A Fast, Spatial Domain Technique for Terrain Corrections in Gravity

A Fast, Spatial Domain Technique for Terrain Corrections in Gravity Field Modelling G. Strykowski KMS, National Survey and Cadastre, Rentemestervej 8, DK-2400 Copenhagen NV, Denmark F. Boschetti, F.G. Horowitz CSIRO Division of Exploration and Mining, 39 Fairway, Nedlands, WA 6009, Australia Abstract. A fast computation of terrain corrections requires (1) a quick forward algorithm, and (2) a strategy for organising the data in a computer program. In this paper we present some new ideas on (1). The discussion of (2) is very brief. The proposed method is a space domain method, similar to prism integration, but quicker and more flexible. We show that the method works both in the flat-Earth geometry and in the spherical Earth geometry. The "elementary body" of the mass density model is an infinitely thin, horizontal and homogenous rectangular lamina. The main speedup comes from replacing the exact formulas for the gravitational attraction of a lamina by an approximation, a polynomial model. We argue that such approximation is sufficient to ensure the high accuracy of the approximation. In fact, we have tried it in numerical simulations (not shown here). We chose a polynomial model, because it is straightforward to use it for derivation of similar models for other types of gravity data. In practice, terrain corrections are computed from height data given on a regular grid in either spherical or Cartesian geometry. The regular grid structure of the terrain information is a particular advantage. For a given grid spacing, and prior to terrain correction computations, one can construct a polynomial model for all the geometrical aspects of the gravity station-mass point configuration. This is valid for any type of gravity data. We briefly mention how such general polynomial model can be used for optimizing the computations of the terrain corrections. Keywords. speed of computations, space domain, terrain correction, homogenous rectangular lamina

1.1 Classical terrain correction In geodesy and geophysics, the terrain corrections are used to reduce the measured surface gravity data for the attraction of the terrain or parts of it, Heiskanen and Moritz (1967), Telford et al. (1976), Forsberg (1994) and Sjöberg (2000). In this paper we denote the location of the gravity station by P and the location of the mass density point by Q. In the spherical Earth approximation, the (Pellinen) terrain correction CP , Moritz (1980), Sjöberg (2000), is: CP =

G ∆ρ 2 R 2

The new method described in this paper is still "work in progress". Our objective here is to outline the main ideas. We still anticipate minor changes to the design of the method.

(H - HP )2 dσ l0

σ

3

(1a)

where G D )D

- gravitational constant - absolute mass density = Dtopo - Dref (e.g. Dtopo = 2670 kg/m3 Dref = Dair = 0 kg/m3 ) F, dF - unit sphere and its area element R - mean sea level radius of the Earth H - topographic elevation R - the spherical distance, Moritz (1980) l0 = 2R sin(R/2) Other notation concerning the spherical geometry: r,8,2 - radial distance, geocentric longitude, geocentric co-latitude (= B/2 -N, N-geocentric latitude); r = R + H For flat-Earth geometry, Sjöberg (2000), we have: CP =

G∆ ρ 2

∫∫ E

1 Introduction

∫∫

(H - HP )2 dE s

3

(1b)

where E - Cartesian plane, dE - the area element

s=

(x − x P )2 + (y − y P )2

Equations (1a),(1b) are approximations. Terrain corrections are defined as gravitational attraction of

the surplus/deficit of mass (in 3D) of the true terrain relative to the Bouguer plate, Heiskanen and Moritz (1967), Torge (1989), Forsberg (1994). For the flat-Earth geometry it yields:     z H   Q P c P = G ∆ρ dE  Q  dzQ 3    2 2 H ref ( x , y ) E Q   z Q - H P + s    

(

χ = Nχ (V)

(5a)

where NP is a linear or linearized operator. For example, for gravity disturbances *g z:

H ( x,y)

∫ ∫∫

linear/linearized relation between various types of gravity data P and the gravitational potential V:

)

(2)

Nδgz =

∂ ∂z P

(5b)

where In particular, one should notice that the operator acts on P, i.e. the location of the gravity station, and not on Q. The inverse operator is of course the integration with respect to the vertical parameter:

