THE USE OF TOPOGRAPHIC HEIGHTS IN GRAVITY FIELD PREDICTION

Transcription

1 513 THE USE OF TOPOGRAPHIC HEIGHTS IN GRAVITY FIELD PREDICTION B. Benciolini, F. Sansò Istituto di Topografia, Fotogrammetria e Geofisica Politecnico di Milano, Milano, Italy Abstract A method for gravity anomalies interpolation and prediction using topographic heights is proposed; it is free from any hypothesis on terrain density. Tests with data from a very rough area are reported. 1. Introduction This is a progress report about thè work performed in Milan about thè gravity field approximation and prediction. The work is done in thè frame of a more generai Italian research program named ITALGEO 90; thè final task of this research is thè computation of a high precision geoid for thè Italian area, thè intermediate steps are thè data collection, organization and validation, and thè development, testing and comparison of various mathematical models and computational procedures. The particular topic of this paper is an interpolation and diction procedure of gravity anomalies on a locai basis; this is of interest f'or thèactjlyity of SSG.s 3.90, 3.88 and 4.70 too.

2 f 514 We will start with a short review of thè relevant methods we know for gravity field interpolatici! and prediction using terrain heights as data, too; these methods are deterministic, stochastic or of mixed type. Note that in this context thè word "stochastic" simply refers to thè use of a mathematica! mode! where some parameters are computed using thè proper best fitting criterion, and that approximations are performed via collocation using an empirically determined covariance function as reproducing kernel. The more classical procedure is completely deterministic, and it is carried out in 3 steps: - thè effect of topographic masses is subtracted from thè data (mainly Ag ) ; - integrai formulas are used to oompute T (or N ) and? and TI ; - thè effect of topographic masses is added back to thè results. Sometìmes an interpolation procedure is applied to densify thè Ag data, by means of a polynomial model of global or regiona- 1ized type. A different possibility is thè use of collocation for estimating thè requested quantities, coupled with thè use of thè classical method to account thè effect of thè topographic masses (C.C.Tscherning, 1984; Forsberg, 1984). An other proposai is contained in Heiskanen and Moritz (1967) where it is suggested thè use of a least squares linear prediction technique based on thè knowledge of thè auto- and crosscovariance functions of thè gravity field (represented by Ag ) and of thè terrain heights. In thè procedure called mixed collocation (Sansò-Tscherning, 1982; Tscherning, 1984) thè density of thè visible masses is an unknown to be determined by a proper least norm criterion. Sùnkel and Kraiger (1984) apply thè collocation method with

3 parameters to thè prediction of terrain corrected Faye anomalies, with a raodel which is linear in thè point height. 2. Pur proposai for gravity modelling using topographic heights We have developed a procedure based on collocation with parameters for gravity anomaly interpolation and prediction; subsequently thè model has been modified to make it physically meaningful and to allow, in thè future, thè computation of different quantities related to Ag. A model similar to that one of Sùnkel and Kraiger has been used first, but its application to uncorrected gravity anomalies requested thè use of parameters to account thè effect of visible masses. The complete model is: Ag = b + b.h + Z a.h + P(N,E) + v (1) o lo 33 2 = s + n (2) where: Ag is thè gravity anomaly H o is thè point height H. are thè mean heights (referred to thè point height) P (N,E) in thè circular zones around thè point (fig. 1) is a polynomial trend of second order, function of thè planimetrie position v, thè residuai, is broken into signal (s) and noise (n) The parameters b, b, a. and thè coefficients of O 1 3 thè polynomial P are- unknown and must be estimated. The numerical application of this model to a sample-zone in Northern Italy shows that ali thè parts of thè model (1) are

4 & 600 m 1200 m 1200 m 1800 m 1800 m 2400m significant for thè reduction of thè residuai variance. The meaning of thè different parts of thè model can be better understood looking at fig. 2. The formula (1) has to be applied on a locai basis (say in a l xl or 30'x30' zone) and we hope it can work in quite rough areas too. In this case, we must consider thè effect of thè nearest topographic masses, which is accounted by thè terms b and b H (corresponding to Bouguer piate reduction) and by thè summation of thè circular zone mean heights (corresponding to thè so-called residuai terrain correction). The presence of big topographic masses in thè immediately neighbouring zone and of mass anomalies have a large but smooth ef-

5 517 fect on thè gravity in thè area under investigation and can therefore be reasonably represented by a polynomial. We choose a polynomial in North and East coordinates only (not H) mainly because of thè shape of thè 3-D region where thè data points are: it is extended about 30 * 50 Km on thè piane, and only 1*3 Km in height. Furthermore thè residuals v can be split into noise (unpredictable) or signal (predictable) to account in a different way (stochastically) ali thè effects not-explicitely considered in thè model; this requests thè determination and use of a suitable covariance function. The main criticism against expression (1) is that it is purely empirical in its form which prevents its use for thè prediction of quantities different from Ag, mainly of geoid ondulation (N) The first modification which can be brought to expression (1) is suggested by geophysics: thè so-called linearized expression of terrain correction is a function of thè squares of thè residuai heights.

