“A Simplified Method for Deducing Subterranean Mass Distribution by the use of the Response for Unit Gravity”(Part I)

Abstract

1) If a unit gravity is represented by Dirac's unit function δ(χ), and if the subterranean mass distribution that is responsible for it is eχpressed by Φ(χ), the subterranean mass distribution for a given gravity distribution g(χ) is given by
ρ(χ)=∫^∞_∞g(χ′)Φ(χ-χ′)dχ′ .
The function Φ(χ) is, according to the potential theory, represented by
Φ(χ)=1/4π²κ²∫^∞_∞e|ω|De^iωχdw
where, D represents the depth of the plane at which the mass is assumed to be condensed and k² the gravitational constant. The above integration does not converge unless the wave number w has an upper limit. Since it is reasonably considered that gravity variations having shorter wave lengths are due to masses situated at shallower depths, we need not take into account those if we are interested in masses at greater depths. Thus we can reasonably impose an upper limit to the wave number which will be denoted byΩ. Then the response functionΦ*(χ) is written as
The mass distribution p*(χ) for g(χ) is therefore given by
ρ(χ)=∫^∞_∞g(χ′)Φ*(χ-χ′)dχ′
This amounts to determine the mass distribution corresponding to g*(χ) instead of tog(χ). Where g*(χ) is such that its spectrum is the same with that of g(χ) for|ω|<Ω, and is zero for |ω|>Ω. Another characteristic of g*(χ) is that it satisfies
ρ(χ)=∫^∞_∞g(χ)-g*(χ))²dχ-amin..
φ*(χ) introduced above is the response function for unit gravity δ*(χ)=sinΩχ/πχ.
2) If the gravity variation along the earth's surface is considered to be periodic, we can use Fourier series instead of Fourier transform. δ*(χ)=sin2M+1/2/2π·sin /2χ is the unit gravity in this case. The mass distribution φ*(χ) for gravity δ*(χ) is given by
Φ*(χ)=1/2π²{1/2 +MΣm=1 e^mD cos mx}
Here, M is the upper limit of the wave number.The mass distribution for gravity, g(χ) is given by
ρ*(χ)=∫^2π₀g(χ′)φ*(χ-χ′)dχ′ .
3) If the gravity variation along the earth's surface is considered to be periodic and its values are given at intermittent points distributed at an equidistant interval, the unit gravity δ(χ)=sin{(2N+1)/2}χ/(2Nsin(1/2)χ will be used. Here 2N is the number of sampling points.
Response function φ(χ) for gravity δ(χ) is
φ(χ)=1/2πκ²N{1/2+N-1 Σ m-1 e^mα cos mχ+1/2e^ND cos Nχ}
and the mass distribution corresponding to gravity g(χ) is
ρ(χ)=2N Σ s=1 g_sφ(χ-Sπ/N),
where g_s is the values of gravity at the sampling points. p(χ) is the overlapping sum of response function values multiplied by gs. The results thus obtained are identical to that of Tsuboi's Fourier series method. The upper limit of wave number, in this case, depends on the numbers of sampling points.
If we are to carry out “low pass : high cut” operation, we can use unit gravity δ(χ)* instead of δ(χ) where
δ(χ)=sin2n+1)/2/N sin1/2χ .
The response function φ(χ) is 1/2πκ²N(1/2+n Σ s-1 e^mD cos mχ .
The mass distribution ρ(χ) is ρ(χ)=2N Σ s=1 g_sφ*(χ-Sπ/