It is usual to interpret gravity anomalies at the surface as due to an anomalous mass distribution condensed on a surface at a certain depth. If we consider an underground mass distribution which is always positive on the surface at a certain depth, we can easily calculate equivalent gravity anomalies, but it is difficult to decide from the surface anomalies whether the underground mass distribution is always positive or not. In this paper, this criterion is investigated from the stand point of the spectrum and it is found that the necessary and sufficient condition for the underground mass distribution to be always positive on the surface at a certain depth D, is written as,
G₀(ω)=G_D(ω)⋅e- ^|ω|D,
where G₀(ω) represents the spatial wave number spectrum of gravity anomalies and G_D(ω) represents an arbitrary auto-correlation function derived from an arbitrary function A(u)as
For simplicity's sake, we illustrate a few example when ℑG_D(ω)=0, that is, the gravity distribution is symmetrical about the origin . The examples are classified into three cases.
1) The example which gives negative mass distribution somewhere on the surface at a certain depth, because G_D(ω) does not satisfy the necessary condition for auto-correlation function. Gravity distribution represented by e-^1/2 (χ/σ is an example of this case (Fig. 1).
2) The example which gives positive mass distribution because G_D(ω) satisfies the necessary and sufficient condition for auto-correlation function. Gravity distribution represented by b/(x²+b²) or 1/{(1+x)²+1}+1{/(1-x)²+1} is an example of this case.
3) The example which does not gives positive mass distribution because G_D(ω) satisfies the necessary condition only but does not satisfy the sufficient condition for autocorrelationfunction. Gravity distribution (sin ^π_hχ^π_hχ)² is an example of this case.