On the use of the profiled singular-function expansion in gravity gradiometry

We demonstrate a method of inverting gravity data, based on profiled singular-function expansions. It is well known that the inverse problem of determining underground density variations from gravity data is severely ill-posed and prior knowledge is needed to restrict the range of possible solutions. Viewed as a linear inverse problem, various standard methods, all of which produce solutions approximating the generalised solution, tend to give density variations concentrated near the surface. To overcome this potentially undesirable trait, various authors have introduced depth weighting. In this paper we carry this idea a step further and introduce a method based on a profiled singular-function expansion which uses the prior knowledge that the underground object is centred at a particular depth. The use of an appropriate depth-weighting profile leads to a solution at the correct depth. Furthermore, we show that if the depth of the object is unknown, a range of solutions at different depths can be produced, allowing other prior knowledge, such as object size or density, to be introduced to determine which solution is the most plausible. The truncation point of the profiled singular-function expansion is determined by the level of noise on the data. We examine how the achievable horizontal resolution varies with this truncation point. A notable property of our approach is that when the centre of the profile corresponds to the true object depth, the solution appears, in a certain sense, to be the most focussed one. Finally we consider a gravimetry example using real data.