Geophys. J. Int. 1990 Schwarz 485 514

Geophys. J. Int. (1990) 100, 485-514

REVIEW PAPER

The use of FFT techniques in physical geodesy

K. P. Schwarz’, M. G. Siderid and R. Forsberg2 ‘Surveying Engineering Department, University of Calgary, Calgary, Alberta T2N 1N4, Canada ‘Geodetic Institute, 2920 Charlottenlund, Denmark

Accepted 1989 July 25. Received 1989 January 31

SUMMARY The fast Fourier transform (FFT) technique is a very powerful tool for the efficient evaluation of gravity field convolution integrals. It can handle heterogeneous and noisy data, and thus presents a very attractive alternative to the classical, time consuming approaches, provided gridded data are available. This paper reviews the mathematics of the FFT methods as well as their practical problems, and presents examples from physical geodesy where the application of these methods is especially advantageous. The spectral evaluation of Stokes’, Vening Meinesz’ and Molodensky’s integrals, least-squares collocation in the frequency domain, integrals for terrain reductions and for airborne gravity gradiometry , and the computation of covariance and power spectral density functions are treated in detail. Numerical examples illustrate the efficiency and accuracy of the FFT methods.

Key words: FFT, physical geodesy, spectral methods.

1 INTRODUCTION

Physical geodesy is the branch of geodesy which uses measured gradients of the anomalou6gravity potential T to determine a unique and coherent representation of the terrestrial gravity field at the Earth’s surface and in outer space. The anomalous potential T is the difference between the actual gravity potential of the Earth and the reference potential of an ellipsoid with the same mass, flattening, and angular rotation rate as the Earth. An approximation of T is needed to model geodetic measurements, to predict perturbations of satellite orbits, to determine global ocean circulation patterns, to assist global geophysics, and to support oil and mineral exploration.

In recent years, the amount of data available for the solution of this problem has increased dramatically, both in quantity and in type. This has made the data processing problems more severe and has created a demand for efficient numerical solutions. Since much of the data is available in gridded form, the use of fast spectral techniques was clearly appropriate. Progress in the application of these methods to geodetic problems has been rapid during the last three years and it is almost certain that, because of their efficiency and accuracy, they will become standard procedures for a number of applications. However, it has also become clear that geodetic and, more generally, geophysical data often present specific problems not usually encountered in typical electrical engineering applications. The problems are with the heterogeneity of the data, the complicated surface on which they are given, the uneven spatial distribution, and the non-uniformity of the data noise. This paper will discuss the use of FFT techniques in physical geodesy with specific regard to these problems. The presentation will thus be application oriented and will progress from the straightforward use of FFT techniques for the treatment of the Stokes and Vening Meinesz integrals to more complicated cases like the Molodensky series and the handling of airborne gravity gradiometer data. In addition, the computation of terrain effects by FFT methods will be discussed, as well as spectral estimation techniques yielding estimates of degree variances based on local data.

2 THE GEODETIC APPROXIMATION PROBLEM

This section summarizes some basic properties of the Earth’s gravity field and briefly reviews the geodetic approximation problem. It closely follows Moritz (1980) which should be consulted for all details.

485

at The R

eference Shelf on May 7, 2016

http://gji.oxfordjournals.org/D

ownloaded from

http://gji.oxfordjournals.org/

486 K. P. Schwarz et al.

The gravitational potential of the Earth is expressed by

U

where P is a point on or outside the Earth’s surface, Q is the centre of the volume element dv, and is variable within the Earth’s body a, I is the distance between P and Q, p(Q) is the mass density at Q considered constant for dv,, and G is the Newtonian gravitational constant. The usual assumptions that the Earth is a rigid body without atmosphere and without temporal variations will be made. These introduce errors of the relative order of lop6 and lop7, respectively, which are negligible for the problem considered here.

In general, an Earth-fixed rectangular coordinate system will be used, which is defined in the following way: the origin is at the Earth’s centre of mass; the z-axis coincides with the mean axis of rotation and is thus normal to the mean equatorial plane which is the xy-plane; within this plane the mean meridian of Greenwich determines the direction of the x-axis and the definition of a right-handed system determines the direction of the y-axis. The mean spin axis and the mean Greenwich meridian plane are used to obtain a time-independent definition.

The gravity potential of the Earth is expressed by

W(P) = V(P) + UP) ,

v,(P) = +w*(x’ + y’)

(24

(2b)

where

is the potential of the centrifugal force and w is the angular velocity of the Earth’s rotation. Again, w can be considered as constant if an error of the relative order of is admissible. An equipotential surface is defined by

W(P) =constant. (3)

The equipotential surface most commonly used as a reference is the geoid which, somewhat loosely, can be described as the idealized surface of the oceans. The gravity vector 2 is the gradient of W, i.e.

b=gradW= W, , (3 where the components Wx, W,, W, are the partial derivatives of W with respect to x, y and z. Of particular interest is the magnitude of this vector because it is a measurable quantity. It is usually denoted by

The direction of 8 is defined by the unit vector ii

where the astronomical latitude @ and the astronomical longitude A are again measurable quantities and refer to a point P on the geoid.

The second-order derivatives of W or the gravity gradients are given by the tensor

w, wxy Wxz

whose diagonal components satisfy the relationships

AW = 20’

outside the Earth’s surface, and

AW = -4nGp + 2w2

inside the Earth’s surface, where

AW = W, + Wyy + Wzz.

at The R



ownloaded from


The use of FFT techniques in physical geodesy 481

The corresponding components of the gravitational gradient tensor xj satisfy Laplace’s equation AV = 0 in the first case, and Poisson’s equation AV = -4lcGp in the second. Because of (Sb) and the symmetry of the tensor, W, has five independent components which are again measurable.

The normal gravity field is defined by

U(Q) = B(Q) + UQ), (6)

where B(Q) is the gravitational potential of the homogeneous ellipsoid which is the best approximation to the actual geoid, and Q is a point on the ellipsoidal surface. U(Q) can be determined by fixing four parameters and by postulating that the resulting ellipsoid is an equipotential surface for (I, i.e.

U(x, y, 2) = U, = constant. (7)

A typical set of parameters which can be derived from measurements is: a, b; semiaxes of the ellipsoid, o; angular velocity of Earth rotation, and M; mass of the Earth.

The deviation between the actual gravity potential Wand the normal potential U is small, as is the deviation between the geoid and a best-fitting ellipsoid, which nowhere exceeds 100 m. Since U(Q) can be expressed by closed formulae, it is often used as a convenient first approximation of W(P). Similarly, the normal gravity vector

?(a) = grad U(Q) (8)

is used as a first approximation for %(P). Its magnitude can again be expressed by a closed formula while its direction is normal to the ellipsoid.

The anomalous gravity potential T is defined as the difference

T(P) = W(p) - u(P),

@(P) = W) - ?(a)

(9)

thus having the anomalous gravity vector 4 (104

with components

Y(A - A) cos @

P( @, A) is a point on the geoid and Q($, A) the corresponding point on the ellipsoid where; 9 and il are the ellipsoidal latitude and longitude, respectively.

The components 5 and q are called deflections of the vertical and usually do not exceed 10-20 arcseconds; Ag is called gravity anomaly. T and its derivatives will be used for the problem definition in the following. In linear approximation, the components of 4 depend on T in the following way:

1 3 T E = --- r a y ’ 1 3 T

= - ;ax9

where x , y are local Cartesian coordinates in the tangent plane to the geoid at P, and r is the distance between P and the Earth’s centre of mass. Similar simple relations can be established for the anomalous gravity gradients zj.

Stated in a somewhat loose fashion, gravity field approximation is the estimation of the anomalous potential Tin the space SZ outside the Earth’s surface S from discrete and noisy data given on S and in SZ. Two major approaches to the solution of this problem are in common use. One considers only data on S, the other data in SZ and on S. In the first classical approach, a boundary value problem for the gravity anomalies is formulated. Its solution is approximated either by a series expansion into solid spherical harmonics or by integrals. In spherical approximation, i.e. when accepting errors of the relative order of the Earth’s flattening in T, the series expansion of the anomalous potential may be written as

at The R



ownloaded from



where P,,, are the associated Legendre functions, M,, and SK,, are the differences of the spherical harmonic coefficients for W and U, a is the semimajor axis of the reference ellipsoid, and r , 8, A are the spherical coordinates of point P. Summation for n starts at 2 in order to constrain the origin of the coordinate system to the centre of mass. Equation (11) is a spectral representation of the anomalous gravity potential, with spectral coefficients SJ,,,, SK,, and with spherical harmonics as the appropriate eigenfunctions.

The corresponding autocovariance function C , is defined as an average M over the sphere

where variances and can be derived from the spectral representation (11) by

and (Y are spherical distance and azimuth, and P,,(cos V ) are Legendre polynomials. The terms a,, are called degree

a, = c (Gi, + SK2,,), m =O

where the overbar indicates fully normalized spherical harmonic coefficients; for a derivation see Heiskanen & Moritz (1967, section 7-3).

In principle, the approximation problem can be solved by determination of the unknown coefficients J,,,, K,, in some suitable manner. In practice, a poor global data distribution and the large number of coefficients needed limit this solution to a truncated low degree expansion, i.e. to the representation of the long-wavelength features of the global gravity field. Series expansions of type ( l l ) , called geopotential solutions, form the basis for all local approximations of the gravity field.

The solution of the boundary value problem by integral equation leads, for a known spherical boundary, to Stokes’ integral and, for an unknown boundary surface, to Molodensky’s problem and the resulting integral series. In its simplest form, the solution is

where R is the mean radius of the Earth, Ag is the gravity anomaly function on the sphere u with radius R, S(V) is Stokes’ function (an analytical summation of an infinite series), is the spherical distance between P and the data points Q. The zero and first-order terms have been fixed in the same manner as above. It should be noted that T is determined from one data type only, the gravity anomalies Ag. The relation between Tat the boundary surface S and Te in the space outside S can be obtained by solving Dirichlet’s problem for the exterior of the sphere. This leads to Poisson’s integral

where 1 is the distance between the surface point and the exterior point P and r, is the distance between the centre of mass of the Earth and P . Both (14) and (15) are convolution integrals and are therefore suitable for treatment by FFT techniques. The effects of projecting data from the sphere to the plane will be discussed later. Both equations can be used to resolve the anomalous potential in a given area to any desired wavelength, provided data with sufficient density are available.

