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Benford’s law (BL) is a mathematical theory of leading digits. This law predicts that the distribution of first digits of real-world observations is not uniform and follows a trend in which measurements with a lower first digit (1, 2, …) occur more frequently than those with higher first digits (…, 8, 9). A data set from earth’s geomagnetic field, the estimated time in years between reversals of earth’s geomagnetic field, the seismic P-wave speed of earth’s mantle below the southwest Pacific, and other geophysical data obey the BL. Although there are other statistical methods for analyzing a data set, we test, for the first time, the analysis of the seismic reflectivity through the Benford distribution point of view. We applied the BL on real reflectivity data from two wells from the Penobscot field and another two from the Viking Graben field. In both data sets, the reflectivity was in conformity with the BL. Moreover, after analyzing the effect of sonic and density logs despiking on Benford’s distribution through the BL, we found an optimum coefficient for the despiking process, which was a common procedure used to edit the well-log data before its use on reservoir studies.

Tying seismic data to well data is critical in reservoir characterization. In general, the main factors controlling a successful seismic well tie are an accurate time-depth relationship and a coherent wavelet estimate. Wavelet estimation methods are divided into two major groups: statistical and deterministic. Deterministic methods are based on using both the seismic trace and the well data to estimate the wavelet. Statistical methods use only the seismic trace and generally require assumptions about the wavelet's phase or assume a random process reflectivity series. We compare the estimation of the wavelet for seismic well tie purposes through least squares minimization and zero-order quadratic regularization with the results obtained from homomorphic deconvolution. Both methods do not make any assumptions regarding the wavelet’s phase or the reflectivity. The best-estimated wavelet was used as input to sparse-spike deconvolution to recover the reflectivity near the well location. The results show that the wavelets estimated from both deconvolutions are similar, which builds our confidence in their accuracy. The reflectivity of the seismic section was recovered according to known stratigraphic markers (from gamma-ray logs) present in the real dataset from the Viking Graben field, Norway.

Well to seismic tie is used to correlate well log information to the seismic data. The main link between a primary reflection signal and the reflectivity from a well log is the seismic wavelet. In this paper we test two different methods to estimate the seismic wavelet for the well tie procedure: one based on both the well log and seismic data, the deterministic approach, and one based only on the real seismic trace, through the predictive deconvolution. Our tests with numerical data show the estimation of seismic wavelet with reasonable accuracy for both cases. The feasibility of this approach is also verified on a real seismic and well data from Viking Graben field, North Sea, Norway.

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