R. RUMMEL and R.H. RAPP Department of Geodetic Science

The Ohio State University Columbus, Ohio 43210

UNDULATION AND ANOMALY ESTIMATION USING GEOS-3

ALTIMETER DATA WITHOUT PRECISE SATELLITE ORBITS

Abstract

The paper describes results obtained from the processing o f 53 Geos-3 arcs o f altimeter data obtained during the J~rst weeks after the launch o f the satellite in April, 1975. The measurement from the satellite to the ocean surface was used to obtain an approximate geoid undulation which was contaminated by long wavelength errors caused primarily by altimeter bias and orbit error. This long wavelength error was reduced by fitting with a low degree polynomial the raw undulation data to the undulations implied by the GEM 7 potential coefficients, in an adjustment process that included conditions on tracks that cros= The root mean square crossover discrepancy before this adjustment was +- 12.4 meters while after the adjustment it was + 0.9 m . These adjusted undulations were used to construct a geoid map in the Geos-3 calibration area using a least squares pdter to remove remaining noise in the undulations. Comparing these undulations to ones computed from potential coefj~tcients and terrestrial gravity data indicates a mean difference o f 0.25 m and a root mean square difference o f +- ].92 m .

The adjusted undulations were also used to estimate several 5 ~ , 2 ~ , and 1 ~ anomalies using the method o f least squares collocatiorL The resulting predictions agreed well with known values although the 1 ~ x ] ~ anomalies could not be considered as reliably determined.

Introduction

In April 1975, the National Aeronautics and Space Administration (NASA) launched the Geos-3 spacecraft. This satellite was to be used for geodetic purposes as had been done with Geos-1 and Geos--2. One of the new types of measurements to be made with Geos-3 was that made by an altimeter capable of determining the distance from the satellite to the ocean surface. This distance and related data can be used by geodesists and oceanographers to determine information related to the gravity field of the earth and to the ocean surface.

For certain geodetic applications the range from the satellite to the ocean surface can be used to derive estimates of sea-surface topography or with some corrections geoid undulations. These undulations along profiles of the satellite arcs can be used to derive undulation maps in the ocean areas. In addition these geoi~al heights can be used to derive gravity anomalies as described, for example by Koch (1970), Gopalapillai (1974) and Rapp (1974).

This paper describes a procedure that can be used to derive geoid heights from

Bull. Geod. 31 (1977)pp. 73-8&

73

R. R U M M E L and R.H. RAPP

altimeter sea surface heights as measured by the Geos-3 altimeter. The method will produce a consistent system of geoid height profiles without the need for precise orbits. The technique developed will be applied to actual'Geos-3 altimeter data.

Error Sources for a Geoid Height Determination from Altimeter Date

The relation of the altimeter data to the geoid heights is described by two simple expressions. First, the geometric sea surface height, i t , can be derived from the altimeter measurement to a point 1= by the approximate formula (Gopalapillai, 1974) :

h = Ps- a - p p , + pP' ( 1 - PP' ) e 4t /n: 2 8 p= ~P'

(1)

The quantities are, as shown in Figure One,

& s

t

r

Figure One. Altimeter Geometry.

i

Ps

Pp,

geometric sea surface height, i.e. the separation of a defined geocentric reference ellipsoid to the instantaneous sea surface,

altimeter measurement

geocentric radial distance of the satellite

geocentric radial distance of the point P' which is the foot point of 1 = on the ellipsoid

eccentricity of the ellipsoid.

Then, the geoid height, N , can be determined from (see Figure One) :

N = h - H , (2)

where H is the instantaneous deviation of the sea surface from the geoid (denoted as orthometric height in the geodetic terminology).

The value of H will depend on such factors as tides, currents, winds, atmospheric pressure, etc... For these studies the only quantity considered known (or modeled) is the tidal effect denoted by t . Knowing t the quantity (H - t) will be treated as noise in later processing. In addition the estimated sea surface height, ~', estimated from the altimeter data and the initial orbit, will deviate from the true value h by the error, d p , of the computed orbit, and by the error n , , of the altimeter measurement.