H(x,y) - height for the horizontal location of Q Href(x,y) - the reference level For example, refine Bouguer anomalies can be obtained from simple Bouguer anomalies using Href(x,y) = HP , Torge (1989). The correction for the residual terrain model (RTM) utilises Href(x,y) = Hsmooth (x,y), Forsberg (1984), where Hsmooth (x,y) is a low frequency model of the terrain. Furthermore, equation (1b) is obtained from equation (2) by approximation, Href(x,y) = 0 and for s>>|zQ -HP |. 1.2 General terrain correction

zQ - H P

(

 z -H P  Q

)2 + s2 

3

(3)

H ( x, y)

∫ ∫∫

Looking at the right hand side of equation (4), the only quantity that depends on P is GP(P;Q). From the "mother" Green's function GV for the attractional potential, GV (P;Q) /G/|P-Q|, a whole class of the associated Green's functions GP can be derived: (5d)

In practise, I and NP can be interchanged, i.e.:  H ( x, y )        c P , χ (P ) ≡ N χ  G ( P ; Q ) ∆ ρ ( Q ) dE dz  V Q Q  =  H ( x , y )  E   Q   ref   H ( x, y )    = N χ   GV ( P; Q )∆ρ (Q) dEQ  dzQ =   E H ref ( x, y)   Q H ( x, y)     = N [ G ( P ; Q ) ] ∆ ρ ( Q ) dE  χ V Q dzQ =   H ref ( x, y) E Q  H ( x, y     =  Gχ ( P; Q )∆ρ (Q) dEQ dzQ  H ref ( x, y) E Q  

∫

∫

From the well known relation between the spherical coordinates (r,8,2) and the Cartesian coordinates (x,y,z) the definition given in equation (3) can be extended to the spherical case. Equation (2) can now be expressed more generally (P=*g z):     c P , χ (P ) =  Gχ ( P; Q)∆ρ (Q)dEQ dzQ   H ref (x , y ) E Q 

(5c)

G χ ≡ Nχ (GV )

A general discussion of terrain corrections involve the concept of Green's function, GP , where P is some gravity field related quantity. For P=*g z, *g z is the z-component of the gravity disturbance vector, and for the Cartesian geometry, G*gz is: Gδ gz (P; Q) ≡ G

N -1δgz = ∫ dzP

(4)

Notice in equation (4) that the anomalous mass density is inside the integral. Furthermore, the structure of equation (4) is also valid for other types of gravity data P. The mathematical foundation of physical geodesy, Moritz (1980), makes use of a

∫

∫∫

∫∫

∫∫

∫ ∫∫

(5e) Equation (5e) shows for various P the relations between cP,P, GP and GV . Similarly, inverse operators, see equation (5b), can be applied to obtain GV from GP . However, this process is not entirely reversible. By comparing equations (5b) and (5c) it is clear that N*gz will "remove" the DC-

component (i.e. the mean value) from V. If one subsequently apply N-1*gz this generally non-zero DCcomponent cannot be restored. It is absent in the gravity disturbances. In geodesy the anomalous gravity potential T is used which has a zero mean globally, i.e. V=T.

2 Discretization of the Green's function In practical computations of terrain corrections Digital Elevation Models (DEM) are used. DEM's are often given on a regular grid in either spherical or Cartesian coordinates. Each grid point represents an average height for a grid cell of size )E: ∆E = ∆ x × ∆ y

(6a)

where )x and )y is the grid spacing in x- and ydirection. Thus, in Cartesian geometry, each grid point Qij , with horizontal location Qij = (xQ (i),y Q (j)), i=1,…,I and j=1,…,J represents an area Eij :  ∆x ∆x  Eij =  x Q (i ) − , x Q (i ) + × 2 2    ∆x ∆x  ×  yQ (i ) − , yQ (i ) + 2 2  

(6b)

Equation (4) can now be discretized using the concept of "elementary bodies", e.g. homogenous rectangular prism. Thus, from equation (5) we get:

3 Terrain Corrections - the new method Instead of equation (7) consider now:

cP , χ (P ) ≈   H (i, j )   ≈ ∑ ∑ ∆ ρij G ( P ; Q ) dE ∫  ∫∫ χ Q dzQ = i j  Href (i, j ) Eij   = ∑ ∑ ∆ρ ij × prismχ (i , j ) ≈ ∆ρ × ∑ ∑ prismχ (i , j ) i j i j

rent (constant) values )Dij for different prismP(i,j) at a cost of one multiplication per prism. Usually, this option is not used in geodesy, except when the objective is to model the gravitational attraction of inland water or ice, e.g. a lake or a glacier. More general models are used in geophysics, e.g. a more complicated mass density distribution in the vertical in each mass column. In equation (7) it corresponds to including more than one prism for each (i,j). The computational costs of prismP(i,j) is high. Closed expressions exist for a number of different quantities P. The reader can inspect these formulas e.g. for P=*g z or P=T in Forsberg (1989, equation (32), (33)) or in any other modern standard book on geodesy or geophysics, e.g. Blakely (1996). The expressions involve sum of functions of the type ln and arctan, each of which is computationally quite expensive. In addition, prism integration requires bookkeeping of the relative location of P with respect to the centre of mass of each prism. To speed up the computations there are some numerical shortcuts. One is the so-called MacMillan expansion, Forsberg (1989). From a large distance the exact prism formula prismP(i,j) can be replaced by a mass point formula, which is much cheaper. Furthermore, attraction of the distant parts of topography can be approximated using a coarser topographic grid, i.e. the heights of a detailed grid can be averaged and replaced by fewer, but larger prisms.

c P , χ (P ) =

(7)

∑∑ ∫ i

2.1 Prism integration- cost of computations Equation (7) shows explicitly how the so-called method of prism integration relates to the discretization of the Green's function. PrismP(i,j) is a volume integral of GP, volume =Eij H [Href(i,j),H(i,j)]. There are few comments to equation (7). The flexibility of the method alows to have diffe-

j

∫∫

H ( i, j )

=

where H(i,j)=H(Qij ) and Href(i,j)=Href(Qij ). Furthermore, in geodesy one assumes that the mass density of the terrain is a constant, i.e. )Dij =)D=constant.

    ∆ρ ij (zQ ) Gχ (P ; Q)dEQ dzQ E  H ref (i , j )  ij  H (i , j )

=

∑∑ ∫ ∆ρ ( ) i

j

ij ( zQ ) ×lamχ

(i, j )dzQ

(8)

H ref i , j

where lamP (i,j) is a gravitational response of a homogenous, rectangular and horizontal lamina, Eij . Clearly, equation (8) is more general (and flexible) than (7). Notice that )Dij , i.e. the mass density contrast, can vary arbitrarily in the vertical within each mass column. Using mathematical tables, the components of the vertical derivative of the gravity disturbance vactor *g P=(*g x,*g y ,*g z) were obtained. We will use the following short notation:

xoff (1) ≡ xQ (i , j ) − ½∆ x − xP xoff (2) ≡ xQ (i , j ) + ½∆ x − x P yoff (1) ≡ yQ (i , j ) − ½∆ y − y P

(9a)

yoff ( 2) ≡ yQ (i , j ) + ½ ∆y − y P zoff ≡ zP − zQ

where, xoff and yoff are the horizontal offsets between the corners of the rectangular area element Eij and P; the vertical offset zoff is the depth/height below the z-level of P. With this notation we get: y '+ x'2 + y' 2 + (zoff )2 xoff (2) ∂ (δg x ) ( zoff ) = G × ln( ) xoff (1) ∂( zoff ) x '2 + (zoff )2 ∂ (δg y ) ∂( zoff )

( zoff ) = G × ln(

x'+ x '2 + y '2 + ( zoff ) 2 y '2 + (zoff

)2

∂ (δg z ) x' y' ( zoff ) = G × arctan( ) 2 ∂( zoff ) x ' + y '2 + 12

)

yoff ( 2) yoff (1)

xoff ( 2 ) yoff (2 ) xoff (1) yoff (1)

