A critical problem in inversion of geophysical data is developing a stable inverse problem solution that can si multaneously resolve complicated geological structures. The traditional way to obtain a stable solution is based on maximum smoothness criteria. This approach, however, provides smoothed unfocused images of real geoelectri cal structures. Recently, a new approach to reconstruc tion of images has been developed based on a total varia tional stabilizing functional. However, in geophysical ap plications it still produces distorted images. In this paper we develop a new technique to solve this problem which we call focusing inversion images. It is based on specially selected stabilizing functionals, called minimum gradi ent support (MGS) functionals, which minimize the area where strong model parameter variations and disconti nuity occur. We demonstrate that the MGS functional, in combination with the penalization function, helps to generate clearer and more focused images for geologi cal structures than conventional maximum smoothness or total variation functionals. The method has been suc cessfully tested on synthetic models and applied to real gravity data. TIKHONOV REGULARIZATION AND STABILIZING FUNCTIONALS Consider the inverse problem d= Am, (1) where A is the forward modeling operator; m = m(r), a scalar function describing geological model parameter distribution in some volume V in the earth (m EM, where M is a Hilbert space of models with L z norm); and d = d(r), a geophysical data set (d E D, where D is a Hilbert space of data).

We develop a method of 3-D magnetic anomaly inversion based on traditional Tikhonov regularization theory. We use a minimum support stabilizing functional to generate a sharp, focused inverse image. An iterative inversion process is constructed in the space of weighted model parameters that accelerates the convergence and robustness of the method. The weighting functions are selected based on sensitivity analysis. To speed up the computations and to decrease the size of memory required, we use a compression technique based on cubic interpolation.Our method is designed for inversion of total magnetic anomalies, assuming the anomalous field is caused by induced magnetization only. The method is applied to synthetic data for typical models of magnetic anomalies and is tested on real airborne data provided by Exxon-Mobil Upstream Research Company.

A practical inversion procedure is used to parametrize an unconfined aquifer on Cape Cod, Massachusetts, using groundwater age and hydraulic head data. The inversion resulted in estimates of the recharge flux, homogeneous but anisotropic hydraulic conductivity, porosity, boundary hydraulic heads, and the aquifer thickness. The range of estimated values agreed well with independent measurements at the site. By themselves hydraulic heads are sensitive to the ratio of recharge flux to hydraulic conductivity. The age data are sensitive to the ratio of recharge to porosity. Together, age and head data provide a constraint on both boundary values and material properties. However, a sensitivity analysis shows that porosity, hydraulic conductivity, and recharge are all correlated, and thus a unique inverse solution requires an independent constraint on one of these parameters. Simulations also show that groundwater ages at a given point on a streamline are sensitive to the recharge flux at the point where the streamline begins at the water table. This implies that a vertical profile of age data will be sensitive to the upstream recharge flux, suggesting that spatial patterns in recharge can be estimated using vertical profiles from multilevel monitoring wells. Because measurements of groundwater age can be made in many shallow aquifers, this approach may be applicable to other sites. PORTNIAGUINE

Thin-bed reflectivity inversion is a form of spectral inversion which produces sparse reflectivity estimates that resolve thin layers below the tuning thickness. The process differs from other inversions in that it is driven by geological rather than mathematical assumptions, and is based on aspects of the local frequency spectrum obtained using spectral decomposition of various types. The resolution of thin-bed reflectivity inversion is far superior to the input data and so makes it very suitable for characterization of thin reservoirs.

An inversion-based algorithm for computing the time-frequency analysis of reflection seismograms using constrained least-squares spectral analysis is formulated and applied to modeled seismic waveforms and real seismic data. The Fourier series coefficients are computed as a function of time directly by inverting a basis of truncated sinusoidal kernels for a moving time window. The method resulted in spectra that have reduced window smearing for a given window length relative to the discrete Fourier transform irrespective of window shape, and a time-frequency analysis with a combination of time and frequency resolution that is superior to the short time Fourier transform and the continuous wavelet transform. The reduction in spectral smoothing enables better determination of the spectral characteristics of interfering reflections within a short window. The degree of resolution improvement relative to the short time Fourier transform increases as window length decreases. As compared with the continuous wavelet transform, the method has greatly improved temporal resolution, particularly at low frequencies.