6 518 We therefore substitute (1) by: Ag = b +bh + E a (H) + P (N,E) + v (3) o 1 o j j 2 Let's recali that H is always referred to H, from point to point. The second attempt to make thè model closer to thè physical reality is in using a harmonic polynomial trend P (N,E) as thè theory would request. This is only a first step in order to have a trend that can be harmonically extended in thè outer space and that can be used for geoid computation. The use of a polynomial, instead of trigonometrie functions, as proposed by some authors, is justified by thè need of a very smooth interpolating function if we want our previous interpretation to be plausible. We shall come back later on this point. A comparative remark is now in order before considering thè first attempt of applying thè proposed procedure to a real case. The classical methods for gravity field modelling using topographic heights require thè knowledge of some geophysical model to derive thè terrain density but such a model may be available or not. Only a few methods among that we know are free of this request; mainly thè Heiskanen-Moritz least squares procedure, thè internai collocation and, partly, thè Sùnkel and Kraiger procedure. Now, our method is an attempt in this direction, hopefully successful. 3. The data used and their covariance functions The test area we have chosen ranges from 45 00' to 45 40' in latitude and from -5 00' to -4 30' in longitude (referred to thè Italian longitude origin, i.e. M.Mario near Rome).

7 519 It is a quite rougged area, where heights range frora 100 m to 3,500, and it is marginally interested by thè so-called Ivrea body. The coverage of gravity data is quite dense and almost regular. Both gravity and height data have been supplied by thè University of Lecce, where a group of researchers is engaged in collecting and homogeneizing ali thè available gravity data relative to Italy. Height and bathymetric data have been prepared by thè sarne group by map digitization (see Carrozzo et al., 1981). The different data samples have been used to test thè gravity anomalies models described in thè previous paragraph. The corresponding zones are shown in fig. 3, and they will be called in thè sequel ^Complete ^one (CZ), Mountainous Zone (M2) and _South-j3ast J5one (SEZ). They differ in many respects: thè CZ is quite inhomogeneous from thè morphological point of view, because it has thè maximum altitude variation; thè MZ presenta almost thè same height variation, but it is much smaller, and this enables us to use a much denser data set; thè SEZ on thè other hand, is only moderately ondulated and presents thè maximum data density. Table 1 contains quantitative Information about thè characterictic of thè various zones. Note that thè values of M(H) and a(h) are computed from gravimetrie points only, and therefore they do not completely account thè true terrain roughness. A first computation has been performed to ensure that ali thè different parts of thè Ag model (1) are useful, and to obtain a first set of v values, from which thè covariance function can be estimated for thè application of collocation (see next paragraph). The estimation of thè covariance function is performed as usuai in three steps: thè empirical estimation, thè choice of a model and thè fitting of thè model to thè empirical function; thè cen-

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9 521 Table 1 CZ MZ SEZ Nr. of points used Mean point spacing (Km2/Nr. of points) M(H) (in m) ff(h) (in m) M(Ag) (in mg ) a ( Ag) (in mg) Latitude ' ' ' 45 00' Longitude ' -4 30' -4 50' -4 30' ' -4 30' trai problem is just thè proper choice of suitable family of covariance functions. The use of global covariance functions, used in a locai mode by removing a certain number of harmonics, is not convenient in such a small and rough area. Several models of locai harmonic covariance function have been studied by various authors; in this case thè most appropriate one is very simple: C(P,Q) a z A 1 J (a s) Jl where z = z + z and J is thè Bessel function of zero order. The good fitting of one of thè empirical covariance functions by thè interpolating function is shown in fig. 4; thè agreement

10 522 is almost as good as in this case for ali thè considered samples. Nevertheless other harmonic covariance functions are under investigation. mg V \l function model function x^n / * f\ \m I \ V / fig.

11 Numerical results Some computations have been performed first in order to check whether ali thè parts of model (1) are effective in reducing thè variance of thè residuals v. In this step thè simple 2 fitting criterion: E v. = min has been used, neglecting any correlation among thè data. The results, summarized in tab. 2, show that thè complete model (1) has to be used. The number of parameters of thè model is 13; if we compare it with thè numbers of points of thè samples and we look at thè prediction results (see below), we can reasonably say that we are modelling thè gravity and not only fitting numbers. Tab. 2 cz MZ SEZ a of data (Ag) 47.9 a of residuals (v) Ag = b + b, H + v 40.7 o 1 Ag = b + b,h + Za H. + v 34.1 o 1 33 Ag = b + b H + P (N,E) + v 16.8 o 1 i Ag = b + b H + Ea.H. + P(N,E) + v 10.8 o Further experiments show that thè polynomial is needed even if expression (3) is used instead of expression (1). The results of thè main experiments are shown in tab. 3; ali thè three samples have been treated with three different models, i.e. expression (1), expression (3) and this last with a harmonic polynomial P(N,E) instead of P(N,E). The choice between thè purely empirical model (mod. I) and one of thè two geophysically meaningful models (mod. II and mod. Ili)