The three types of errors, i.e. the orbital error d p , the instrumental noise n , , and the unmodeled components of the orthometric height (H - t ) , may be characterized by their presumed approximate wavelength features. In Table One these features are expressed in dependence of n , where n is the degree of a Legendre polynomial expansion between ( - ~r, + ~r ) .

The purpose of our procedures will be to remove the three error components in

74

UNDULATION AND ANOMALY ESTIMATION ...

a reasonable manner and by this to obtain reliable estimates for the geoid heights.

Data C ~ r ~ ' l ~ r i s t i c s

The main pert of the processed data is taken from two tapes in the BCD format received from the NASA Wallops Flight Center in February, 1976. The tapes contain a total of 64 arc segments with altimeter data from the GEOS-3 satellite collected between April 21, 1975 and May 20, 1975. We selected out of these 64 arc segments the 48 arcs in the Atlantic Ocean area inside the geographic area with [ x [ 260 ~ <" x <~ 12 ~ ] and [ ~ I 0 = ~ ~ <~ 68 ~ ] . Five additional arc segments with preliminary data,, distributed in Octobre, 1975, were added to give a total of 53 arc segments for our analysis. Their angular length varies between 1 ~ and 72 ~ . The observations were partly gathered in the short pulse mode and partly in the long pulse mode (Leitao et al., 1975).

The preprocessing of the altimeter data and the information provided on the tapes in the BCD format are described in (ibid). For our purpose the important quantity on the tape is the estimated geometric sea surface height, -s which is different from the true geometric sea surface height, h , due to the orbit error d p and the altimeter error n=, as previously explainecL The sea surface height ~- is provided at sample rates of 2.048sec (-" 0.12~ angular length) and :3.277sec ( - '0 .19 ~ angular length) . The estimates ~ are already corrected for tropospheric refraction and altitude bias. The altimeter data, ] , used for the determination of T is a mean of the ori0ihal measurements which have to fulfill certain quality criterions inside a time frame of length 2.048 sec and 3.277 sec, respectively as described in (ibid). The geometric sea surface height, "~, gives an estimate of the separation between the sea surface and an adopted reference ellipsoid as shown in equation (1) and Figure One. The adopted ellipsoid has the dimensions

a e = 6378145.0 m for the semi-major axis

and

f = 1. / 298.255 for the flattening.

Besides the information on the tapes we used the distributed listing of the GEOS-3 equator crossing times and coordinates.

Some preliminary processing steps were performed before starting the error analysis. The sea tide correction, t , also provided on the tapes was subtracted from the geometric sea surface height. Obviously incorrect sea surface heights were removed. This was mainly data gathered over continents and over islands and in addition a few sea surface heights which were unreasonably off from the neighboring heights. The geocentric coordinates of the crossover points of the satellite tracks in ocean areas were determined. For these points the geometric sea surface heights, "~, were linearly interpolated for each of the crossing tracks from neighboring data.

To a certain degree apriori information is available about the three considered types of error influences, i.e. orbital error d p , instrumental noise n=, and the residual orthometric height (tl - t ) . Such information can be described as follows :

1. The considered 53 arc segments have 76 crossover points in the ocean area. At the crossover points the estimated geoid height, derived at each of two intersecting tracks

75

R. RUMMEL =rod R.H. RAPP

should be identical. Thus, the actual differences of the unprocessed interpolated geometric sea surface heights ~" provide an error indication. A histogram of the discrepancies for the 76 crossover points prior to any adjustment is shown in Figure Two. The maximum crossover discrepancy is - 50.8 m. Especially noteworthy is the high epriori r.m.s, discrepancy of + 12.43 m.

2. The differences between the geometric sea surface heights, ~, and the geoid heights as computed from the potential coefficients of the Goddard Earth Model 7 (Wagner et al., 1975) provide some indication about the gravity information surplus implied by the altimeter data and about the amount of noise.

3. Finally, the standard deviations of the altimeter measurements are given on the observation tape. They are estimated from the deviations of the accepted observations from their mean �9 inside each of the mentioned time frames. From them we obtain some insight into the magnitude of the random errors of the altimeter measurement itself and the high frequency sea surface fluctuations.

Treatment of Long Wavelength Distortions

The first part of our procedure will be concerned with the removal of the long wavelength errors in the given estimated geometrical sea surface heights, ~'. As shown in Table One, we expect these errors to be due to orbital errors and instrumental bias.