In Stokes’ integral, equation (14), all data must be given on the surface of a sphere R. In reality, observations are made on the surface of the Earth which is considered unknown with respect to the reference sphere R. This leads to the well-known Molodensky problem where the Earth surface is considered as the unknown boundary surface, see e.g. Moritz (1980). T can then be written as a series of integrals

of which Stokes’ integral, (14), is the first term

The other terms are of the form

at The R



ownloaded from



where

G, = R 2 / / xo da, U

G2 = R 2 / / 7 x1 d a - 3R @ y,hPp xo da + 2nx0 tan2 /3, 4 " 0

3R' x1 d a - - // (h ;:'I3 xo d a + 2nx1 tan' /3, etc., G3 =I?'\/ Yx, do - - 4 13 2 U U U

where /3 is the terrain inclination angle, and

0

The formulae show clearly that instead of a straightforward integration of Ag, an iterative process involving Ag and the height differences (h - hp) is needed. The terms involving (h - hp) are no longer simple convolution integrals, and additional steps are required to obtain expressions which are suitable for treatment by FFT techniques. Similarly, the terrain correction integral, which may be viewed as an approximation of the Gl term, has to be expanded into series in order to obtain a form which can be managed by FFT techniques.

Boundary value problems are important in geodesy because they define the minimum information needed to construct an approximation of T outside S. However, in terms of data availability, these formulations are not realistic. A more general approach has been proposed in Krarup (1969) who sought an approximation to T in a reproducing kernel Hilbert space. The collocation solution which he pioneered is of the general form

K(P) = [LoK(P, Q)]'[LPLFK(P, Q)]-'6bj(Q), (19)

where Li and Lj are bounded functionals representing gravity field data, 66, are measurements, and K(P, Q) is the reproducing kernel

m RZ n + l

K(P, Q) = a;(-) Pn(cos W ) , (20) n = 2 rprQ

which implicitly defines the norm of the Hilbert space. The terms a; are dependent on the choice of the base functions. They are called model degree variances and describe a kind of decay rate for increasing degree n; for details see Tscherning & Rapp (1974). In practice, the model degree variances are made to follow the trend of the physical degree variances. They are obtained from (13) using the coefficients jn,, En,,, from an actual series expansion. This relates least norm collocation in a reproducing kernel Hilbert space to statistical least-squares collocation; see Moritz (1980). 'The advantage of the solution by (19) is that it allows all functionals of T to be treated as observables and thus makes it possible to handle the actual data more easily. It also is not restricted to data on the boundary surface but allows the consistent treatment of data on S and in Qoutside S. It will be shown later that (19) can be considered as a multiple-input-single-output model within the usual systems engineering terminology. It is therefore again suitable for treatment by FFT techniques if data are available in gridded form. Airborne gradiometry will be discussed as a specific case where this technique is used.

3 DEFINITION AND APPLICATION OF THE 2-D FOURIER TRANSFORM

This section presents the basic formulae for the definition of the 2-D Fourier transform and outlines its practical evaluation. The fundamental mathematical properties are given, and serve as references for the development of frequency-domain equations used in the various problems of physical geodesy discussed in subsequent sections.

3.1 Definition of the direct and inverse 2-D continuous Fourier transform (CFT)

The 2-D Fourier transform or spectrum of a function h(x, y) is defined (Papoulis 1977) to be the integral

H(k, , k,,) = I / h(n, y)e-i(kxx+kyy) dx d Y = F[h(x, Y)l, m m

-m -m

where F is the 2-D Fourier operator, H is the spectrum of the function h(x, y ) , k,, k,, are the wavenumbers (spatial circular frequencies) corresponding to the x and y spatial coordinates respectively, and i is the imaginary unit (i = fi). A function

at The R



ownloaded from



h(x, y ) in the space domain can be obtained from its spectrum H(k,, k,) by means of the inverse 2-D Fourier transform

where F-' is the inverse 2-D Fourier operator. The functions h(x, y ) and H(k,, k,) form a Fourier transform pair denoted by

h(x, y ) -H(k, , ky) Or h(x, Y ) - W u , v) , (23)

where u and u are the spatial frequencies (in cycles per distance unit) in the directions of x and y respectively, related to k, and ky by the equations

k, = 2nu and ky = 2nv.

By substituting (24) into (21) and (22), the definition integrals become

h(x, y ) = [ [ H(u, u)e2ni(m+Vy) du du = F-l[H(u, u)] .

Throughout this paper capital letters will denote spectra, except when stated otherwise.

-m -m

3.2 Properties of the 2-D Fourier transform

In the following, some properties of the Fourier transform which will be needed in subsequent chapters are listed. The proofs, which follow directly from the definitions, can be found in numerous textbooks, such as Papoulis (1977) or Bracewell (1978), and will not be given here. (a) Linearity

ah(x, y ) + bg(x, y ) - aH(u, u ) + bG(u, u). (27)

(b) Scaling

(c) Shifting

h(x - a, y - H(u, u) . b ) f, e-2zi(au+bu)

(d) Derivatives

( f ) Frequency domain convolution

h(x, y ) g ( x , y ) - H ( u , v )*G(u, v) . (33)

h(x, y ) @ g ( x , y ) - H * ( u , u)G(u, u), (34)

(g) Correlation

where H*(u, u ) is the complex conjugate of H(u, u) , and the correlation, denoted by El, is defined as follows:

at The R



ownloaded from


The w e of FFT techniques in physical geodesy 491

(h) Relationship to the Hankel transform If h(x, y) is a radially symmetric function, i.e. h(x, y) = h(r), r2 = x z +y2, then

H(u, v) = H(q) ,

where H ( q ) is the Hankel transform of h(r) , given (Bracewell 1978) by the relation

H ( q ) = 2n$ rh(r)Jo(2nqr) dr = H[h(r)].

H is the Hankel transform operator. Jo is the zero-order Bessel function defined by the integral

q2 = u2 + v2,

The inverse Hankel transform is defined as m

h(r) = 2nl, qH(q)Jo(2Jv) dq = H-1[H(q)17

where H-’ is the inverse Hankel transform operator.

(37)

(39)

3.3 Fourier transform pairs of interest

For quick reference, the pairs needed for the developments that follow are given here. Each pair can be obtained by application of the definitions of Section 3.1 and/or the properties of Section 3.2.

(i) ( x z + y2)-ln e (u2 + v2)-ln. (40)

(iv) a6(x, y) * a , where 6(x, y) is the Dirac delta function. (43) sin ( m u ) sin (nbv)

1x1 < a /2 ,

(v) n - n - e a b = ab sinc (au) sinc (bv),

lyl < b / 2

(b’) nau nbv

(3 {i: elsewhere. where ll - n - =

3.4 The 2-D discrete Fourier transform (Dm)

Stationary random data are available in practice only in a finite area, say - X / 2 5 x S X / 2 , - Y / 2 S y 5 Y / 2 , and consequently the definitions of (21) and (22) cannot be applied. Instead, an estimate of the spectrum can be obtained by computing the finite integral

x/2 Yl2

HF(u, v) = I 1 hF(x, y)e-2ni(ux+uy) dx d Y. -x/2 -Y/2

(45)

Furthermore, data are assumed to be known only at the discrete points of a regular grid with sampling intervals Ax, Ay. The record lengths may thus be expressed by

X = M Ax, Y = N Ay, (46)

where M and N are the number of points along the x and y directions, respectively. With the discrete given data, the spectral estimation integral of (45) may now be approximated by a sum

M-1 N-1

&(Urn, V,) = c 2 h(Xk, y~)e-2ffi(ud*+u ny‘)AxAy, m = 0 , 1 , 2 , . . . , M - 1 , n = 0 , 1 , 2 , . . . , N-1 . (47) k = O I = O

This approximation forms the basis for connecting the continuous and the discrete Fourier transform. Assuming the data to be periodically extended in the plane, the spectrum becomes discrete with frequency spacings

at The R



ownloaded from



Due to the discreteness of the data in both domains, the spectral estimation of (47) will only allow frequencies up to

to be recovered. The frequencies uN and vN are called Nyquist frequencies and are the highest frequencies that can be resolved from the given data. In terms of a discrete Fourier transform pair, the spectral transformations may be written as

M-1 N - 1

H ( m A u , n A v ) = A x A y x x h ( k A x , I A y ) e x p ( - Z n i ( $ + $ ) ) , m=0,1 , . . . , M-1, n=0 ,1 , . . . , N-1, (50) k = O 1=0

M-1 N - 1

h(k Ax, 1 Ay) = Au Av 2 k = 0 , 1 , . . . , M-1, I=O,1, . . . , N-1. (51) m=O n = O

In the form of (50), the spectrum is obtained for frequencies from zero up to fuN and fv,, the basic spectrum periods being 2U, and 2vN.

Usually the DFT pair of equations (50), (51) is represented only by the integer wavenumbers k, 1 and m, n, as

h(k, 1) * H(m, n ) , (52)

corresponding to Ax = Ay = 1 and thus Au = 1/M, Av = 1/N. This is the basic form of the Fast Fourier transform algorithm used to evaluate the DFT. The FFT is extremely fast, which is a major reason for the popularity of spectral over space domain methods. Basically, the FFT relies on prime factorization of N and M, and is fastest when N and M are powers of 2. The computation times are typically of the order MNlog(MN) while direct evaluations of the DFT by (50) or (51) would have computation times increasing like (MN)’.

3.5 Evaluation of the DFI: and the need For windowing

If ~ ( x , y ) is a (boxcar) function such that w(x, y) = 1 except when 1x1 > X / 2 , JyI > Y / 2 where w(x, y ) = 0, then the finite Fourier transform of (45) can be viewed as the CFT of the function hF(x, y ) = h(x, y )w(x , y ) obtained from (25). According to property (f), the finite spectrum is HF(u, v) = H(u, v) * W(u, v), where W(u, v) = F[w(x, y ) ] is obtained from the Fourier transform pair (v) with a = X, b = Y. As Fig. 1 indicates (in 1-D), the true spectrum of the h(x, y) function is ‘blurred‘ by the side lobes which ‘leak’ from the main lobe of W(u, v). This effect is called spectral leakage and is due to the finite extent of the record lengths. In other words, the long wavelengths (longer than 2X and 2 Y ) cannot be resolved from the finite-length data set. Since the DFT is periodic, the implicit use of the rectangular window w(x, y) introduces discontinuities at the edges, responsible for the leakage. To smooth out the discontinuities, other windowing functions are used which taper off to zero at the edges thus minimizing the spectral leakage. In this case, the side lobes of W(u, v ) tend to be insignificant when compared to the main lobe which becomes wider. It must be mentioned here that leakage is inherent only to the DFT and not to the ClT, depends on the record lengths and window used, and is not caused by the sampling. The choice of good window functions is important for many engineering and geophysical applications, e.g. when looking for peaks in the spectrum. For gravity field applications, however, spectra tend to have a very smooth decay, so the ‘blurring’ of the spectra due to leakage tends to be less critical.

