GENERALIZED WAVELETS AND INVERSION OF THE RADON TRANSFORM ON THE LAGUERRE HYPERGROUP

Let X=Rn +×R denote the underlying manifold of polyradial functions on the Heisenberg group H n. We construct a generalized translation on X=Rn +×R, and establish the Plancherel formula on L2(X,dμ). Using the Gelfand transform we give the condition of generalized wavelets on L2(X,dμ). Moreover, we show the reconstruction formulas for wavelet packet trnasforms and an inversion formula of the Radon transform on X.

Sensitivity-based upscaling for history matching of reservoir models

Simulation of reservoir flow processes at the finest scale is computationally expensive and in some cases impractical.Consequently,upscaling of several fine-scale grid blocks into fewer coarse-scale grids has become an integral part of reservoir simulation for most reservoirs.This is because as the number of grid blocks increases,the number of flow equations increases and this increases,in large proportion,the time required for solving flow problems.Although we can adopt parallel computation to share the load,a large number of grid blocks still pose significant computational challenges.Thus,upscaling acts as a bridge between the reservoir scale and the simulation scale.However as the upscaling ratio is increased,the accuracy of the numerical simulation is reduced;hence,there is a need to keep a balance between the two.In this work,we present a sensitivity-based upscaling technique that is applicable during history matching.This method involves partial homogenization of the reservoir model based on the model reduction pattern obtained from analysis of the sensitivity matrix.The technique is based on wavelet transformation and reduction of the data and model spaces as presented in the 2Dwp-wk approach.In the 2Dwp-wk approach,a set of wavelets of measured data is first selected and then a reduced model space composed of important wavelets is gradually built during the first few iterations of nonlinear regression.The building of the reduced model space is done by thresholding the full wavelet sensitivity matrix.The pattern of permeability distribution in the reservoir resulting from the thresholding of the full wavelet sensitivity matrix is used to determine the neighboring grids that are upscaled.In essence,neighboring grid blocks having the same permeability values due to model space reduction are combined into a single grid block in the simulation model,thus integrating upscaling with wavelet multiscale inverse modeling.We apply the method to estimate the parameters of two synthetic reservoirs.The history matching results obtained using this sensitivity-based upscaling are in very close agreement with the match provided by fine-scale inverse analysis.The reliability of the technique is evaluated using various scenarios and almost all the cases considered have shown very good results.The technique speeds up the history matching process without seriously compromising the accuracy of the estimates.

INVERSE RADON TRANSFORM WITH ONE-DIMENSIONAL WAVELET TRANSFORM

In this paper, the wavelet inverse formula of Radon transform is obtained with onedimensional wavelet. The convolution back-projection method of Radon transform is derived from this inverse formula. An asymptotic relation between wavelet inverse formula of Radon transform and convolution-back projection algorithm of Radon transform in 2 dimensions is established.

Determination of regional earthquake source parameters in wavelet domain

Investigating source parameters of small and moderate earthquakes plays an important role in seismology research. For small and moderate earthquakes, the mechanisms are usually obtained by first motion of P-Wave, surface wave spectra method in frequency-domain or the waveform inversion in time-domain, based on the regional waveform records. We applied the wavelet domain inversion method to determine mechanism of regional earthquake. Using the wavelet coefficients of different scales can give more information to constrain the inversion. We determined the mechanisms of three earthquakes occurred in California, the United States. They are consistent with the previous results (Harvard Centroid Moment Tensor and United States Geological Service). This proves that the wavelet domain inversion method is an efficient method to determine the source parameters of small and moderate earthquakes, especially the strong aftershocks after a large, disastrous earthquake.