Rapid 3-D forward modeling of gravity and gravity gradient tensor fields
Abstract:Three-dimensional inversion are the key process in gravity exploration. In the commonly used scheme of inversion, the subsurface of the earth is usually divided into many small prism blocks (or grids) with variable density values. A key task in gravity inversion is to calculate the composite fields (gravity and gravity gradient tensor) generated by all these grids, this is known as forward modeling. In general forward modeling is memory-demanding and time-consuming. One scheme to rapidly calculate the fields is to implement it in Fourier domain and use fast Fourier transform algorithm. The advantage of the Fourier domain method is, obviously, much faster. However, the intrinsic edge effect of the Fourier domain method degrades the precision of the calculated fields.
We have developed an innovative scheme to directly calculate the fields in spatial domain. There are two key points in this scheme. One key point is spatial discretization. Spatial convolution formula is discretized using an approach similar to normal difference method. A key idea during discretization is to use the analytical formula of a cubic prism, and this makes the resultant discrete formula have clear physical meaning: it embodies the superposition principle of the fields and is the exact formula to calculate the fields generated by all grids. The discretization only requires the grids have the same dimension in horizontal directions, and grids in different layers may have different dimension in vertical direction, and this offers more flexibility for inversion. Another key point is discrete convolution calculation. We invoke a high efficient two-dimensional discrete convolution algorithm, and it guarantees both time-saving and memory-saving. Its memory cost has the same order as the number of grids. Numerical test result shows that for a model with a dimension of 1000x1000x201 grids, it takes about 300s to calculate the fields on 1000x1000 field points in a personal computer with 3.4-GHz CPU (Code is carried out on Matlab), and the calculated fields is almost identical with the analytical solutions.
The above spatial scheme can also be used to calculate geomagnetic field and its gradient tensor. With limited extra-effort, it even can be used to conduct the forward modeling for the electromagnetic field.