12 524 Tab. 3 Sample CZ MZ SEZ Model I II III I II III I II III Nr. of data o of data a(v) a(n) Nr. of data a(v) o(n) Gravity values (in mg) MODEL I : Ag = b + b H + E a H + P(N,E) + V o lo D_j MODEL II : Ag = b + b H + Za.H2 + P(N,E) + v MODEL III:. A g = b +bh + X a H2 + P(N,E) #- v (P is harmonic) o lo j j ~==~--^7 is questionatale. As a matter of fact model I and model II (which have exactly thè same number of parameters) seem to give almost thè same performance in terms of r.m.s. of thè residuals. On thè other hand if we want to use these data in different contexts than thè pure gravity interpolation (e.g. geoid estimation) we have to stick to thè completely consistant model III. The correlation coefficients between thè residuals v and thè noise n of models I and II have been computed, and thè results are in tab. 4. The large values of p teli us that thè models have a very similar behaviour. The unaspected result p s p nn vv, and mainly thè large difference presented by thè results of thè SEZ, are /

13 525 probably due to thè different performances of thè covariance functions used in thè various cases. Tab. 4 CZ MZ SEZ p vv P, nn 5. Perspectives The main problem when modelling gravity with a purely statisticai procedure is that we are not able to compute other quantities related to Ag, as already mentioned (unless thè harmonic character of thè field is a-priori guaranteed). In our model thè difficulty lies on thè transform of thè polynomial into an expression that can supply thè correspondent geoidal height. On thè other hand thè usefulness of a polynomial trend to interpolate Ag has already been recognized by other authors (Sùnkel Kraiger, 1983), and corresponds, as we already mentioned, to thè right degree of smoothing of thè data to allow for thè interpretation given in 2. Another type of trend, i.e. by means of trigonometrie functions, would allow a much easier harmonic prolongation, as first step for geoid computation; but Fourier series are too slowly convergent for rough functions and therefore it is difficult to use them directly. A solution of this problem is perhaps thè following: - to interpolate thè data with an harmonic polynomial,

14 526 - to transform thè polynomial into a troncate Fourier series, - to built up thè harmonic prolongation of thè Fourier series. In this way thè data are interpolated by a much more significant function, and thè polynomial interpolation termediate step, mainly to smooth thè data. We have used a second order harmonic polynomial serves as in- ^ 2 2 P(N,E) = an + be + c(n -E ) + dne in a rectangle of sides A and B. The most convenient way of transforming o. P into a series of trigonometrie functions is to doublé thè rectangle in each di- % rection and to consider P as an even function of wave-lengths 2A and 2B (fig. 5), since in this way we avoid Gibbs phenomena related to boundary discontinuities. B fig. 5 Once Ag is given as sum of functions of thè form -kz e cos K x cos K Y x y (K = /K + K '

15 527 thè corresponding geoidal terra is -kz N = - = Y K cos K X cos K x y Y as one realizes after observing that such a function is harmonic, regolar to infinity (z -> ) and satisfies thè proper Newman condition Aknowledgment The cooperation of thè colleagues of thè University of Lecce is kindly aknowledged. Financial support to this work has been partially supplied by thè Italian "Ministero della Pubblica Istruzione", under contract "MPI-40%-Sansò-1983". Aknowledgment The cooperation of thè colleagues of thè University of Lecce is kindly aknowledged. Financial support to this work has been partially supplied by thè Italian "Ministero della Pubblica Istruzione", under contract "MPI-40%-Sansò-1983".

16 528 References (1) Barzaghi, R., Betti, B., Sansò, F. (1984): Integrateci Geodesy: A Purely Locai Approach; Proc. of thè "3 Simposio GNGTS-CNR", Rome, Nov (2) Carrozzo, M.T., Chimenti, A., Luzio, D., Margiotta, C., Quarta, T., Zuanni, F. (1981): Realizzazione di un archivio delle quote medie; Proc. of thè "1 Simposio GNGTS-CNR", Rome, Nov (3) Forsberg, R. (1984): A Study of Terrain Reductions, Density Anomalies and Geophysical Inversion Methods in Gravity Field Modelling; Reports of Department of Geodetic Science and Surveying, O.S.U. Rep. n (4) Groten, E. (1966): Bestimmung der Schwere an der Erdoberflàche aus Fluggravimetermessungen; D.G.K., Reihe C, Munchen. (5) Heiskanen, W.A., Moritz, H. (1967): Physical Geodesy; Freeman & Co., S. Francisco-London. (6) Sansò, F., Tscherning, C.C. (1982): Mixed Collocation: A Proposai; Quaterniones Geodaesiae, 3, 1, p (7) Sùnkel, H., Kraiger, G. (1983): The Prediction of Free-Air Anomalies; M.G. 8/3, p (8) Tscherning, C.C. (1984): Locai Approximation of thè Gravity Potential by Least-Squares Collocation; Proc. of thè International Summer School on "Locai Gravity Field Approximation - Beijing, Aug. Sept. 1984", The University of Calgary, Calgary, 1985.