The influence of DFT periodicity errors and of windowing is conceptually clearer when using FFT for evaluating convdutions through multiplication of spectra in the frequency domain; see equations (31) and (32). Using FFT methods, the implicit periodicity assumption means that the convolution will be substituted by a cyclic convolution, which the DFT approach turns into (in integer wavenumber notation)

For a point near an edge of the area, points near the opposite edge will thus have a large (erroneous) influence. By multiplying the data h with a window function w, this influence is diminished, as the products to be summed in (53) are downweighted near

t””’ t rinc(u

(a I (b)

Figure 1. The rectangular (boxcar) function and its spectrum, the sinc function.

at The R



ownloaded from



Figure 2. Aliasing due to insufficient sampling rate.

the edges. However, no windowing scheme can make up fully for the 'wrap-around' effects in (53), affecting all points in the area except at the centre. For more details on the cyclic convolution and its treatment, consult Bracewell (1978).

The sampling rate introduces other types of errors into the spectrum. High-frequency components can impersonate lower frequencies if the sampling rate is too low. This effect is called aliasing and can be removed by sampling a function at a rate at least twice as high as the highest frequency present in the data; see equation (49). This is because a function may be represented exactly by discrete values only when its highest frequency component is less than the Nyquist frequency (Shannon's sampling theorem). This fact is demonstrated in Fig. 2 in one dimension.

From the preceding discussion it is obvious that in order to reduce the aliasing error, a higher sampling rate must be used, and to improve frequency resolution and decrease leakage, larger record lengths together with an appropriate window are required. Obviously, such requirements cannot always be met in practice, especially for gravity field applications. Therefore, a trade-off between accuracy and feasibility is often necessary. This is, however, common geodetic practice independent of the specific method applied. Bergland (1969) gives an excellent pictorial overview of these problems which occur in the transition from the CFT to the DFT. For additional details on aliasing, leakage, and the use of windows, Papoulis (1977), Bendat & Piersol (1971), Harris (1978) and Eren (1980) should be consulted for 1-D signals, and Sideris (1984, 1987a) for 2-D signals.

3.6 Covariance and power spectral density functions by the FWFT

The FFT technique provides an important tool for estimating covariance (CV) functions and power spectral density (PSD) functions, because the CV and the PSD function form a Fourier transform pair.

For stationary 2-D data, the correlation (CR) of h(x, y ) and g(x, y ) is defined as

Rhg(x? Y ) = E [ h ( x O , YO)g(x + x O P Y +YO)]

Y-m

Using (34), (35), Rhg(x, y ) in (54) can be expressed as the correlation of h(x, y ) and g(x, y ) :

1 R h g ( x , y ) = lim - h h ( x , y ) W g ( x , y ) .

X-m XY Y-m

The spectrum of the CR function is the PSD function

1 Shg(u, v) = F[Rhg(x, y ) ] = lim - H*(u, v)G(u, v).

x - m X Y

The CV function is defined as

chg(x, Y ) =E{[h(xO? Y O ) - phLhl[g(xO+x, YO+Y) -

= Rhg(x, Y ) - phpg?

where ph and pg are the mean values of h(x, y ) , g(x, y ) . The ph, using (25), is

H(O,O) h(x, y ) dr d y = lim -. X-m X Y

Y-- Y--

From equations (54), (%), (56) and (57) follow the important relations:

Rhg(x, Y ) = F-'[Shg(u, v)],

chg(xj Y ) = F-'[Shg(u? v) - Shg(0, O)a(u, v)]? which allow the computation of CR and CV functions from the PSD functions, using the Fourier transform.

Some very useful properties of the functions discussed in this section are given below:

Rhg(-x, -Y) = Rgh(xt Y ) , Rhg(O, 0) = q h g = E[h(xo, Yo)g(xO? YO)]?

Rhg(OO, 03) = phpgr

(55)

at The R



ownloaded from



All the formulae in this section hold when h(x, y) =g(x, y ) . In this case the functions are called autocovariance (ACV), autocorrelation (ACR) and auto power spectral density (APSD). Also, I),,, and a,,, are the mean square value and variance, respectively, and are simply denoted by I# and u2. When h(x, y) # g ( x , y ) , they are often called cross covariance (CCV), cross correlation (CCR) and cross power spectral density (CPSD).

Estimation of the PSD

In practice the data are known in a finite X, Y domain. When v sample records are available, an unbiased estimate for (56) is given by a kind of averaging (Welch 1967) over all records:

Since the spectra of h(x, y), g(x, y ) are computed from windowed data, the constant c in (69) accounts for the loss of power due to the window w(x, y) and is computed by the formula

The normalized standard error E of &(u, v) computed from v sample records, or more generally, using Y number of averages is

Thus, 100 averages are required for a 10 per cent error. When just one sample record is available, i.e. v = 1 in (69), the estimated PSD is called the periodogram. The periodogram approach is generally considered to be very noisy but is often necessary when the stationarity of the signal is questionable.

More details on the estimation of PSD functions can be found in Bendat & Piersol (1971, 1980), Rabiner & Rader (1972), and Childers (1978).

3.7 Input-output relations-iiltering by FIT

As it can be seen from property (e), a convolution of two functions in the space domain. is equivalent to filtering of one function by the other (multiplication of spectra) in the spectral domain:

g ( x , y ) = f ( x , y ) * h ( x , y ) o G ( u , v ) = F ( u , v)H(u, v). (72) In terms of linear systems, h(x, y) is named the input function, g(x, y ) the output function, and f (x , y) the filtering or system response function. Its spectrum F(u, v) is called the transfer function. Forming the CR functions R&, y) and Rhg(x, y) and taking the Fourier transform results in

which describe the equations of a single-input-single-output linear system (Bendant & Piersol 1980). From the known or measured PSD functions of the input and output functions, the system transfer function can be determined in amplitude and phase by solving (73) and (74). This procedure is called system identification since the unknown system response function is determined. In case the input signal is corrupted by noise n(x, y), with PSD function Snn(u, v), and the output signal is corrupted by noise e(x, y), with PSD function See(u, v), equations (73) and (74) become

where h(x, y ) , n(x, y ) and e(x, y) are assumed to be uncorrelated.

at The R



ownloaded from



n1 1x.y) I

t "z 1x.y)

Figure 3. A two-input-single-output linear system.

A single-input-multiple-output system can be treated as a set of single-input-single-output systems (e.g. input = gravity anomalies, outputs = geoid heights, deflections of the vertical) and will not be discussed further here. However, multiple-input-single-output systems address the problem of heterogeneous data in physical geodesy. This problem is usually discussed in the context of least-squares collocation and some important analogies between the two approaches have been shown by Vassiliou (1986).

The equations for a multiple-input-single-output system will be given here using the simple example of a two-input-single- output system shown in Fig. 3, with noise present in both the input and output stages. In the spectral domain, the system is described by the equation

Assuming uncorrelated input signals and noise and also uncorrelated input and output noise, the PSD functions are

where

Sn,(u, v ) = O for i # j . (80)

In order for the system to be optimum, it must minimize See(u, v) for all possible choices of 4(u, v). This optimality criterion leads to

Shr(u, v) = 0, i = 1, 2, (81)

i.e. the input signals and the output noise must be uncorrelated. Solving (79) with the condition of (81) results in

After the system identification has been completed, the output noise PSD function can be obtained from (78):

equations (82) become

An extended discussion on input-output relationships, filtering by FFT, and system identification can be found in Bendat & Piersol (1971).

at The R



ownloaded from



4 SPECTRAL SOLUTIONS TO THE CLASSICAL BVP

In this section the equations for the solution of the classical boundary value problem are reformulated and evaluated in the frequency domain. The use of geopotential models in addition to gravity anomalies is briefly outlined. All derivations are given in planar approximation. A discussion of formulae that correct FFT results for the Earth’s curvature concludes this section.

4.1 The Stokes and Vening Meinesz integrals in the spectral domain

The disturbing potential T a t a point P on the surface of a sphere can be computed, using free-air gravity anomalies, from (14), which in expanded form is

R 2 X X

T(P) = -& / I A d 3 , n N 3 ) sin 3 dV d f f ? (86) a = O y=o

where

3 2

- 6sin-+ 1 - 5 cos I) - 3 cos 3 In 1

S ( 3 ) = . sin ( 3 1 2 )

According to the Bruns theorem, the geoidal undulation N can be computed from T by the formula

(87)

When the data are given in a limited area E, the spherical surface can be approximated by the plane tangent at P. In this case 3 is small and Stokes’ function in (87) can be approximated by

Using the planar distance s and the azimuth a as coordinates, + = s / R and consequently

2R S(s) = -.

S

The differential area d a = sin 3 d 3 d a becomes

Thus, in planar approximation, Stokes’ formula is

Using rectangular instead of polar coordinates to describe the area E , (91) takes the form

Now the Vening Meinesz integral, to determine deflections of the vertical, can be derived. Equations (1Oc) and (88) yield

which, using (92), result in

Formulae (92) and (94) are convolution integrals (see 32) and thus can be evaluated in the frequency domain. In convolution

at The R



ownloaded from



form they are

1 N ( ~ , y ) = - A d g ( x , y ) * l , ( x , Y ) , (95) 2RY

where

“ ( x , y ) = (x’ + y2)-lI2, (97)

(98)

Making use of the spectra of the functions involved in (95) and (96), i.e. AG(u, v), L,(u, v), LE(u, v), L,(u, v), the geoid undulations and deflections of the vertical are computed from the following equations:

The spectra of the kernel functions are given analytically (see 40 and 41) and thus N , 5 and t,~ are obtained from

1

The above formulae can be used with either point or mean gridded anomalies. In the spectral domain, it is very easy to unsmooth the mean Ag so that the results correspond to those that would be obtained from point values. A 2-D sinc function is used for this purpose; detailed derivations can be found in Sideris & Tziavos (1988).

It must be noted that the Stokes and Vening Meinesz kernels of (97) and (98) are infinite at x = y = 0. In the practical implementation of the FFT (equations 99 and 100), this singularity affects only the geoidal undulations since only the Stokes kernel spectrum is infinite at u = v = 0. To avoid the singularity, the effect on N of the innermost circular zone of radius so around the computation point P can be computed separately (Heiskanen & Moritz, 1967, p- 122) by the approximate formula

6N(P) = Ag(P). Y

(101a)

Assuming for the FFT that the grid element area Ax Ay is equal to the area of a circle of radius so, (101a) becomes

(101b)

This correction should be added to the results of (99b). It may also be implemented into the FFT procedure (Sideris 1987b) by modifying slightly the l,(x, y ) kernel in (97):

I&, y ) = 2 f i ( A x A Y ) ” ’ ~ ( x , y) + (x’ + y2)-’”[1 - 6 ( ~ , y)].