We emphasize that we do not expect to gain any information about.the low degree potential field of the earth from the altimeter data considering the limited coverage of the ocean areas by the altimeter arcs at the present time. With this in mind, we switch for the analysis from the adopted reference ellipsoid to the equipotential surface generated by the GEM-7 coefficient set up to degree 16. In other words, we subtract from the geometrical sea surface heights, ~-, the corresponding geoid heights, NGEM7 with the same flattening f , as the adopted ellipsoid, where

= GM N ~ (Cn=m cos m X + Snm sin m Z) Pnm (sin ~) 131 NGEM7 Op')' n=2 m=0

and pp is again the geocentric distance to the point of which the geoid height is computed. All other terms are explained, for example, in Rapp and Rummel (1975, p. 2). The resulting quantities are the residual geometric sea surface heights, dE,

ds' = s"- NGEM7 (4)

This and all further quantities referred to NGEM7 will be denoted by "d." . After our

processing, NGEM7 will be added back again.

All long wavelength components remaining in di after subtraction of NGEM7 can be identified as orbital errors, 4 o , instrumental bias and errors in the GEM-7 set of potential coefficients. These distortions are modeled separately for each track by a first degree Legendre polynomial with unknown phase. The low degree of the polynomial prevents an effect on the medium wavelength gravity information in the data.

Let us consider the model for a particular observation at the point Pl of track

number j with nj data points, [ i l i = 1 . . . . . n j ] , [JlJ = 1 .... k ] and ~ k nj = n j= l

76

UNDULATION AND ANOMALY ESTIMATION . . .

where k is the number of tracks ( = 5 3 ) and n is the total number of considered observations ( = 7 2 9 7 ) . We have

d;j i - ~ j l = X j l4" y j I c~ 4" y j 2 ) , (5)

where dhji iS the vector of residuals in our least squares polynomial f i t and is the desired trendfree residual sea surface height ; # j i is the spherical distance of Pi from the equator crossing of this arc.

After applying the transformation x2 = Yl co t Y2 and x3 = - Yl sin y= the final model with three unknowns x , , x2 , x3 becomes

,- d'~i - dhji = x j l - I - cos~p j I xj2 "l ' ' in~j i xj3

or as a complete system for all tracks in matrix form

d~ dh = A X (6) d i m : n . l n , l 'n,3k 3k.1

For tracks shorter than 22.5 ~ only one unknown parameter, x j , , is introduced. The

normal matrix AI"A has a block diagonal structure with submatrices of dimension (3 �9 3) . This expresses the fact that at this stage the same result can be obtained by a separate least squares polynomial fit to each track.

The situation changes after considering the crossover discrepancies. Equation (6) does not take care of the already mentioned condition for the crossover points. Therefore, additional "pseudo" observation equations are introduced for the crossover points. The purpose of these equations is to force the system of footprint tracks, after a proper Weighting with the system of equations (6), to fulfill in tendency the condition of identical (adjusted and trendfree) residual sea surface heights dh" at the intersection points for each of two crossing tracks. After the least squares adjustment only medium and short wavelength noise shall contribute to the remaining crossover point discrepancies.

We assume for example track number two intersects track number four in the point Ps with [s ls = 1 .... m ] where m is the number of crossover points (=76 ) .

Thus we obtain for the "pseudo" observation equation

d ; 2 • - d~'4s - d h 2 s dh4s = x2! - X41 4" COS~2~X22 - cos ~4s

J," sbi @ 2.~ X23 -- SEn 4, 4, ~ X43 (7)

The complete system for all crossover points becomes with ~24s = d~2s - d;'4Q and

~h24e = dh2e- dh4e

,,'~ AT, B X = - -- (8) dim : m �9 1 m �9 1 m . 3k 3k .1

For the weight matrix P, for the system (6) we choose P, = ! the identity matrix, and for P__2 for the system (8) we use P_3% = 400 I . The factor 400 is derived empirically and is in essence a function of the number of crossover points per arc. For the ratio 1:400 between the along track observation equatio n weights and the crossover point equation weights the residual crossover point discrepancies are in proper balance to the r.m.s, value of the adjusted residual sea surface heights, d'h .

