RESULTS

The optimization problems described by E₁ and E₂ were solved using a conjugate gradient method. I chose to constrain the first filter in each of the series of filters to unity. Thus I chose the first shot gather to be my reference gather. Figure and show results obtained during testing the nonstationarity of the recorded data on a few shot gathers. I tried different window sizes and shapes, but the answers, after about 40 iterations, were always very similar to and . This behavior encouraged me to apply the equalization process about 50 shotgathers in a two second time window, knowing that this time window averages over a fair amount of energy that is radiated from the source with different emergence angles. I chose to use a medium length filter of about 40 points (160 msec). Results of those equalizations are shown in two groups. First a minimization described by E₂ for the components Xx, Yy and Zz (capital letters denote source components, lower-case letters denote receiver components). The prediction error filters obtained are shown in Figures , and . The filter obtained after about 40 iterations is minimum phase. There seem to be hardly any time shifts between different source points. The amplitude in the main peak and lower energy wavelet characterize all the filters consistently. Figure shows a comparison between the raw prestack time slice and the filtered version. The differences in that time slice are small but noticeable. The continuity in character of the time slice is increased. Figures to are obtained from maximizing symmetry in the prestack data using the objective function E₁, where all components are equalized simultaneously. The offdiagonal elements in the three-by-three filter set show identical behavior. Again the filter are consistent along the line and exhibit the similar properties as the filters in Figures to . The short 1-D filters equalize by averaging over an angular distribution of radiation pattern. The amount of averaging is determined by the time window and number of different arrivals within the time window.

In the future I plan to investigate how strong the influence of such an equalization is on the results produced by a simple stack, velocity analysis and migration, thus determining when it is necessary to estimate the radiation pattern in an absolute manner instead of, as shown here, a purely relative manner.

 

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Figure 7 Prediction error filters at 10 locations estimated from 0-3 seconds in one-second-long windows of the data.

 

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Figure 8 Prediction error filters at 10 locations estimated from 3-7 seconds in one-second-long windows of the data.

 

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Figure 9 Prediction error filters at 50 locations estimated for X source component and x receiver component in a two-second-time window. Minimizes objective function E₂.

 

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Figure 10 Prediction error filters at 50 locations estimated for Y source component and y receiver component in a two-second-time window. Minimizes objective function E₂.

 

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Figure 11 Prediction error filters at 50 locations estimated for Z source component and z receiver component in a two-second-time window. Minimizes objective function E₂.

 

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Figure 12 Raw and equalized time slice of the part of the data for which reciprocal data exist for 50 source points (Xx-component). To go through time slices, press the button.

 

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Figure 13 Prediction error filters at 50 locations estimated for X source component and x receiver component in a two-second-time window. Minimizes objective function E₁.

 

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Figure 14 Prediction error filters at 50 locations estimated for X source component and y receiver component in a two-second-time window. Minimizes objective function E₁.

 

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Figure 15 Prediction error filters at 50 locations estimated for X source component and z receiver component in a two-second-time window. Minimizes objective function E₁.

 

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Figure 16 Prediction error filters at 50 locations estimated for Y source component and x receiver component in a two-second-time window. Minimizes objective function E₁.

 

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Figure 17 Prediction error filters at 50 locations estimated for Y source component and y receiver component in a two-second-time window. Minimizes objective function E₁.

 

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Figure 18 Prediction error filters at 50 locations estimated for Y source component and z receiver component in a two-second-time window. Minimizes objective function E₁.

 

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Figure 19 Prediction error filters at 50 locations estimated for Z source component and x receiver component in a two-second-time window. Minimizes objective function E₁.

 

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Figure 20 Prediction error filters at 50 locations estimated for Z source component and y receiver component in a two-second-time window. Minimizes objective function E₁.

 

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Figure 21 Prediction error filters at 50 locations estimated for Z source component and z receiver component in a two-second-time window. Minimizes objective function E₁.