L,(u, v) = 2 f i ( d r Ay)”’ + (u’ + v’)”’[~(u, V) - 11.

(102)

(103)

Now the spectrum of IN@, y ) is

The above modified Stokes kernel and its spectrum are both non-singular at the origin. Geoidal undulations computed from (99a) and (103) include the effect of the innermost zone.

4.2 Computational procedure: preprocessing and postprocessing

The gravity anomalies used in (99) and (100) are given in a limited area around the computation point and thus cannot resolve the long wavelengths of the gravity field. These long wavelengths are contained in geopotential model expansions, of the form of ( l l ) , representing a smooth approximation of the geoid. In a preprocessing step, the free-air gravity anomalies AgFA are

at The R



ownloaded from


498

referenced to the gravity field implied by the model, as follows:

K. P. Schwarz et al.

= AgFA - AgGM, (104)

where, from (1Oc) and (11) with a = r = R

1 nMi n

AgGM = - (n - 1) 2 [Mnm cos (mA) + 6Knm sin (mA)]Pnm(sin 9). (105) R n=2 m =O

nmax is the maximum degree of coefficients in the geopotential model, and determines the size of the area E, say b" x b", in which local data are required: for an errorless geopotential model, b" = 180°/nmax. Naturally, the results of the FFT will not contain the geopotential model contribution, which is added in a postprocessing step. The formulae, which can also be evaluated by FFT (Rizos 1979; Colombo 1979, 1981), are

1 gGM = - - 2 [Mnm cos (mA) + 6Kn, sin (mA)]dPnm(sin 4 ) / d &

R y n = = ~ m = ~

The results of the planar formulae for 5, v, N can be corrected for the curvature of the Earth. Using matched asymptotic expansions, Jordan (1978) computed the following spherical corrections:

v v cN( v) = 2 cosec - , 2

ca,,(v) = -sin v2 cosec39 8 2 '

where v is the spherical distance from the centre of the local x,y-system. The cN, cE, , correcting terms are factors by which the FFT results are multiplied. Their effect, however, is very small; for example, for N, cN = 1.001 and cN = 1.05 for q = lo" and ly = a", respectively. In practice, W < 10" in most cases and thus these corrections can safely be neglected; see discussion in Sideris (1987a) and (1987b).

To account for the high frequencies of the gravity field, terrain corrections have to be applied. In the preprocessing steps, the refined Bouguer correction-or some other terrain reduction-is applied to the gravity anomalies. Then, in the postprocessing stage, the effects of the topographic masses are added back to E, q and N . FFT can be used to evaluate these terrain correction integrals, as discussed in Section 7. Such terrain reductions result in a smoothed gravity field. They should not be regarded, however, as an approximation to Molodensky's corrections, presented in the, next section, which account for the fact that the gravity data are given on a non-level surface.

As a final note, it is mentioned that an interpolation procedure is necessary to evaluate the geoidal undulations and deflections of the vertical on points not belonging to the FFT grid.

5 SPECTRAL SOLUTIONS TO THE MOLODENSKY BVP

Molodensky's BVP, i.e. the gravimetric determination of the Earth's physical surface, can also be reformulated and solved in the frequency domain. The solution, given by (16), (17) and (18), will first be simplified using planar approximation (R+m). The first two correcting terms, neglecting the second integral in (17c), become

TI = I] ?dr dy, 2n E

where

(110a)

(1 lob)

( l l l a )

( l l lb )

at The R



ownloaded from



In the above expressions, the following formulae have been used:

and

dh dh ’ tan’ /? = tan2 /?, + tan’ Py = (z)’ + (G) = h: + h;.

For the sake of a simple notation, all functions in this section will be used without their arguments (x , y) or (u, v).

5.1 Preliminary mathematical derivations

A few useful formulae that are repeatedly used in the following sections will be discussed here. (a) For a harmonic function f i n space, i.e.

Af = L x + f Y y +fZ, = 0,

the spectrum of the Laplacian is (see equation 30)

F(Af) = [(2niu)’ + (2niv)’ + (2niw)’]F(f) = 0,

where w is the frequency corresponding to z. Since, in general, F(f) # 0, (115a) yields

u2 + u* + w’ = 0- -w2 = u’ + u2 = 42,

from which iw = -4. Thus the vertical derivative spectrum is

(114)

(1 15a)

(115b)

Such a function is the Ag-function in planar approximation; since Ag - - d T / d z and AT = 0, it follows easily that AAg = 0. The terrain h(x, y ) is also considered as having a harmonic extension in space above the xy-plane for the developments of Section 5.2.

(b) For the product of two harmonic functions f and g , (114) gives

( fg) , , = - ( f g ) x x - (fg),, = - fgxx - gfxx - 2 g x f x - fgyy - g f y y - %yfy

= fg,, + gf,, - 2 ( g x f x + g y f y ).

(c) For two functions h and g , it is (h - hp)g = -h,(g - g p ) + hg - (hg)p and thus

(1 18a)

The integrals on the right-hand side of (118a) express vertical derivatives (Heiskanen & Moritz 1967, p. 38). Thus, (118a) can be written as ’ 11 y g dr dy = -hgz + (hg),. 2n E

(d) If d, is the vertical derivative operator, i.e. f, =f * d,, and 1, is the Stokes kernel, then

f, * 1, = f * d, * l , = F-l[ F(f)( -2nq d)] = -2nf

(1 18b)

5.2 The Molodensky series terms in convolution form

The formulae for the Molodensky second-order solution in planar approximation are reformulated here as convolutions so that they can be evaluated in the spectral domain by FFT; see Sideris (1987b) for detailed derivations. The G, and G2 terms using (118), take the form

G1 = -hAg, + (hAg), = -h(Ag * d,) + (hAg) * d,, (120a)

G2 = -hGl, + (hG,), + Ag(h$ + h t )

= -h(G, * d, ) + (hG1) * d, + Ag[(h * d,)’ + (h * dy)2] , (120b)

at The R



ownloaded from


500 K. P. Schwurz et al.

where d, and d,, are the two horizontal derivative kernels. After computing the Gl and G, terms, their effects on T (see equations 110) are given by

(121a)

1 - -_ G2* I N - {;hz(Ag * d,) - h[(hAg) * d,] + f (h2Ag) * d , } , (121b) 2n

where (121b) was derived using (118) and (h - h,)’ = (h2 - hc) - 2hp(h - h,). Equations (120), (121) contain the terrain inclination explicitly and are rather complicated, even for FFT evaluation.

However, they can be significantly simplified by evaluating directly T,, T, from h and Ag, without computing Gl and G, by equation (120).

The q term, using (121a), (120a) and (119), is

(122a)

The T, term, using (121b), (120b) and (119), becomes

1 2n

1 2n

T, = - {G2 + [ih2Ag, = h(hAg), + ~ ( h 2 A g ) , } , * 1N

=- {-hG1, + (hG,), + Ag(h: + h;) + &h2Ag,), - [h(hAg),], + ~(hzAg) , , } * 1,.

The above expression is greatly simplified using (117) and (120a), and the terrain inclination term is eliminated. The final result is

1 2n

T, =-[- i (h2Agz) , + h(hAg,), - $hzAg,,] *lN

Equations (122) are much simpler to compute by FFT than the combined equations (120) and (121). Moreover, they can be written as

1 2n T2 = -g, * “ ,

where, from (122) and (119),

(123a)

(123b)

(124a)

This simplified formulation can be derived starting from the analytical continuation to point level solution (Moritz 1980) instead of the original Molodensky solution (Molodenskii, Eremeev & Yurkina 1%2), as shown in Sideris & Schwarz (1986, 1988) and Sideris (1987b). Equation (123) and (124) are the most suitable for FFT evaluation. The analytical continuation solution can be split up into (i) ‘downward‘ continuation to sea level and (ii) ‘upward’ continuation from sea to point level. However, only the values gz at sea level and their derivatives are needed for the computations (Sideris 1987b). This will become evident when the formulae for computing E, 9, 5 are developed.

at The R



ownloaded from



5.3 Formulae for computing deflections of the vertical and geoid andulations

The height anomaly 5, which may be viewed as the harmonic extension of the geoid undulation N in space (Heiskanen & Moritz 1967, p. 326), will first be computed. By Bruns’ formula and (123a) and (124a), it is given by

and using (125a), the corresponding deflections are (Moritz 1964)

Noting that

gn-- aCn - -_ Y

and

(125a)

(125a) becomes

c1 = hacg/az + 5:. (125b)

This clearly indicates that first 5, is evaluated at sea level (term cy) and then it is continued upward to point level (term h d c / d z ) . Similar explanations can be given for the computations of the deflections by (126a).

The second-order corrections are

and can easily be given a form similar to (125b) but including second-order derivatives as well. The proposed procedure is the following. First, evaluate Ag at sea level by computing all the g: terms which have a

significant contribution. Then, evaluate c, 5 and q at sea level and continue them upward to point level; general formulae can be found in Sideris (1987b).

The same preprocessing and postprocessing steps as those described in the solution of the classical BVP can be applied also -in the solution of the BVP of Molodensky. A terrain remove-restore technique in the context of Molodensky is discussed in Moritz (1968), and is expected to improve the convergence of the series.

6 TERRAIN EFFECT COMPUTATION BY FFT

In rugged terrain, the major part of the short-wavelength gravity field variation is caused directly by the topography. Assuming that the density of the topography is known, the terrain effects may be eliminated, thus producing a smoother residual field, more suitable for gravity field modelling. The gravity field modelling may be done by e.g. integration formulae, least-squares collocation, or the FFT methods outlined in this paper. After prediction of the wanted gravity field quantities, the terrain effects must then be added back to obtain the final predictions. This general remove-restore technique is indispensable when working with gravity field modelling in rough topography, as the gravity data spacing is hardly ever sufficient to represent the short-wavelength gravity field variations. Thus, large aliasing errors due to undersampling are always a problem.

The computation of terrain effects lends itself readily for use with FFT methods. Height or bathymetry data in the form of digital terrain models (DTMs) are acquired in gridded form. Considering the amount of data contained in detailed DTMs, the FFT methods outlined in the following sections provide substantial gains in computation speed over space-domain based integration methods such as the prism integration. In these methods, the terrain can be considered as consisting of either point heights or prisms. The spectra of the two representations are then related via a 2-D sinc function (Sideris & Tziavos 1988).