77 6

R. RUMMEL e~d R.H. RAPP

The complete adjustment model is then, with equations (6) and (8)

I; ~ [; o] [;] dtm:(n+m).(n+m)(n+ra).t ( n + m ) . ( n + m ) ( n + m ) ' ! (n+m). (n+m) (n+m).3k 3k.[

(9)

The system of equations for the individual tracks becomes linked by the crossover point equations, i.e. the normal matrix looses its block diagonal structure. The linear system to be solved has a maximum dimension of (3k- 3k) where k is the number of trackrh Since in the average SO ~ of the arcs are shorter than 22.5 ~ angular length, for which we solve only for one unknown instead of three, the linear system to be solved has usually a dimension of about ( 2k. 2k) . When the number of tracks to be adjusted becomes too large -- for example more than 500 arcs at a time -- the arcs are divided into different geographical areas and processed separately. Thereby the overlapping and already adjusted arcs from neighboring areas will be kept fixed.

Results of the Lust Squares Polynomial Fit with Cro=over Point Cmtditions

The desired quantity from the adjustment is not so much the trend parameter vector X but the vector dh which represents the trandfree residual sea surface height and the vector A_~.h of the remaining discrepancies in the intersection points. In our procedure, the quantity

_~h = d h "1" N_.GE M 7 (10)

represents the best estimate of the geometrical sea surface height (compare equation (1)) as derived from the provided sea surface heights ~- The best estimate of the sea surface height h (except for the tide effect, t , subtracted previously} is considered to differ from the best estimate of the geoid height, N , only by medium wavelength and short wavelength random errors due to instrumental noise and the residual orthometric height ( H - t) .

A good impression about the improvement gained by the adjustment is obtained by comparing the discrepancies in the crossover points prior to and after the adjustment. A histogram of the discrepancies ~ (aposteriori values) and of the A;" (apriori values) is presented in Figure Two. The standard deviation for these values dropped from + t2.43 m prior to the adjustmept to +0.94 m afterwards. This remaining standard deviation is of the order of magnitude of the expected noise level for the GEOS--3 altimeter data

The root mean square (r.m.s.) v~ue of the residual sea surface heights, d;', dropped from • t t.2 m to • 5.5 m for the adjusted and trendfree residual sea surface heights, dh. Without the crossover point condition, using solely system equation (6), the r.m.~ value for dh was • 5.0 m. The r.m.s, dh values, which contain the medium and short wavelength noise, agree well with the undulation information above degree 16 of • 4.4 m implied by the anomaly degree variance model suggested in Tscherning and Rapp (1974). This agreement is also reasonable considering that the statistical expectations over the analyzed area may deviate considerably from the global expectations. The r.m.s. estimates are collected in Table Two.

By assumption, the adjusted trandfree sea surface heights h differ from the best

78

UNDULATION AND ANOMALY ESTIMATION ...

estimate for the geoid heights N only by random noise. Thus, this data already serves as input quantity for least squares methods of gravity anomaly recovery such as least squares collocation. The necessa~/ noise matrix D can be designed as a diagonal matrix with the given altimeter noise variances as diagonal elements. For deterministic methods of gravity anomaly recovery the geoid heights have to be recovered from the trendfree sea surface heights h in a second procedure in which the remaining error components in ~ will be removed.

~v~o

14 14 12

e v t - q l 4 range : 4 U - ~1, .0 - i ~ . 0

- 3 0 . 0 - Z l I . O - ZlI.O

- :P4.0 I~+ J l l 0

- 20 .0 - I ILO - I 1 .0

- 14.0 - 12.0

- IO.O - I1.0

- ILO - 4 .0

2 . 0 4"

| . 0 " - "

4 2 3.4 - I 3.,1-4

2 . I - I 2 . 0 -4 1 .1-1 1.1-1 1.4 - I

m]2": ~111/////i///,~ ~_'~. w/////////////,~

�9 m ; t ~

of ewmts I

W E I G H T R A T I O O F C R O S S O V E R CONOITIOi~ T O T H E , ~ . T I M E T E R

T ~ OATI : P=/ P , , * 0 0 .

es l t l l4e r m q l l : - 50.11

Figure Two. Histogram of the Crossover Point Discrepancies Prior To and After the Least Squares Adjustment.

