· . '\1 J

dA == - ! F- :I/� ('\1Fl log Z + 'VZ (2 50) v'l 2 VFZ .

with

'\I Z == ! X- 112'\1 X + � F- 1 1:1 'V F 2 2 . + 2F- I/�'\1 E - EF-31:" F . ,

'\I X == '\1D. + 11. '\1 E + .\2'\1F.

Where 8 == C == O. (2.35) simplifies to

and 2A'VE '\1 (IT) ; = - -- vD + EA E2

+� (A'VE) (D + EA) - If�.

For 8 = C = 0 (2.3i) simplifies to

(2.51 )

(2.52 )

(2.53 )

(2.54)

(2.55)

W= 2A.1A +� .111.:1 [ 3AE: _ AF ] + ..!. �A5 AF� 3 D: 2D 20 D:

(2.56)

'V AX - lI:ld A = D - II:'V W - - W D- :1!:'\1 D flo\.

. 1 - .10\ 2

(2·.57 )

. . } [

E A ] 'VW==g �A:' 6A D 'V (E/D) - "2 'V (F/D?

3. POINT MASS AND NUMERICAL CUBATURE ALGORITHMS

The simplest possible numerical cubature scheme was used by Koch and �forrison 11970) and Koch and Witte (1971 ). The constant density blocks were divided into four 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 wOllld corre­spond to some array of point masses. One sets up the array of sampling points in the

f/>. A coordinate system and applies the numerical cubaiure formula to

T= J J�7 (3.1) to obtain

!J.T,, = f . . WI)( f/>,. A,) cos �acb�A i - , v'(Zj -%. ) � + (ri - r. ) � + (ZI -Z.)%

(3.2) (Xi) (COS A; cos f/>i)

r, == 'i (f/>, . Aj ) sin �j cos f/>j %i sm f/>i

T= � ATn• n

where Wi are the cubature formula weights.

(3.3)

(3.-1 )

For the formula actually used. N = 1 . and lUI �f/>.1A cos tb , == area of the block or sub·block. Hence. 13.2) simplifies to

(3.5) and

(3.6)

+ :0 .111.5 At 'V(F/D) . (2.58 ) with

'V (E/D) = (D'\IE - E'iJD)!D� '\I ( F/D) = CD'\1F- F'\1D)ID:l.

(2.59) (2.60)

2 1

1 ( ".-:t. ) '\1 ( 1/,! ) = r*' r. - r. . " Z,, - =. (3.7)

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