The FIT methods for terrain effect computations all rely on series expansions of the basically non-linear terrain effect integrals. The terrain effects are obtained as a sequence of convolutions involving powers of the given height data, with the first two terms often being sufficient. Two important cases can be distinguished: the computation of terrain effects at a level surface,

at The R



ownloaded from



Fcye 4. Geometry for the terrain effects on a level surface and on the surface of the Earth.

and the computation of terrain effects at the physical surface of the Earth. The first case is typically applicable to ocean and airborne data, while the second case includes the classical terrain correction for terrestrial gravity data. This terrain correction has a special significance, as it provides an approximation to the effect of the Molodensky G1 term of (17b) on ij, 17 and <when gravity anomalies and heights are linearly related.

6.1 Terrain effects for a level surface above the terrain

Consider a computation level z, > h,,, where h,,, is the highest topographic elevation in the area of interest (Fig. 4). The gravity anomaly Agh at Po due to the topography may in the flat-earth approximation be written as

where p is an assumed uniform density and G the gravitational constant. Taking the Fourier transform (see 25) of (130)

p) ak dy dz dr, dy,, e--Zni(=,+vy

yields after integration with respect to xp, y , and z

On expansion of the exponential ezzqh into series, this gives a sum of Fourier transforms

This expression was given by Parker (1972), who has also shown that the convergence of the series is uniform and better than the series

Improved convergence of the series may therefore be obtained by splitting the topography into two mass bodies; a Bouguer plate of thickness ha,, where h, is an average height between the maximum and minimum elevation, with gravity effect 2nGphav, plus the effect of the residual topography relative to the level ha,. This will make the ratio in (134) smaller, as h,,, and hp are now relative to ha,.

Instead of computing effects of all topography above sea level, it is often advantageous in gravity field modelling to consider only the effect of topographic irregularities relative to some smooth mean height surface h,&, y ) , i.e. residual terrain model (RTM) effects (Forsberg & Tscherning 1981). In this case, the Parker formula of (133) takes the general form

In addition to airborne applications, the Parker formula of (133) or (135) may readily be used for computing effects of Ocean bathymetry, as well as computing isostatic effects. Gravity field quantities other than Ag, may be obtained by transforming F(Agh) using the methods of Section 4 for deflections of the vertical and geoid undulations. Formulae for the terrain effects cn,

at The R



ownloaded from


The w e of FFT techniques in physical geodesy 503

txy, t,,, tyy, ty,, and f,, on second-order gradients are readily obtained from (135) or (133):

For details and numerical examples, see Tziavos et af. (1988). Often the first two terms of the Parker expansion provide sufficient accuracy. In that case, equivalent space-domain

expansions (Dorman & Lewis 1974) may be used with advantage, as this allows use of multiple-grid FFT methods (Forsberg 1985). In the simplest case of these so-called multipole expansions, the integrand of (130) is expanded about z = 0 as follows:

z - 2, z, r3-3.z; 2 3 n Z - - , + T Z + * . * ,

[(XP - X I 2 + (YP - Y)” + (z, - 2) I ro ro

where

r, = [(xp - x ) ~ + ( y p - y ) 2 + 23]ln.

Integrating (130) with respect to z, Agh is obtained as a sum of convolutions

Ag, = Gp[(f, * h) + (f2 * h2) + * -1 with convolution kernels

(137)

Examples of applications of this approach to other gravity field quantities may be found in Forsberg (1985) and Tziavos et al. (1988).

6.2 Terrain corrections at the surface of the Earth For gravity field data collected on the surface of the Earth, the methods of the previous sub-section cannot be applied. Instead, an expansion of the classical terrain correction integral is possible. Consider a point P at the, surface of the topography (Fig. 4). The total gravity topographic effect at P may be split into a Bouguer plate effect and the terrain correction c:

dgh = 2nGph - C, (140) z - h p

2 3/2 l7k dY d z . = G P l l C E [(n p - x)2 + ( Y p - y ) 2 + (h, - 2) I When expanding the integrand of (140) about the level z = h , analogously to (137), the second-order term drops out, and by integration of z

(142a)

r, = [(np - x)’ + ( y p - Y ) ~ ] ” ~ . (142b)

This approximation represents the so-called ‘linear approximation’ to the terrain correction, cf. Moritz (1968). The accuracy of the linear approximation is usually sufficient (sub-mgal) in an rms sense, although large errors are possible for gravity points with large terrain slopes in the vicinity of the point (Forsberg 1984a). The linear approximation in (142) may readily be written in convolution form (Sideris 1984, 1985a; Forsberg 1985)

c = iGp[f * h Z - 2h(f * h) + h2g] (143)

with

Equation (143) is easily implemented by FlT, and with a finite evaluation area E, the term g becomes g = F(0, 0) according to

at The R



ownloaded from



(25) with u = v = 0. Also, the singularity off at x = y = 0 is eliminated as it can be shown that (143) is insensitive to the actual value of f ( 0 , 0) in the discrete case, and thus f(0, 0) = 0 may be substituted in practice.

The singularity off may be circumvented by expanding the integrand of (140) about a constant elevation z = h, instead of about z = h,. In this case, the f ( x , y) function becomes

1 = ( x 2 + y 2 + h33nJ

in which case the Fourier transform of (144) is given analytically by (42) as

2n Znho(uz+d)’n F(u, v) = F(f) = - e- h,

(145)

with 2n

g = F ( 0 , O ) =-. h0

Besides the existence of an analytical transform, the expansion around an h, smaller than the grid spacings may also provide improved approximations compared to the simple linear approximation (Comer et al. 1986; Tziavos et al. 1988). The same technique has been applied for the FFT evaluation of isostatic correction integrals; see Sideris (1985b) for explicit formulae.

In addition to providing a part of the gravity terrain effect, the classical terrain correction has often been used in (110a) instead of the G1 term of (l l la) . Although ( l l l a ) involves both h and Ag, substituting c in place of G, is possible because locally the (free-air) gravity anomalies are nearly perfectly correlated to local topography (Moritz 1968). However, with the outlined FFT methods for directly computing the G1 term and its effect on T (see Section 5), no real computational saving results from using c instead of G1 in (110a), provided a proper interpolation of gravity values is performed at the nodes of the given height grid. For a numerical example, see Sideris & Schwarz (1986).

For quantities other than gravity anomalies, analogous linear approximations may not be sufficiently accurate, and higher order expansion terms are required. This is, for example, the case for deflections of the vertical. Deflection terrain corrections are given by

1 Y P - Y

E

This represents the total terrain effect, as the Bouguer plate contribution is zero. Expanding the denominator as follows:

with r, given by (142b), gives upon integration with respect to z the convolution expression (Forsberg 1985)

(;) = -qc * h - 3hZ(d *h) + 3h(d *h’) - d *h3], Y

with convolution kernels

c =$ (3, d =‘ r)- r = ( ~ ~ + y ’ ) ~ ~ . 2r5 ’

(147)

(149)

Similar expressions for the terrain effects on the geoid can be found in Forsberg (1985). Alternatively, to avoid the use of such higher-order expansions, the computed total gravity terrain effect of (140) may be

transformed to the required quantity by the previously outlined frequency domain methods. However, as the computed gravity effects represent samples of a harmonic function on an uneven surface, the Molodensky approach (see Section 5) should formally be used. In practice, reasonable results may be obtained using even the ordinary Stokes or Vening Meinesz FFT methods (Kearsley et al. 1985).

As a final remark regarding the practical computation of terrain corrections by FFI’ methods, it should be noted that the correction is very dependent on local topographic irregularities, and dense height data are often needed to make the computations sufficiently accurate. Thus, DTMs with resolution of about 1 km, available for many areas of the world, are often not sufficient. The large requirements on computer storage for the FFT methods with dense height data may be circumvented by two means: either the 2-D FFT algorithm may be broken down into a sequence of 1-D transforms, yielding slower computations, or a two-grid procedure may be used, where a detailed grid is only available for the inner zone computations, while a more coarse grid is used for distant topography effects. This approach requires convolution kernels given in the space domain, as closed analytical expressions of the spectra of the required truncated kernels generally do not exist; for details see Forsberg (1985).

at The R



ownloaded from



7. LEAST-SQUARES COLLOCATION

This chapter takes up the discussion of multiple-input-single-output systems in Section 3.7. As has been mentioned there, some important analogies between this method and least-squares collocation exist. Least-squares collocation is the only geodetic method that allows the consistent treatment of heterogeneous gravity field related data. On the other hand, it is a method that requires excessive amounts of computer time if large data sets are involved. By making use of the structural similarities between the two approaches, a fast numerical technique for the solution of least-squares collocation is obtained.

The collocation solution given in Section 2 is of the form

1;:(P) = [LoK(P, Q)]’[L:LFK(P, Q)]-’6bj(Q) (19)

where 1;: is a functional of the anomalous gravity field, Li and Lj are bounded functionals, 6bj are measurements, and K(P, Q ) is the reproducing kernel which defines the norm of the Hilbert space. The choice of the norm is critical in this method. In practice, it is made by replacing the model degree variances a; in equation (20) by empirical degree variances a,. The estimation of a, involves statistical assumptions, such as stationarity of the gravity anomalies on the sphere, and has therefore led to a statistical interpretation of formula (19). Using matrix notation, the statistical least-squares collocation estimate is

1;: = C..Cr16b. ZJ I1 I’ (151a)

or with

(151b)

(151c)

Cij is the cross-covariance matrix of the functional 1;: and the functional s underlying the measurement 6bj, while Cjj is the autocovariance matrix of 6bj. Note, that Cjj will usually contain a noise component and will therefore be of the form

cjj = ( c s s + c n n ) j j , (152) where C,, and C,,, are the autocovariance matrices of signal and noise, respectively, and s and n have been considered as uncorrelated. The step from equation (19) to equation (151) is not without difficulties; for a discussion, see Moritz (1980), Tscherning (1986) and Sanso (1986). In the following, equation (151) will be used as a starting point and the signal and noise processes involved will be considered as stationary and ergodic. These assumptions underly all applications of least-squares collocation which use an empirical, isotropic covariance function.

Formula (151b) which contains the least-squares estimator aij can be rewritten as a convolution integral for gridded data in the plane. It is of the form

+-

Cij(Xi - x i , yi - yj) = l l f 2 ( X j - x , yi - Y)Cj j (X - x i , y - y j ) dx dy. (153) -m

The spectral equivalent of this equation is

Sij(u, V) = Aij(u, v)$j(u, v), (154) where S is again the PSD function of the covariance function C with the same subscripts. Thus, the least-squares estimator A, can be obtained by applying an inverse transform to the following equation

Aij(u, v) = Sij(u, v)/Sjj(u, v). (155) The obvious advantage of this approach is that the inversion of a large matrix Cjj in the spatial domain is replaced by a quotient of PSD functions in the spectral domain. In many cases, this is the only way to treat the large amounts of data in a consistent way.

