THE OPTIMIZATION PROBLEM

I have set up a least-squares optimization problem in which I minimize the difference between data and its reciprocal counterpart. Analyzing the solution of that problem shows that none of the eigenvectors has a pure DC component. Thus the absolute radiation pattern cannot be determined. However, relative changes are perfectly resolvable within a cable length. This relativity requires us to specify a reference with respect to which the prediction error filters are computed. There are a multitude of choices. One possibility would be to average the sources over some range and establish the average as a reference source. The disadvantage of this method is that a relatively noisy (bad) common source gather biases the estimation procedure unfavourably. Another choice would be to design or give some training data as reference, but that would have the effect of wavelet shaping. Instead, I took the first source as a reference source, all other filters show differences with respect to that source. My initial guess is a single spike at the zero lag filter location. In the minimization process the filters are not constrained at all (except for being specified as to the number of filter coefficients). Once I decided on the filter shape (1-D, short), I tested the following objective functions governing the minimization process.

First, we find the global minimum of the squared difference for all the components at the same time. This requires simultaneous accessibility of all 9 component data. For each iteration that data must be read once.  

$$E_1 = \sum_{i,j,k,l}\{ d_{ji}^{lk} * s_j^{lk} - d_{ij}^{kl} * s_i^{kl} \}^2 .$$ (5)

E₁ represents the complete misfit of filtered data samples; i stands for the source and j are reciprocal source coordinates. The indices i and j trace all different source and reciprocal source locations. The indices l and k (source and reciprocal source components) both stay within the range 1,2,3. The values of i and j may describe source location in 2-D or 3-D. Convolution is denoted by *. For each source location, the general shape of s^kl in E₁ is

$$\pmatrix{ s_{11}(t) & s_{12}(t) & s_{13}(t) \cr s_{21}(t) &... ...}(t) & s_{23}(t) \cr s_{31}(t) & s_{32}(t) & s_{33}(t) \cr } .$$

For pressure data, the index range for l and k degenerates to just being unity. For vector wave field data these indices indicate the type of component (vertical, horizontal, inline, crossline). To each ensemble a filter pair s_i^kl and s_j^lk is applied. In equation (5) s is estimated, for each source component pair, as a short function in time.

Instead of equalizing all components simultaneously, we can treat each reciprocal component pair separately. The advantage is smaller data sets to be worked on. Equation (6) gives the appropriate objective function. The filter s^kl is then a single scalar value for each separate minimization process. E₂^kl represents the objective function to be minimized.  

$$E_2^{kl} = \sum_{i,j} \{ d_{ji}^{lk} * s_j^{lk} - d_{ij}^{kl} * s_i^{kl} \}^2$$ (6)

A third minimization alternative, however, is to find a global minimum for a filter that consists of a scalar array multiplied by a source location filter, as follows  

$$E_3 = \sum_{i,j,k,l} \{ d_{ji}^{lk} * (s_j^{lk} f_j^l) - d_{ij}^{kl} *(s_i^{kl} f_i^k) \}^2$$ (7)

$$s_i^{lk} f_i^l = \pmatrix{ f_{1}(t) & 0 & 0 \cr 0 & f_{2}(t... ... s_{21} & s_{22} & s_{23} \cr s_{31} & s_{32} & s_{33} \cr }$$

This filter decomposition honors the fact that for a single experiment with a given source orientation, the vector source function is separable and consistent in time for all source components. Such a filter compensates for mispositioning of the source vector by a rotation and for differences in the time history.