Removed of Medium and Short Wavelength Noise

The remaining noise components in the adjusted data, dh, are due to instrumental effects and due to the unmodeled or.thometric height, (H - t ) , as can be seen in Table One. Supposingly, the major amount of the noise will be concentrated in the short wavelength close to the shortest resolvable wavelength but also medium wavelengths will be influenced. Realistic information about the noise is provided through the already mentioned standard deviations of the altimeter measurements. Their determination is described by Leitao e t e l . , (1975). The standard deviations have only small fluctuations inside each track but the mean standard deviation % j varies from

nj o:,jl) I n ]2 J- is the standard deviation of track to track, where oa, j = [ (2 ;1=1 and oa,ji the altimeter measurement in the point Pi of track number j . As expected, a significant difference appears between the mean standard deviations for the short and for the long

_L pulse mode. This is shown in Table Three, where o a = [( ~jk__ s %2j ) /k ]2 .The expected value (ibid) are given in parenthesis.

79 6,

R. RUMMEL and R.H. RAPP

best method for removing random errors in the data is least aTuares filtering if reasonable information about the error structure is available. We will use the least squares filter technique assuming the remaining error components in dh to be random with constant variance o~j for each track. As an alternative the smoothing of

the data by the moving average method has been considered as described in (Rummel, 1976). The latter procedure is extremely simple and economical, but it has the severe

2 drawback that specific noise features such as those given through the variances oaj cannot be reflected properly.

The model for least squares filtering is given in equation (11). Assuming the determined data ah to be a mth order random Markov sequence a filter model of the form

")t ' i l t

dN" i = ~ W k dhi+ k (11) k=-m

is appropriate. The estimated residual geoid height at a point Pi is denoted dN 1 . The filter coefficients _w k are derived from the discrete Wiener - Hopf equation by

wT = s Cdh I = ---dN (s - t -D) - 1 (12)

which includes the matrix for the random error model D . CdN outside the pararenthesis

is the autocoveriance vector between the signal dN at the observation points Pi +k and

the signal dN at the estimation point El �9 The matrix elements ere taken from the global

covariance model of Tscherning and Rapp (1974) using the subroutine COVA for n > 16. The matrix Cdh = CdN + D consists of the autocovariance matrix between

the geoid heights in the points Pl+k and of D , the corresponding noise matrix. For

each track a constant noise matrix D = aa=,j [_. was applied. From numerical experiments

the filter width (2 r e + l ) = 11 was chosen. Because of this narrow filter width no large linear system has to be solved.

Geoid Results

As already stated, the data obtained from the least squares filtering represents our best estimate of the residual geoid heights. Together with the geoid heights as computed from the GEM-7 potential coefficients using equation (3) the final estimated geoid heights are

= d N + NGEM. / (13)

An example for an arbitrary data profile in its different stages of processing is presented in Figure Three. Shown is an arc segment of GEOS-3 orbit no. 410 with a mean standard deviation of oa,41 o = + !.'/8 m. The filtered and adjusted profile shows

medium wavelength variations ( - ] = ) . To identify these variations as geoid height fluctuations additional information from oceanographers, or by repetition of altimeter arcs, or by precise sea gravimetric measurements would be very helpful.

Since the least squares polynomial fit was performed with respect to the GEM-7 equipotential surface the estimated geoid heights obtained from equation (13) refer to

80

U N D U L A T I O N A N D A N O M A L Y E S T I M A T I O N ...

- 0

- I 0

-20,

~ -30

g

�9

- 4 0

-50

-60

-70

Figure 3: arc segment from GEOS-3 orbit no.410, long pulse mode, for different stoges of processing

~ ~ , . ~ g e o i d heights from GEM-7 - - "~.~ J / - adjusted and filtered seosurface heights

~ . ( , / (=estimated geoid heights) .. ~ " ~ ~ adjusted seasurface heights -o-o--o-- -

+~+~+. ~."~c---raw seasurface heights (from WFC) +,§

"14" + +4-

+§ % 4. +

~ § 4- §247

I I I I I

0 4 8 12 16 20 Ps'r (degrees)

[sphericol distonce from the first point]

24

81

R. RUMMEL and R.H. RAPP

an ellipsoid defined by an unknown best set of constants with the same flattening as the adopted set of constants. They no longer refer to the initially adopted reference ellipsoid.