Even more important, however, is the capability to combine in the spectral domain different gravity field related measurements for the estimation of a specific gravity functional. To show this, a somewhat intuitive argument will be used. First, the equations for stepwise least-squares collocation will be transformed to the spectral domain. Then, it will be shown that they correspond to a specific form of the multiple-input-single-input equations discussed in Section 3.7.

In stepwise collocation the total measurement vector is split up in sub-vectors in order to reduce the size of the matrices to be inverted or to add additional observations to improve a set of initial estimates. The approach follows standard least-squares procedures. A detailed derivation can be found in Moritz (1980, Section 19). By partitioning the vectors and matrices in equation (151a)

cjj = (Z;; c”)cij = (ClC2), c22

at The R



ownloaded from



the following formulae can be derived:

g = (CIC;,’ - C2C2;1C21C;;1+ ClC;,1C12C’2;’C,;1C;)6bl+ (CZC,;’ - C1C;;’C12C’,-:)662

where

c;; = (C, - c21c;,1c12)-1

and g is the functional to be estimated. Using the substitutions

fl = clc;,’ - c2C;1c21c;;1 + c,c;~c;.~.,-:C;;c;;

and

(156a)

(156b)

(157a)

f2 = c2C,;l - ClC;l lCl2C~~

equation (156a) can be written as

g =f1661 +f@2 (158a)

(157b)

Considering that the right-hand side of equation (151a) approximates an integral operator, it can be shown that the products in equation (158a) are the result of a discrete convolution. Thus, the PSD function of g can formally be written as

G = F1 Hl + F2 H2 (158b)

where Hi is the Fourier transform of 6b,. As a comparison with equation (77) shows, this is the basic form of the multiple-input-single-output equation for two linear inputs without noise. The relation between least-squares collocation for gridded data and the multiple-input-single-output equations can be established by showing that the Fourier transforms of 8 given by equations (157) correspond to the 5 given by equation (82). Considering the covariance matrices in equation (157) again as infinite matrices and denoting their PSD functions by S, the following formulae are easy to verify

(159b)

(159a)

Considering the basic definitions underlying the two methods, and since all covariance functions are real and even, it can be shown that

Si = Sh&, v)

Sii = Sh&(ll, v) = S,,

thus establishing the equivalence of equations (82) and (159).

given on a grid. For a more detailed discussion and the treatment of noisy data, see Vassiliou (1986). Data combination by least-squares collocation is therefore equivalent to the multiple-input-single-output method for data

If the signals represented by the measurements 6bl and 6b2 are linearly correlated, i.e. if

HZ(u, v) = D(u, v)Hl(U, v)

a much simpler expression for 5 is obtained, namely

compare equations (85) and the equivalences (160). In the applications of least-squares collocation to gravity field approximation, a relation of the form (161) is always assumed. Thus, in stepwise collocation of gridded data the cumbersome formula (156a) can be replaced by the much simpler set of equations

at The R



ownloaded from



Table 1. RMS accuracy and CPU time required for the estimation of first-order gradients from second-order gradients using 167000 measurements.

Measurements RMS Error of CPU time required (seconds)

0.70 0.46

Txz*Tyz’Tzz 0.41

TZz

Txz ‘Tyz

Tx( I 0-5m s-’)

0.79 0.87 0.79 0.63 0.62

TZ z Txz

Txx Txy Txx 2Txy STxz

Txx * Txy * Txz TZz

0.49 TZZ T 0.69

0.76 T 0.60

YZ Txx *Tyy

545 560 578

584 583 601 617 6 32

583

584 601 615 634

where

cj = (C1, C2), 6bj = (;;;). In summary, the following can therefore be stated.

(a) Least-squares collocation can be treated by FFT techniques as long as data are given on a regular grid. (b) Different gravity field related data types are combined by multiple-input-single-output equations in exactly the same

(c) If the input signals, i.e. the observables, are linearly correlated, a major simplification of the formulae results in both

(d) Measurement noise can be treated as well in the spectral domain as it is in the spatial domain, especially when white

way as by least-squares collocation.

the spectral and the spatial domain.

noise assumptions can be made; see e.g. Vassiliou (1986) and Sideris (1987a,b).

A typical example where data combination is needed is airborne gravity gradiometry. In this method, the five independent second-order gradients of the gravitational potential are measured on a regular grid at aircraft altitudes and are used to compute first-order gravity gradients at ground level. The solution must therefore consider data combination as well as downward continuation of gravity field related data. Table 1 shows results of a simulation study which used different gradient combinations and evaluated their effect on the components T,, T,, T, of the gradient vector of the anomalous gravity potential T. The total data set contained 167000 second-order gradients, i.e. about 33500 measurements of each independent gradient. This means that the minimum matrix to be inverted in least-squares collocation is of the order 33500. Inversions of this size obviously present time problems even on a supercomputer and results will suffer from round-off errors. In contrast, the CPU times for the FFT method on a mainframe computer are quite reasonable; see the last column of Table 1. When using a supercomputer with vector algebra, these times would be reduced to about 30 CPU seconds.

A detailed discussion of the results given in the above table can be found in Vassiliou (1986); especially the effects of gradient combination, windowing and downward continuation are discussed at length.

8 DEGREE VARIANCES AND COVARIANCE FUNCTIONS B Y FFT

So far in this paper, with the exception of the last section, all methods presented have only used the Fourier transform as a fast way of performing convolutions. However, the power spectrum of the gravity field is useful by itself, as the shape of the power spectrum puts constraints on the possible density anomaly variations inside the Earth, and also because the power

at The R



ownloaded from



spectrum may, in a simple way, be related to the spherical harmonic degree variances (see equation 13). The degree variances play an important role in error studies and in the selection of proper analytic covariance function models in collocation.

The 2-D power spectral density Sgg and the covariance function Cgg are as outlined in Section 3.6 (equations 59 and 60), related by the Fourier transform

Sgg(x9 Y ) = FtCgg(u, v)l, (164) where it has been assumed for simplicity that the mean of the signal g is zero. The estimation of the 2-D PSD may be done by the periodogram approach or by Welch's method, as outlined in Section 3.6, equation (69), or by improved spectral estimation methods such as autoregressive techniques (Cadzow & Ogino 1981; Kay & Marple 1981; Haykin 1983).

Of special interest is the isotropic case

Cg& Y ) = Cgg(r),

Sgg(u, v) = S,,(q),

r = (x2 + y2)ln,

q = (u' + . 2 y .

In this case, (164) becomes a Hankel transform relation (see 37)

(165a)

(165b)

On a spherical earth, C,,, and the corresponding degree variances c,, are related by a Legendre transform given by (12) (Heiskanen & Moritz 1967)

m

cn = - 2n; 6 Cgg(~)Pn(cos I#) sin 3 d q . (167b)

For the signal g being gravity anomalies, the degree variances c, are related to the potential degree variances a, by the expression

Equations (166) and (167b) may be used to relate the degree variances and the PSD function. Using the asymptotic relationship

(Gradshteyn & Ryzhik 1980, 8.722) and assuming the covariance function Cgg(r) to be near zero at great distances (e.g. use of a low-order spherical harmonic reference field), the relationship

(170a)

(170b)

is obtained (Forsberg 1984b). Equation (170) may be used together with the equation

Cg&, Y ) = F-"Sgg(u9 v)l (171)

to obtain degree variances and covariance function from the PSD estimate. In general, the estimated 2-D PSD will be non-isotropic. In order to obtain isotropic estimates, the following circular averaging process may be used:

1 r2n

1 '= Cig(r) =% Cgg(r cos 8, r sin 8 ) do.

(172a)

(172b)

This circular averaging process improves the stability of the PSD estimate, thus making the simple periodogram estimator somewhat more reliable.

Examples of the use of the above mentioned principles for the study of the spectral behaviour of the local gravity field may

at The R



ownloaded from



be found in Forsberg (1984b) or Vassiliou & Schwarz (1986). There seems also to be a good agreement between space-domain estimates of the covariance function and spectral estimates, see e.g. Schwarz (1985). In general, the empirical results follow more or less the classical Kaula rule (Kaula 1966)

indicating a power spectral decay for gravity anomaly PSD function of the form

where K is a constant. Empirical results tend to indicate a slightly different exponent in (174), with values around -2.6 being quite common. The constant K varies especially according to degree of roughness of the local topography. Using terrain reductions (see Section 6), the stationarity of the reduced field is much improved (Forsberg & Tscherning 1987). Due to the smoothing produced by the terrain reduction, the remaining power will usually be less than indicated by the Kaula rule, which represents a kind of global mean of spectral behaviour of the gravity field including average topographic effects.

Finally, it should also be mentioned that the 1-D PSD estimated from track data, e.g. sea gravity or satellite altimetry, is not identical to the PSD of (172b), even when tracks with many different azimuths are analyzed and averaged. The relationship of the 1-D and 2-D PSD is given by Nash & Jordan (1978) as

m

(175) S;iD(u) = SiiD(u, v ) dv, -m

while the covariance functions are the same

C&D(r) = C&D(r). (176)

It thus follows that trackwise FIT analysis of data does not provide a simple means of estimating degree variances. Rather complicated conversions are necessary to achieve this, cf. Wagner & Colombo (1979).

9 PRACTICAL EXAMPLES OF USING FFT METHODS

In this section, the implementation of the spectral expressions for computing gravimetric quantities is presented. The applications include terrain effects on gravity and airborne gradiometry, and the use of Stokes’, Vening Meinesz’ and Molodensky’s integrals for the computation of geoid undulations and deflections of the vertical. All tests were performed in the Kananaskis area, located about 100 km west of Calgary, in the Canadian Rocky Mountains. Due to its alpine topography, this area presents a challenge to any gravity field modelling method. Elevations range from about 1400 m to more than 3400 m (see Table 2). The area is well covered by gravity and height data and is shown in Fig. 5. A detailed 100 x 100 m DTM is available for most of the area (56 X 36 km). Astrogeodetic deflections of the vertical have been observed by zenith camera and geoid undulations have recently been determined by the University of Calgary using GPS and levelling measurements. Besides the results presented in the following, more results in the same region can be found in earlier studies which included the effect of topography on airborne gravity and gradiometry (Tziavos et al. 1988), and inertial gravimetry (Forsberg 1986; Forsberg & Wong, 1987).

Table 2. Statistical analysis of topography and terrain corrections in the 36 km x 56 km test area of rough topography using a 0.1 km X 0.1 km grid of point heights.