xoff (1 ) yoff (1) − zoff − zoff xoff (2 ) yoff ( 2 ) − zoff − zoff

(9b) where the four limits of the double integration should be inserted in place of x' and y'. At a first glance not much seem to be gained from these expressions. Both atan and ln appear in the formulas. Nevertheless, when we plot the values as a function of the vertical offset, zoff, these functions are very smooth, see figure 1. They are clearly suitable for the approximation by a smoother function, e.g. a piecewise polynomial. Figure 1 shows examples for the spherical case for different horizontal offsets. The examples are given for HP =0, i.e. the depths represent the vertical offset with respect to the zero level. In section 3.3 we will return briefly to figure 1. Skipping all the details, the polynomial approximation in Cartesian geometry is equally good approximation. 3.1 Horizontal homogenous lamina and the polynomial model - cost of computations Replacing lamP(i,j) in equation (8) by a polynomial model, p P,(i,j)(zP -Q ), yields:

(

)

H (i, j ) c P, χ (P ) ≈ ∑ ∑ ∆ ρij ( zQ ) pχ , (i , j ) zP − zQ d zQ ∫ i j Href (i, j )

(10)

Fig. 1 Spherical geometry. Justifying the approximation of lam*g in equation (8), (9b) and (10), see sec. 3.1, by a polynomial model p*g, for different horizontal offsets. Top: small horizontal offset |x P-x Q| < ½)x and |y P-yQ| < ½)y. The thick dots are the exact values computed by equation (9b). The model p*g consists of "the half-step value" and the coefficients of a polynomial. Middle: medium horizontal offsets |x P-xQ| > ½)x and |yP-y Q| > ½)y. Notice that p *g and lam*g are indistinguishible on the figure. Bottom: large horizontal offset |x P-x Q| >> ½)x and |y P-yQ| >> ½)y. The relation is almost linear. The model crosses zero at a height of some 600 m (negative depths=positive heights). Thus, the sphericity of the Earth is included in the polynomial model.

Focussing only on one (i,j), i.e. just on one node of the horizontal grid, for )Dij (zQ ) . constant, and using a notation PP,(i,j), PP,(i,j) =Ip P,(i,j)dzQ , we get:

(

)

H (i, j ) ∆ρ ij ( zQ ) pχ , (i, j ) z P − zQ dzQ ≈ ∆ρ ij ( zQ ) × ∫ Href (i, j )

[

(

(

)

× Pχ , (i, j ) zP − H (i, j ) − Pχ , (i , j ) zP − Href (i , j )

)] (11a)

If )Dij (zQ ) in a mass column [associated with Eij ] is piecewise constant, e.g. ice on top of homogenous topography, equation (11a) can be split into intervals where )Dij (zQ ) is a constant. Thus, the attraction from a mass column is a sum of values of PP,(i,j) for different vertical offsets (to the top and to the base of each geological unit) multiplied by the mass density value. Notice that in a mass column with many overlying homogenous prisms, the price of computations is approximately one polynomial value per prism. The reason is that the base of one prism is the top of the other. Thus, the same polynomial value is used for both prisms. Unlike the prism integration, the bookkeeping of the relative location of P and the centre of mass of the lamina is not necessary. For a general )Dij (zQ ) the integral in equation (11a) is a 1-D convolution (denoted by *), i.e.: H ( i, j)

∫ ∆ρ

H ref (i , j )

(

i , j ( z Q ) pχ ,(i , j ) z P

)

− zQ dzQ ∝ ∆ρ i , j * pχ ,(i , j )

(11b) Notice that the transfer function is pP,(i,j) . Furthermore, at a cost of 1-D convolution per mass column, the attraction in P and in a number of points above and below P can be obtained.

Fig. 2 Spherical geometry. Polynomial coefficient an , n=25 independently estimated for different horizontal offset locations (xoff, yoff) and plottet as a function of: "(xoff)2 +$(yoff)2 . Top: (",$)=(1,1) small misallignment because of assymetry )x…)y. Bottom: "Fine tuning", better alignement for (",$)=(1.034,1)