These geoid heights serve as input quantity for deterministic methods of gravity anomaly recovery. They may also be used for geoid interpolations. From the resulting geoid heights of equation (13) a geoid map has been interpolated in the calibration area of the ~EOS~3=~tellite, ['~J12 ~ <~ ~ < 40 ~ and [x I 278 ~ <~ x < 300~ Because of the still limited number of available arc segments the map is considered to be :~reliminary. The contour line map is shown in Figure Four.

A comparison was performed with the geoid as computed by Rapp and Rummel (1975) and with the detailed geoid as computed by Marsh and Vincent (1975) by sampling corresponding geoid heights in a 2 ~ x 2 ~ grid. The number of sampling points is 90"and 83 respectively.

The mean difference and the standard deviation of the centered geoid height differences is given in Table Four. The mean and the standard deviation are also given for the differences NG~M 7 minus H from (Rapp and Rummel, ibid) to provide some

insight about the gravity information surplus gained by the altimeter derived geoid heights.

Although the number of altimeter arcs used for the geoid computation was limited the comparison shows a very promising picture. The results of the geoid comparison indicated that the quality of the altimeter measurements as well as the used processing procedure are satisfactory.

Anomaly Recovery -- Theow

If we assume that the geometrical sea surface heights as given by equation (10), corresponds to geoid undulations, we can use these values to estimate gravity anomalies u sing least squares collocation as suggested by Smith (1974), Rapp (1974) and others.

The prediction of a gravity anomaly, As, from the altimeter data, h', can be written as :

•g = C=h (C_.hh +_.D) -1 b_-- (14)

2 m s = -CI~i~ - ~i~h (Chit "J'-D)"I Chs (15)

where m s is the standard deviation of the prediction. The C values represent the wollowing covariances :

C=a : covariance between the mean anomaly to be predicted and the point altimeter (undulation) data ;

Chh : covariance between the measured altimeter (undulation) data.

D is the error--covariance matrix of the observations which was taken in these computations to be a diagonal matrix with elements taken to be the o=j I values provided

by NASA. The covariances for Ci~ = and C== ware determined from subroutine COVA

described in Tscheming and Rapp (1974). A numerical integration procedure similar to that described in Heiskanen and Moritz (1967, p. 277) was used to determine the appropriate covariances between the mean blocks and the point altimeter data.

82

UNDULATION AND A N O M A L Y ESTIMATION ...

280 4-

290

- N

\

!

I I I I I

F i g u r e Four . Geoid C on t ou r Line Map in the GEOS-3 C a l i b r a t i o n A r e a In te rpo la ted f r o m the Es t ima tmt Geoid Heights . T he c o n t o u r l ine i n t e rva l is 2 m. In a r e a s w h e r e not enough da ta was a v a i l a b l e the c o n t o u r l ines a r e do t ted o r omi t ted . The u sed GEOS-3 a l t i m e t e r p ro f i l e s a r e shown by the th in l i n e s .

83

R. RUMMEL and R.H. RAPP

Anomaly Recovery- Results

For these tests we chose to estimate three 5 ~ equal area anomalies, two 2 ~ x 2 ~ anomalies and two 1 ~ x 1 ~ anomalies. The location of these blocks wes chosen to maximize the number of altimeter tracks in and around the blocks. The blocks, with nearby altimeter tracks, are shown in Figure Five.

In carrying out the actual predictions an ellipsoidal reference model and a model defined by the GEM 7 potential coefficients taken to degree 12 was used. In the latter case, the undulations implied by the GEM 7 coefficients were removed from the values obtained from (10) ; the covariances from COVA were computed with respect to this reference field ; and the final estimated anomaly was obtained by adding to the predicted anomaly, the anomaly implied by the GEM 7 coefficients.

The actual data used was selected to be in a cap surrounding the center of the block whose mean anomaly was to be determined. For the 5 ~ blocks this cap size was 6 ~ while for the 2 ~ and 1 ~ block size, results for a cap size of 5 ~ will be given here although other cap sizes were tested.