Ag corrections corr. for t at z = 4.5 km (zf=l.l km) il 0

topo-

0 txx XY tXZ YY YZ tZZ t t t Stati- graphy z=h z=z

ticts P

(meters) ( I O - ~ ~ s-2) ( Ed t V B s )

min 1378.0 -313.1 -46.6 -76.7 -40.7 -74.9 -45.9 -62.8 -129.7

max 3413.0 -141.0 14.6 72.0 38.9 82.4 70.5 69.2 86.5

mean 2129.6 -226.7 -15.1 0.0 0.3 4.6 0.8 -3.5 -0.8

s.d. 350.2 37.3 12.2 35.7 14.6 37.2 18.5 23.9 45.5

RMS 2158.2 229.7 19.4 35.1 14.6 37.5 18.5 24.1 45.5

at The R



ownloaded from



-115X - 1 1 5:O

51:O

50: 5

Fisw valley)

5. The topography in the Kananaskis area; x , y scales in km. (The deflection and geoid stations are located along the 50 km-long N-S

9.1 Terrain corrections by FFT

The effect of the topography on surface gravity and on airborne gravity and gradiometry was studied for various grid spacing using the results obtained with the 0.1 km x 0.1 km height grid as control values in the comparisons. Table 2 contains the statistics of the terrain effects in the test area. In the third column in the table, the total topographic effect above the geoid is presented, as computed by formulae (140) and (143). The fourth column contains the same effect at flight level q,=4.5 km above mean sea level, or zf= 1.1 km above the maximum elevation in the area, as computed by (133). Due to attenuation with flight level, this effect is smaller than the one on the Earth’s surface. The remaining columns in Table 2 contain the effect of the terrain on the gradiometry tensor, computed by (136). As an indication of the speed of the FFT evaluations, the computation for 566 x 366 grid points of all the quantities in Table 2 took less than 20 CPU seconds.

at The R



ownloaded from



RMS diffennces

. ~. a[ Right Level - at Eanh's Surface

Figure 6. Effect of grid spacing on the gravity terrain correction accuracy (point heights and z, = 1.1 km were used).

RMS differences

5 f

0 .1 0.5 l:o Lrn'

Figpre 7. Effect of grid spacing on the gradient terrain correction accuracy (point heights and z, = 1.1 km were used).

In order to obtain an estimate of aliasing effects occurring from using grid spacing larger than 0.1 km x 0.1 km, the computations were repeated on grids of 0.2 km x 0.2 km, 0.5 km X 0.5 km, and 1 km x 1 km, and the results were compared to those of the 0.1 km X 0.1 km grid. 748 common points in a 33 km X 31 km inner area were used in these comparisons. The RMS differences at flight level and on the Earth's surface are plotted in Fig. 6. As expected, they are larger at the Earth's surface, and increase from 0 to 0.7mgal almost linearly with the increase in the grid spacing. It sheuld be noted that for the terrain effects at the Earth's surface the differences between the 0.1 and the 1 km grid can reach up to several mgal RMS in the mountainous border zones of the test area, with maximum values being as high as few tens of mgal. Fig. 7 illustrates the differences of the effect of the terrain on the gravity gradients. The comparisons between the grids with spacings of up to 0.5 km give RMS differences smaller than 1E. These differences increase up to 4E RMS for the 1 km grid, with maximum values of the order of 16E. These results are expected, since gravity gradients are strongly dependent on the high frequencies of the terrain. It is interesting to note that the x-components of the gradient tensor are generally more affected than the y-components due to the higher frequencies contained in the topography of the test area in the x-direction. It is also evident that for RMS accuracies better than 1 mgal for gravity and better than 1E for gradiometry, the terrain has to be sampled at spacings of 0.5 km or less, in areas of similarly rough topography.

~

9.2 Prediction of deflections of the vertical and geoid undulations by FFT

In this example, the available gravity data were gridded and transformed to deflections of the vertical and geoid undulations (or, rather, height anomalies), and compared to the observed astrogeodetic deflections of the vertical and GPS-derived relative geoid undulations.

A remove-restore technique has been used to handle the terrain effects. With this method, a consistent model of the attraction of the topography relative to a slowly varying mean height surface is removed from the original gravity anomalies prior to gridding, and restored in the FlT-predicted values of deflections and geoid undulations, as briefly outlined in Section 6, equation ( l a ) , (143) and (149). The terrain effects have been evaluated using a combination of FFT and space domain methods, using height data within a relatively limited region (49.S0-52"N, 116.75"-113.55"W) due to lack of height data east of the Rockies. The residual terrain model effects were computed relative to a surface of about 70 km resolution (corresponding to 1" in longitude) using 1 km X 1 km or 100 m X 100 m height data. In addition to the terrain reduction preprocessing step, also a spherical harmonic reference field evaluated to degree and order 180 (OSU86F, Rapp & Cruz 1986) was removed. Table 3

at The R



ownloaded from



Table 3. Effect of topography and reference field on observed data. *

I% f n % e l Data Type

( I ~ - ~ m s-2) (arcsec) (meter)

No. of p o i n t s 413 15 10

S t a t i s t i c s mean o mean o mean o mean o

Observed values 20.5 51.2 1.31 3.31 .95 6.28 1.15 .46

T e r r a i n e f f e c t s -14.9 47.2 .86 2.91 -.22 5.14 .27 .21

Terrain-reduced 35.4 29.8 .51 1 .60 1.16 2 .15 .88 .25 observa t ions

T e r r a i n and re ference 2.9 23.4 2.99 1.35 -1.16 1 . 6 1 (12.78) .33 f i e l d (n -180) reduced %gervat ions

* Geoid undulat ion a t northernmost point a r b i t r a r i l y set t o zero.

Table 4. F+T prediction results (observed minus predicted geoid undulations relative to northernmost point).

n 5 're1

Data Type (Arcsec) (a rcsec) (m)

No. of p o i n t s 15 10

S t a t i s t i c s mean a mean o mean o RHS

P r e d i c t i o n us ing 1.92 .66 -1.57 .48 .15 .09 .17 OSU86F t o nmax=180

P r e d i c t i o n without .92 . 6 5 .31 .57 .22 .09 .24 re ference f i e l d

Predic t ion wi th .91 .69 .33 .60 .23 .09 .25 Molodensky g l c o r r e c t i o n s

illustrates the effects of the topography and reference field reductions. The use of these re&ctions may be seen as a way of reducing aliasing and periodicity errors of the FFT techniques. If terrain reductions were not used, a very dense gravity grid (100 m resolution) would have to be computed to represent the most local features of the gravity field; and if a high-degree field was not subtracted, systematic trends (e.g. a tilt) in the gravity anomalies would produce periodicity errors which could only partly be controlled by windowing.

To provide the required uniform gravity anomaly grid, the reduced anomalies were transformed onto a 1.5' X 2.5' grid, covering the 2.5" X 3" area, using a fast quadrant-search local collocation algorithm. The gridding of data was the most time consuming step in the FFT gravity field modelling process. It took 111 CPU seconds on a CDC 175 to create the 100 x 72 grid of gravity anomalies.

The deflections of the vertical and geoid undulations were subsequently predicted on the same grid by FFT, using formulae (95) and (96). The FFT convolution only required 5 seconds CPU time. Periodicity and spectral leakage errors were controlled through the use of a 5 per cent cosine taper window.

The final comparison to the observed deflections of the vertical and relative geoid undulations was performed using a windowed spline interpolation routine, to obtain prediction results at the observation points. The statistics for the comparison is shown in Table 4.

methods function extremely well, with deflections predicted at the 0.5"-0.6" level, and geoid undulations at the 10-20 cm level, For deflections, the bias values are not significant, due to uncertainties in the datum transformation. For the geoid undulations, the bias values represent a general tilt in the geoid solution of around 20cm over the 50km length of the Kananaskis Valley, corresponding to a bias in N-S deflection component around 1". This tilt is probably a consequence of insufficient gravity data coverage in the surrounding regions and of the small size of the computation area (2.5" X 3"), although possible errors in the orthometric heights may also affect the results.

In addition to the 'simple' FFT formulae (95) and (96), a prediction was done using the first-order Molodensky continuation terms (124), (125) and (126), applied to the terrain-reduced data which still refer to an uneven surface. The computation time

Considering the rugged topography, the comparison results indicate that the

at The R



ownloaded from



for computing the complete FFT transform for g,, C1, El, ql increased to 22s. No improvement in prediction results was observed. In practice, the ‘simple’ FFT approach is therefore a powerful method, which means that the minor approximation of referring data to a common level has essentially no impact. However, since the test area has a very rough gravity field, the inclusion of higher order Molodensky terms could provide better results. This, though, is a subject of further research.

10 CONCLUSIONS

The main conclusion of this paper is that Fast Fourier transform techniques can be applied with advantage to the numerical solution of gravity field approximation problems. The transformation of the standard space-domain solutions into the spectral domain is shown in detail for Stokes’, Vening Meinesz’ and Molodensky’s solutions of geodetic boundary value problems, for least-squares collocation, and for different approaches to the terrain reduction problem. In all cases, an appropriate transformation can be made and the approximations necessary are within the limits of present data accuracy.

The application of FFT methods to geodetic or geophysical data sets presents some problems usually not encountered in engineering applications. They have to do with the heterogeneity of the data, the non-uniformity of the data noise, the complicated surface on which they are given, and the uneven spatial distribution. The first problem can be solved by using multiple-input-single-output techniques and leads to a surprisingly simple solution for the combination of different types of gravity field data. The second problem, the variation of the data noise from one area to the next, can be handled by FFT techniques. It is a non-standard feature rather than an actual problem. Adequate solutions to the third problem seem to be available if the numerical convergence of the Molodensky series can be taken as an indicator. Data gridding, the fourth problem, has to be done for most of the terrestrially measured geodetic data while it is not necessary for much of the airborne and satellite data. Comparisons have shown that the use of gridded data instead of original observations does not affect the results in a negative way. Thus, the accuracy of the numerical solution will not suffer when gridded data sets are used.

Among the solutions presented, the recasting of the Molodensky solution and the spectral formulation for least-squares collocation are of specific interest. The first, because it shows that a two-step solution, consisting of downward continuation to sea level and subsequent upward continuation to point level, not only reduces computation time dramatically but also gives additional insight into the effect of different data groups. The second, because it opens the door to handle very large data sets by least-squares collocation and thus removes a major obstacle for the application of this method. This is of specific interest for the computation of a detailed geoid over areas of continental extent and for data coming from airborne sensors, as e.g. airborne gravity gradiometry. These two applications will also greatly benefit from the methods of FFT terrain reductions discussed in this paper. Especially, the computation of orthometric heights in mountainous terrain from a combination of GPS-satellite heights and gravimetrically determined geoid heights is not possible without fast and accurate methods of terrain reduction.

