Many seismic data processing applications use Fourier Transform over time but not in space. This is often a natural choice because the time axis is always uniformly and adequately sampled, the seismic velocity does not change with time so some operators become time invariant and can be applied economically in the frequency domain (sometimes with the aid of tricks like the log-stretch transform). Fourier transform over space is more problematic because of the irregular space sampling and because velocity changes with space.

Therefore, models are often built, and wave-fields are propagated in the Frequency-space (F-X-Y) domain. Examples are Log Stretch DMO and Inversion to Zero Offset (IZO), Spatial Prediction for signal/noise separation (FXY-decon), Spatial Prediction for interpolation (FXY-Interp), and migration (FXY finite differencing and Kirchoff).

Seismic data are acquired in irregular spatial sampling, Data(shot location, receiver location, time). The spatial sampling is regularized in the processing. For example,

- The zero-offset model (so called `stack') is regularly sampled in X, Y, and time, ZeroOffsetModel(X,Y,Time). (So are migrated data in X, Y, and Z or time. It is only later when the migrated data are interpreted that an irregularly sampled earth model is built.)
- Prestack interpolated data may be in ShotGather(Receiver Y, Receiver Y, Time) or Receiver Gather(Shot Y, Shot Y, Time) or MidPointGather(Offset, Azimuth, Time)

The regularly sampled models have grids which define their X,Y sampling. A wave-field in the (X, Y, Frequency) domain requires a small spatial sampling interval for high temporal frequencies while a large sampling interval will suffice for low frequencies. It is well known that the required minimal spatial sampling interval is inversely proportional to the frequency.

Using the same sampling interval for low and high frequencies is over parameterization which obviously means a waste of storage and processing resources. It is less obvious, but more important, that frequency dependent grids also have quality advantages. Namely

- Unless carefully constrained, an over-parameterized model may contain illegal components that will fit (noisy) data better than a better model which does not include high wave-numbers for low frequencies.
- Spatial prediction filters become the same for all frequencies. This is an important observation (that Dave Nichols made soon after I talked about the first reason to use frequency dependent grids). It means that Spitz's method of using the same prediction filter for double the frequency - half the sampling interval, or triple the frequency - third the sampling interval, are special cases in a continuum of constant spatial sampling interval times temporal frequency.

The promise of frequency dependent grids for spatial interpolation and signal/noise separation are therefore immense; both quality and economy.

The immediate challenge is that data in frequency dependent grids, cannot be described as header-ed seismic traces, nor in cubes. Rather it is like a pyramid in F-X-Y or a code in F-Radius-Azimuth.

The more serious challenge is that F-X-Y operators applied on sparse grids and the complex interpolation involved in going back to the time domain when everything is done are touchy.

I will show some figures I made in Gatwick, which show artifacts which are probably due to suboptimal operators and complex interpolation.

Artifact free time slice with frequency independent grids

Time slice with frequency dependent grids

Another time slice with frequency dependent grids

Another time slice with frequency dependent grids

- Load regular data (start with synthetic data with coherent events including flat and dipping reflectors and point diffractors and added random noise).
- Fourier transform over time
- Interpolate (subsample) the cube into a pyramid
- Solve for a single prediction filter for all frequencies.
- Apply the prediction filter to separate noise and signal
- Interpolate (oversample) the frequency dependent grid signal and noise pyramids to frequency independent grid signal and noise cubes.
- Inverse Fourier transform

- Load irregularly sampled data
- Fourier transform over time
- Solve an inversion in which the unknown is the signal pyramid:

Data(F,Xi,Yi) = Interpolate { Model(F,X,Y) }

With good and/or bad pass constrained as described in my last week msgs,

// Constrained Inversion ///////////////////////////////////////////////////

//

// 1. d = F m // Data d fits model m via forward modeling F

// 2. 0 = B m // Bad pass B on model m gives 0

// 3. m = G x // Model m is good part of unknown x

//

// (d) = (F)

// (-) = (-) G x // 1, 2, & 3 together

// (0) = (B)

//

//

// (d) = (FG)

// (-) = (--) x // Equivalent to above

// (0) = (BG)

//

////////////////////////////////////////////////////////////////////////////

F is (bilinear) interpolation

G is prediction and B is unprediction - Interpolate (oversample) the frequency dependent grid model pyramid to a frequency independent grid model cube.
- Inverse Fourier transform