Not all available data points within the cap were used because the size of the matrix to be inverted could be too large for economical processing, and because the use of a large number of closely spaced data points can lead to instability in the inversion process. For the results reported here every third data point along a track was used, although other data point selections were tried.

The actual predictions and their accuracy are shown in Table Five. The terrestrial anomalies listed in this table are values obtained from actual gravity measurements in 1 ~ x 1 ~ blocks using the data collection described by Rapp (1975). We see that the results from the predictions with the two reference models agree well considering the accuracy of the prediction. In addition the predicted anomaly values show reasonably good agreement with the values based on actual gravity measurements.

Concludom

In this paper a procedure for processing undulations implied by measured altimeter data with approximate orbits has been described that attempts to remove long wavelength error components due to systematic altimeter error, and orbital error. This procedure incorporates a simple error model combined with crossover conditions to determine adjusted undulations that still are contaminated by noise effects related to the actual altimeter measurement, sea state, etc... This latter noise can be filtered out by using least squares filtering. In this" paper this filtering was accomplished using the given standard deviations for the altimeter measurements and a fi lter width comprising eleven data points. Such a technique was applied to tracks in the Geos-3 calibration area with the results being used to cons~uct a limited geoid undulation map based on altimeter measurements, A comparison of these undulations with values obtained from a combination of potential coefficients and 1 ~ x 1 ~ gravity anomalies showed a systematic difference of only 0.25 meters with a root mean square difference of + 1.9 meters. This good agreement points out the success of the proposed method of processing the altimeter measurements without the need for precise orbits.

The unfiltered, but adjusted undulations, ware used to determine gravity anomalies within several 5 ~ , 2 ~ and 1 ~ areas near the calibration area. The predicted anomalies showed good agreement with known values when the accuracy of the data was considered.

84

U N D U L A T I O N AND A N O M A L Y ESTIMATION ...

5 0 ~

4 0 ~

(3

\ /

2 9 0 ~ 3 0 0 ~

\ /

310 ~

Figure Five. Location of Test Blocks and Related Al t imete r Tracks in the Anomaly Estimation Area

85

R. RUMMEL and R,H. RAPP

In future work, more data will enable more accurate undulation and anomaly information to be determined in additional geographic areas. The new work will need to examine sea state effect, potential coefficient models, covariance models, cap sizes and the various other computational parameters. There is every indication at this point that the Geos-3 altimeter is. going to provide the scientific community with increased knowledge related to detailed geoid undulations, and to the earth's gravity anomaly field in the ocean areas.

Acknowledgement

This study was supported by the National Aeronautics and Space Administration through NASA Contract NAS6-2484 administered through the Wallops Flight Center, Wallops Island, Virginia 23337.

0

O, 0

TaMe One

Approxinmte Wmmkmglh Behlwioy of Different Type= �9 of ComidenKI Noise

d'p

Ol - t )

%

Ions n<20

/////////

FJiJfh

rr~dium "20"_~ n <200

short

200 <n

TaMe Two

Root Mean Square Values of the Geometric Sea Surface Height= Prior To sad After the Adjustment

m

o(dh) .(dh) - ~ >le

Quantity Value (n~ters)

(unadjusted} (adjusted with cro,,over cond.) (adjusted, no crossover cond.) Tscherning--Ripp model

+11.2 i 5.5 + 5.0 • 4.4

86

U N D U L A T I O N A N D A N O M A L Y E S T I M A T I O N ...

Table Three

Mean Standard Deviations for Long and Short Pulse Mode

Arcs

Total (k = 53)

Long Pulse (k = 32) Short Pulse 0(=21)

O a

+1.15m

• 1.84 m (1.0) m + 0.78 m (0.6) m

TaMe Four

Al t imeter Geoid Height Comparison with Respect to Two Combinat ion Geoids :

Rapp and Rummel, (1975) end Marsh and Vincent, (1975)

Quanti ty Differenced

N (altimetry) - N (Rapp - Rummel)

N (GEM 7) - N (Rapp- Rummel} N (altimetry) -- N (Vincent -- Marsh)

Mean Diff. R.M.S. Diff.

