The spatial distribution of the South Australian gravity station network (over 65 OOO stations) can be approximated by a fractal point set of correlation dimension 0, = 1.4. The fractality is established over more than 2 decades of distance. The fractal nature of the grid is possibly due to the multistage decisions involved in establishing a network; in each step, previously unexplored areas are dissected by geophysical traverses, as in the classical fractal fragmentation process. It is shown that we cannot observe the short-wavelength components of the gravity field if the dimension of the network is less than two and any attempt to interpolate onto a regular grid could lead to spurious anomalies due to aliasing.

S U M M A R Y Treitel & Wang (1976) observed that autocorrelation matrices, as used for time-domain design of digital deconvolution filters, are ill-conditioned in certain cases. They present an example, where the solution of such a system of linear equations results in significantly different filter points, when the solution is performed on different computers. This paper presents a survey of the causes of such problems from a mathematical point of view. Later, the effect of pre-whitening on numerical stability is examined.For the purposes of discussion here, deconvolution isDeconvolution attempts to produce a function g(s) E G given f(t -s) E F and y ( t ) E Y, where G, F, and Y are the respective function spaces to which the functions belong. Such a problem would be said to be correctly posed if (i) for every function y ( t ) there corresponds a solution g(s) to the problem, (ii) the solution g(s) is unique for a given y ( t ) , and (iii) the solution g(s) is continuous with respect to y ( t ) .Equation (1) is a special case of a Fredholm integral equation of the first kind: b y ( t ) = 1 K(t, s)g(s) ds, c 5 t I d.a As stated by Tihonov (1963), it is not true that a solution g(s) may be produced for any given y ( t ) for equations of this type. So there may be no function, g(s), which, when convolved with a given filter, f, will yield a desired output y ( t ) , which means that deconvolution is incorrectly posed. If the left-hand side of equation (1) is only known to a finite accuracy, the different numerical methods to solve equation (1) lead to quite erratic results. Phillips (1962) presents some interesting numerical examples, and attributes this phenomenon to the fact that the integral operator with kernel K(t, s) generally has no bounded inverse. Franklin (1970) noted these effects, and discussed the use of stochastic processes to provide information about ill-posed linear problems.

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1976) noted that, when autocorrelation matrices are illconditioned, elements of Wiener filters are significantly different when the normal equations are solved on different computers. They presented an example in which the Wiener-Levinson algorithm produced a prediction filter exhibiting significant error. In recent years there has been controversy in mathematical literature relating to stability of algorithms, such as the Wiener-Levinson algorithm, for solving linear systems of equations with a Toeplitz coefficient matrix. In this paper, it is argued that poor-quality results produced by the Wiener-Levinson algorithm, when applied to problems exhibiting an ill-conditioned autocorrelation matrix, may be attributed to stability properties of that algorithm. An example is presented, comparing the results of Gaussian elimination, the Wiener-Levinson algorithm, and the conjugate gradient algorithm. The use of intermediate results of the Wiener-Levinson algorithm to detect ill-conditioned normal equations is discussed.