The numerical examples discussed in the last section of this paper give an indication of the practical advantages of the method. Its most outstanding feature is numerical speed which, in many cases, improve? by at least an order of magnitude when compared to competing methods. A second positive feature is the large amount of data that can be handled in one run. This gives consistent numerical solutions over large areas and eliminates the need for surface patching necessary in some other methods. The combination of these two features will lead to improved operational methods because even sophisticated terrain reduction methods can now be used without an undue demand on computer time. It can therefore be expected that FFT methods will become standard procedures in a number of geodetic applications.

REFERENCES

Bendat, J. S. & Piersol, A. G., 1971. Random Data: Analysis and Measurement Procedures, John Wiley & Sons, New York. Bendat, J. S. & Piersol, A. G., 1980. Engineering Applications of Correlation and Spectral Analysis, John Wiley & Sons, New York. Bergland, G. D., 1969. A guided tour of the fast Fourier transform, IEEE Spectrum, 6 , 41-52. Bracewell, R., 1978. The Fourier Transform and its Applications, 2nd edn, McGraw-Hill, New York. Cadzow, J. A. & Ogiono, K, 1981. Two-dimensional spectral estimation, IEEE Trans. Acoust., Speech, Signal Proc., ASSP-29, No. 3. Childers, D. G. (ed.), 1978. Modern Spectrum Analysis, IEEE Press, New York. Colombo, 0. L., 1979. Optimal estimation from data regularly sampled on a sphere with applications in geodesy, OSU Report NO. 291,

Colombo, 0. L., 1981. Numerical methods for harmonic analysis on the sphere, O W Report No. 310, Department of Geodetic Science and

Comer, R. P. LeSchack, A. R., Marks, K. M. & Tail, K. S . , 1986. Topics in geodetic algorithm research, Repori TR-5186-1, The Analytical

Dorman, L. M. & Lewis, B. T. R., 1974. The use of non-linear functional expansions in calculation of the terrain effect in airborne and marine

Eren, K., 1980. Spectral analysis of GEOS-3 altimeter data and frequency domain collocation, OSU Repori No. 297, Department of Geodetic

Forsberg, R., 1984a. A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modeling, OSU Report

Department of Geodetic Science and Surveying, Ohio State University, Columbus, Ohio.

Surveying, Ohio State University, Columbus, Ohio.

Sciences Corporation, Reading, MA.

gravimetry and gradiometry, Geophysics, 39, 33-38.

Science and Surveying, Ohio State University, Columbus, Ohio.

No. 355, Department of Geodetic Science and Surveying, Ohio State University, Columbus, Ohio.

at The R



ownloaded from



Forsberg, R., 1984b. Local covariance functions and density distributions, OSU Report No. 356, Department of Geodetic Science and

Forsberg, R., 1985. Gravity field terrain effect computations by FFT, Bull. Geod., 59, 342-360. Forsberg, R., 1986. Inertial geodesy in a rough gravity field, UCSE Report #30009, Department of Surveying Engineering, University of

Forsberg, R. & Tscherning, C. C., 1981. The use of height data in gravity field approximation by collocation, J . geophys. Res., 86, B9,

Forsberg, R. & Tscherning, C. C., 1987. Spectral properties of the gravity field in the Nordic countries, Bull. Geod. Teor. Appl., in press. Forsberg, R. & Wong, R. V. C., 1987. Methods to extract gravity information from inertial data, in Proc. of the IAG Symposia, XIX IUGG

Gradshteyn, I. S. & Ryzhik, I. M., 1980. Table of Integrals, Series and Products, corrected and enlarged edition, Academic Press, Orlando,

Harris, F. J., 1978. On the use of windows for harmonic analysis with the discrete Fourier transform, Proc. IEEE, 66, No. 1, 51-83. Haykin, S. (ed.), 1983. Nonlinear Methods of Spectral Analysis, 2nd edn, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo. Heiskanen, W. A. & Moritz, H., 1967. Physical Geodesy, W. H. Freeman and Company, San Francisco. Jordan, S. K., 1978. Fourier physical geodesy, Reporr No. AFGL-rR-78-rn56, the Analytic Sciences Corp., Reading, Massachusetts. Kaula, W. M., 1966. Theory of satellite geodesy, Blaisdell Publishing Company, Waltham, Massachusetts. Kay, S. M. & Marple, S. L. Jr, 1981. Spectrum analysieA modem perspective, Proc. IEEE, 69, No. 11. Kearsley, A. H. W., Sideris, M. G., Krynski, J., Forsberg, R. & Schwarz, K. P., 1985. White Sands revisited-A comparison of techniques to

predict deflections of the vertical, UCSE Report # 3 0 7 , Division of Surveying Engineering, University of Calgary, Calgary, Alberta. Krarup, T., 1%9. A contribution to the mathematical foundation of physical geodesy, Publ. 44, Danish Geodetic Institute, Copenhagen. Molodenskii, M. S., Eremeev, V. F. & Yurkina, M. I., 1962. Methods for the Study of the Gravitational Field of the Earth, Translated from

Moritz, H., 1964. The boundary value problem of physical geodesy, OSU Report No. 46, Institute of Geodesy, Photogrammetry, and

Moritz, H., 1968. On the use of the terrain correction in solving Molodensky’s problem, OSU Report No. 108, Department of Geodetic

Moritz, H., 1980. Advanced Physical Geodesy, Herbert Winchmann Verlag, Karlsruhe, Abacus Press, Tunbridge Wells, Kent. Nash, R. A. & Jordan, S. K., 1978. Statistical geodesy-an engineering perspective, Proc. IEEE, 66, 532-550. Papoulis, A., 1977. Signal Analysb, McGraw-Hill, New York. Parker, R. L., 1972. The rapid calculation of potential anomalies, Geophys. J. R. astr. SOC., 31, 447-455. Rabiner, L. R. & Rader, C. M., (ed), 1972. Digital Signal Processing, IEEE Press, New York. Rapp, R. H. & Cruz, J. Y., 1986. Spherical harmonic expansions of the earth’s gravitational potential to degree 360 using 30’ mean anomalies,

OSU Report No. 376, Department of Geodetic Science and Surveying, Ohio State University, Columbus, Ohio. Rizos, C., 1979. An efficient computer technique for the evaluation of geopotential from spherical harmonic models, Austral. Geod.,

Photogram. Sum., No. 31, 161-170. Sanso, F., 1986. Statistical methods in physical geodesy, Mathematical and numerical techniques in physical geodesy, in Lecture Notes in Earth

Sciences, vol. 7, pp. 49-155, ed. Sunkel, H., Springer-Verlag, Berlin. Schwarz, K. P., 1985. Data types and their spectral properties, in Proceedings of the International Summer School on Local Gravity Field

Approximation, Beijing, China, UCSE Report #60003, Division of Surveying Engineering, Universify of Calgary, Calgary, Alberta. Side&, M. G., 1984. Computation of gravimetric terrain corrections using fast Fourier transform techniques, UCSE Report #20007, Division

of Surveying Engineering, University of Calgary, Calgary, Alberta. Sideris, M. G., 1985a. A fast Fourier transform method of computing terrain corrections, Manuscripra Gkodaetica, 10, No. 1, 66-73. Sideris, M. G., 1985b. On the relationship between spectra of mean and point data with applications in computing, inverting, and updating

gravity field quantities, in Proceedings of the International Meeting on Potential Fields in Rugged Topography, Lausanne, Switzerland, July 30-31 and August 1, 1985. IGL Bulletin No. 7, pp. 82-87, Institut de Geophysique, Universite de Lausanne.

Sideris, M. G., 1987a. On the application of spectral techniques to the gravimetric problem, invited paper, in Proceedings of the IAG Symposia, XIX IUGG General Assembly, Vancouver, BC, Aug. 9-22, pp. 428-442, IAG, Paris.

Sideris, M. G., 1987b. Spectral methods for the numerical solution of Molodensky’s problem, UCSE Reporf #20024, Department of Surveying Engineering, University of Calgary, Calgary, Alberta.

Sideris, M. G. & Schwarz, K. P., 1986. Solving Molodensky’s series by fast Fourier transform techniques, Bull. Geod., 60, 51-63. Sideris, M. G. & Schwarz, K. P., 1988. Advances in the numerical solution of the Linear Molodensky problem, Bull. Geod., 62, 59-69. Sideris, M. G. & Tziavos, I. N., 1988. FFT-evaluation and applications of gravity-field convolution integrals with mean and point data, Bull.

Geod., 62, 521-540. Tscherning, C. C., 1986. Functional methods for gravity field approximation, in Mathematical and Numerical Techniques in Physical Geodesy,

pp. 3-47, ed. Sunkel, H., Lecture Notes in Earth Sciences, vol. 7, Springer-Verlag, Berlin, Heidelberg. Tscherning, C. C. & Rapp, R. H., 1974. Closed covariance expressions for gravity anomalies, geoid undulations and deflections of the vertical

implied by anomaly degree variance models, OSU Report No. 208, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, Ohio.

Tziavos, I. N., Sideris, M. G., Forsberg, R. & Schwan, K. P., 1988. The effect of the terrain on airborne gravity and gradiometry, J . geophys. Res., 93, 9173-9186.

Vassiliou, A. A., 1986. Numerical techniques for processing airborne gradiometer data, UCSE Report #2(x117, Department of Surveying Engineering, The University of Calgary, Calgary, Alberta.

VassZou, A. A. & Schwarz, K. P., 1987. Study of the high frequency spectrum of the anomalous gravity potential, J . geophys. Res., 92,

Wagner, C. A. & Colombo, 0. L., 1979. Gravitational spectra from direct measurements, J. geophys. Res., 84, 4699-4712. Welch, P. D., 1967. The use of the fast Fourier transform for the estimation of power spectra; a method based on time averaging over short,

Surveying, Ohio State University, Columbus, Ohio.

Calgary, Calgary, Alberta.

7843-7854.

General Assembly, Vancouver, BC, Aug. 10-22, pp. 391-404, IAG, Pans.

Florida.

Russian (1960), Israel Program for Scientific Translations, Jerusalem.

Cartography, Ohio State University, Columbus, Ohio.

Science and Surveying, Ohio State University, Columbus, Ohio.

609-617.

modified periodograms, IEEE Trans. Audio Electroacoust., AU-15, 70-73.

at The R



ownloaded from


Documents

Geophys. J. Int. 1990 Schwarz 485 514