wave field = waves up + waves down

One could also attempt to separate the data into two wave fields in a pre-processing step rather than the signal/noise framework described above. Seismologists face this problem when analyzing so called ``receiver-function'' data. Kennet (1991) explains using three-component receivers to remove the free surface interaction. We know that the displacements at the surface (u) are caused by a wavefield (w_o) that is the sum of both up-going ($\bf{w_u}$) and down-going ($\bf{w_d}$) component fields. Thus the $\bf{w_u}$ will be a combination of the direct and the reflected event, but $\bf{w_d}$ will contain only the ``source'' wave field that propagates down to excite subsurface reflectors.

We know that the wave field will give rise to displacements at any boundary. So we can construct a relation between the displacements and the wave field such as

$$\bf{u_o}=\bf{E}\; \bf{L} \; \bf{w_u}$$ (2)

where: $\bf{E}$ is the eigenvector matrix that relates the Fourier components of the wave fields to physical parameters; $\bf{L}$is composed of reflection and transmission matrices within the layer just below the surface; $\bf{u_o}$ is the column vector composed of the compressional and two shear propagation modes (each a function of frequency and ray parameter).

Thus, if we consider a very thin layer just below the surface the all energy is transmitted through, and no energy is reflected from, $\bf{L}$ becomes identity, and we only need a form for $\bf{E^{-1}}$to solve (estimate) the up-going wave field. Because $\bf{E}$ arises through solving the ODE's that relate displacement and tractions to a wave field, we can find expressions for it and evaluate them at the special case of the free surface.

The utility in this argument outlined above is in the ability to estimate the down-going wave field from the free surface that is the actual source wave form to convolve our traces with. Unfortunately, this process has analogous features to a rotation aligning recieiver axes with the angle and azimuth of the incoming energy source. Thus we may not be successful in generalizing this sufficiently when we hope to utilize truly random incident energies.

This treatment highlights our fortune in having three-component data available to us in our test data sets (outlined in Artman (2002)), as well as the need for three component acquisition in the future.