Based on the tabulation of the coefficients of the polynomial model for discrete horizontal offset locations, we found it most efficient to construct a 3D-model of the polynomial coefficients, i.e: N

pδ g (z P − z'Q ) =

∑ a (x n

P

) (

− xQ , yP − yQ × zP − z'Q

)

n= 0

(12a) 3.2 General setup The general setup of the new method is such that the Earth's model is defined on a fixed horizontal grid, either in spherical or in Cartesian geometry. The location of the gravity station can be arbitrary (also inside the masses). Thus, the horizontal offset between a grid node and a gravity station is not necessarily a multiplum of ()x,)y). Consequently, it must be possible to obtain coefficients of a polynomial model for an arbitrary horizontal offset.

where z'Q is the variable vertical parameter and where the coefficient an (xP -xQ , yP -y Q ) for fixed horizontal offsets are real numbers. As a function of the horizontal offset the coefficients are polynomials:

( ) m ∑ Amn [(α mn × {xP − xQ })2 + (αmn × {yP − yQ })2 ] an x P − xQ , y P − yQ =

M

m=0

(12b)

where Anm are real numbers and " nm, $nm are "fine tuning parameters" (see figure 2). By inserting equation (12b) into (12a) the resulting model is a polynomial in 3D. Figure 2, which is also valid for other coefficients, justifies the construction of the 3D polynomial model. 3.3 Spherical geometry - special problems

4

Other refinements and conclusions

We have shown that a polynomial approximation to the discretized Green's function, see equations (8) and (10), yields a considerable computational speedup as compared to the standard prism integration. We have confirmed these ideas by random numerical simulations (not shown here), both in Cartesian and in spherical geometry. It is confirmed that the prism responses (in 3D) can be approximated arbitrarily well by the proposed method, i.e. the accuracy is mentained and depends only on the degree of approximation of the exact values by the model, see figure 1. We argue that the method is more general and more flexible than the prism integration, and that all the computational optimizations for the prism method will also be valid for the proposed method. For example, adding responses of several mass columns corresponds to adding polynomial coefficients associated with the corresponding grid nodes and replacing their individual heights by an average. Another advantage of the polynomial model is that the operation by linear/linearized functionals is simple, see section 1.2. One even reuses the coefficients. Thus, the corresponding polynomial models for other types of gravity data can easily be derived. We have tried it for V and compared to the exact, see section 2.1 (also for the spherical case). Consequently, the polynomial approximation captures the fundamental interrelation between different types of gravity data.

References

Fig. 3 In spherical geometry the horizontal offsets are lengths of arc along the surface of the sphere. Top: A regular grid in spherical coordinates, ()N,)8)-grid, will not be length-of-arcequidistant, ()x,)y)-grid. The parallels are not great circles. The offset between the grid points is not a problem. It can be added to the offset between P and the nearest grid point. A continuous offset model, equation (12a)-(12b), can handle that. The area mismatch, if it is not too big, can be handled approximatively. Bottom: The figure shows how the exact values in figure 1 were obtained using a Cartesian approximation to the small spherical area element. Notice that because the local directions of the vertical in P and Q are different the computation of a z-component in P is a 3D-problem. Notice also, that the thin horizontal lamina in different radial distances has different area size. This is included in the polynomial approximation for the spherical case. The trick is that once the polynomial model has been constructed, there is no difference in the computational procedure between the Cartesian case and the spherical case.

Blakely, R.J. (1996). Potential theory in gravity & magnetic Applications. Cambridge University Press, 1996. Forsberg, R. (1984). A Study of Terrain Reductions, Density Anomalies and Geophysical Inversion Methods in Gravity Field Modelling. Reports Dept. Geodetic Science & Surveying, No. 355, Ohio State University, Columbus, Ohio. Forsberg, R. (1994). Terrain Effects in Geoid Computations. In: International School for the Determination and Use of the Geoid. International Geoid Service, DIIAR -Politecnico di Milano, Italy, pp. 101-134. Heiskanen, W. A. and H. Moritz (1967). Physical Geodesy. W.H.Freeman & Co., San Francisco. Moritz, H. (1980). Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe. Sjöberg, L.E. (2000). Topographic effects by the StokesHelmert method of geoid and quasi-geoid determination. Journal of Geodesy, vol. 74, 255-268. Telford, W.M, L.P. Geldart, R. E. Sheriff, D.A. Keys (1976). Applied Geophysics. Cambridge University Press. Torge, W. (1989). Gravimetry. de Gruyter & Co., Berlin.