0.25 m • 1.92 m

0.60 m • 3.42 m

4.02 in + 2.09 m

TaMe Five

Anomaly Prediction Remits

Block Size Terrestrial Predicted Anomaly Nun~ of Anomaly (regals) E* N = 12 + Pc/nts

A S~ E.A. B 5 ~ E.A.

C 5 ~ E.A.

D 20X2 ~

E 2~ ~

F 1 ~ x 1 ~

G I ~ x 1 ~

- 1 7 • 3

- 2 4 + 2

- 8 + 3

- 3 1 + 7

- 8 + 7

- 2 5 + 14

- 8 + 1 5

- 1 6 + 6

- 2 4 + 5

- 3 + 6

- 3 3 + 7

- 2 •

- 3 0 + 1 1

- 3 + 1 0

- 2 0 + 3

- 2 5 + 3

- 1 1 + 3

- 3 2 + 5

- 8 + 5

- 2 9 + 9

- 9 + 8

198

225

164

161

142

165

145

* Prediction made w/th the eIlipsoidal zeference model.

t Prediction made with GEM 7 to degree 12 as the reference model.

0

0 0

8 7

R. RUMMEL and R.H. RAPP

REFERENCES

S. GOPALAPI LLAI : Non--Global Re~bwry of Gravity Anomalies from a Comb/nation of Terrestrial and SBtallite Altimetry Data, Department of Geodetic Science, Report No. 210. The Ohio State University, Columbus, 1974.

W. HEISKANEN and H. MORITZ : Phydcel Geodesy, W.H. Freeman, San Francisco, 1967.

K.-R. KOCH : Gravity Anometles for Ocean Areas from Sat#lire Altimetry, Proceedings of the ~_~eco__nd Marina Geodesy Symposium, Marina Technology Society, Washington, D.C., 1970.

C.D. LEITAO, C.L. PURDY and R.L. BROOKS : Wallops GEOS--CAltimeter Pra-~oceuing Report, NASA Wallops Flight Center, Wzdlopz Island, Virginia, 1975.

J.G. MARSH and S. VINCENT : Grwirnetric Geoid Computa~ons and Comparisons with Skyiab Altimeter Octa in the GEOS-C CIIibration Area, paper presented at the 16 th General Assembly of the International Union of Geodesy and Geophysics, Grenoble, 1975.

R.H. RAPP : Gravity Anonwly Reoovery from Smtelllte Alt imetry Deta Using Lemt Squaraz Collocation Techniques, Report No. 220, Department of Geodetic Science, The Ohio State University, Columbus, 1974.

R.H. RAPP : The Gravitational Potential of the Earth to Degree 36 from Terrestrial Gravity Data, paper presented at the International Association of Geodesy Assembly, Grenoble, France, 1975.

R.H. RAPP and R. R UMMEL : Methods for the Computation of Detailed Geoidsand Their Accuracy, Department of Geodetic Science, Report No. 233. The Ohio State University, Columbus, 197K

R. RUMMEL : Geos-3 Altlmetar Data Pronsning for Gravity Anomaly Recovery, abstract, EOS, Trens. Amer. GeophyL Union, Vol. 57, No. 4, p. 234, April, 1976.

G SMITH : Meen Anomaly Prediction from Terra~rlal Gravity Data and Satellita Altimeter Data, Report No. 214, Department of Geodetic Sdence, The Ohio State University, Columbus, 1974.

C.C. TSCHERNING end R.H. RAPP : Oosed Covarlance Expressions for Gravity Anomalies, Geold Undulations, and Deflections of the Vertical Implied by Anornaly Degree Variance Models, Report No. 208, Department of Geodetic Science, The Ohio State University, Columbus, 1974.

C.A. WAGNER, F~I. LERCH, J.E. BROWND end J.A. RICHARDSON : Improvement in the Geoponmtial Derived from Satellite and Surface Data, NASA preprlnt X--921--76- 20, 11 p., Goddard Space Flight Center, Greenbelt, Maryland, 1976.

R e c e i v e d : 1 8 . 0 8 , 1 9 7 6

Rev lxed : 1 0 . 1 1 . 1 9 7 6

88