ge oph ] 2 S ep 2 01 2 Automated seismic-towell ties ?

The quality of seismic-to-well tie is commonly quantified us ing the classical Pearson’s correlation coefficient. However the seismic wavelet is time-var iant, well logging and upscaling is only approximate, and the correlation coefficient does not follo w this nonlinear behavior. We introduce the Dynamic Time Warping (DTW) to automate the tying process , accounting for frequency and time variance. The Dynamic Time Warping method can follow th e nonlinear behavior better than the commonly used correlation coefficient. Furthermore, th quality of the similarity value does not depend on the selected correlating window. We compare the de v loped method with the manual seismic-to-well tie in a benchmark case study. Introduction Well logs are commonly used as ground truth to correlate the s eismic signal with the earth’s stratigraphy (White and Simm, 2003). In this process, a wavelet is first est imated from the seismic trace, and then it is convolved with the reflectivity calculated from the well l ogs (sonic log and bulk density log). Iterative techniques are used to estimate the wavelet with correct pha se and amplitude spectrum by matching the actual seismic trace at the well position (Hampson-Russell , 1999). Tying the seismic traces to the well logs aims to minimize the differences in the way seismic data and well logs measure the same parameters (Burch, 2002), but with different resolution. The quality of the tie between the synthetic and the seismic t race is based on the correlation coefficient, which is limited to linear features. The time-variant natur e of the seismic wavelet adds nonlinearities to the trace which cannot be followed by a linear metric. Thus, w avelet phase mismatches frequently occur between the final processed seismic data and the synthetic se ismograms created from well logs. This fact leads to potential complications in stratigraphic and structural interpretation (Van der Baan, 2008). We propose a new method to match these time series accounting for frequency and time variance. We argue that Dynamic Time Warping (DTW) method can follow thes e changes and furthermore the quality of fit is not limited to the selected correlation window. Nonlinearities in time series representing physical proce sses are common in many areas (speech processing (Rabiner and Juang, 1993), medicine, industry and fi nance (Keogh, 2002)). More recently, these concepts have been improved by Keogh and Kasetty (2003) and K eogh and Ratanamahatana (2004). DTW is a robust tool to match time series even if they are out of phase or time shifted (Keogh, 2002). Related works using DTW in seismic applications come with th e attempts to automate well-to-well log correlation (Lineman et al., 1987; Steven et al., 2004). Wel l logs from different wells are correlated to infer common earth features. Steven et al. (2004) found that the cross-correlation was unable to follow local distortions such as stretching or shrinking of strati gr phic intervals, typical of logs collected even from closely spaced wells. Anderson and Gaby (1983) seek for correspondence between features in the logs of various wells using dynamic programming tools. To ou r knowledge, there are no previous reports of dynamic programming applied to the seismic-to-well tie p roblem. Method and Theory The correlation coefficient is commonly used to measure the q uality of the seismic-to-well tie (Hampson-Russell, 1999). Comparing two (time-dependent) sequences S := [s1,s2, . ,sn] andT := [t1, t2, . , tn], both of lengthn, will give a correlation coefficient at the time lag τ : γST (τ) = ∑i=1[S(i)−μS][T (i+ τ)−μT ] (∑i=1[S(i)−μS] ∑i=1[T (i)−μT ]2)1/2 , (1) whereμx is the average of trace x. This measure works well if a constant time shift τ characterizes both signal. But the majority of geophysical applications have time alignment problems (Anderson a nd Gaby, 1983). When this time alignment is constant, the problem is reduced to the correction of the t im lag by cross correlation. But this measure fails to find the best matching in nonlinear cases. An alternative to the cross correlation is to find the Euclide an distance ( L2−norm) between the two time series (Keogh and Kasetty, 2003):