• Source is nearly impulsive
• Received signal contains multiple reflections
• Arrival times of reflected signals are of interest
• Data is "stacked" so that the received signal is "as if" source and receiver are at the same point

• We can compensate for this effect if the acoustic velocity is known
• Determining the acoustic velocity is itself a difficult problem

• Measurements made at surface

• A reflection from a point sets off an expanding spherical wavefront
• This propagates according to the acoustic wave equation
• The actual recorded signal is complicated

• Propagation governed by the 2-D hyperbolic wave equation

• The wave field becomes less complicated and easier to interpret, if we could look at it at a greater depth. This is equivalent to looking at it in an earlier time
• s(x,0,t) can be measured at the surface—serves as a boundary condition for the partial differential equation
• We want s(x,z_o,t), which is the wave profile at depth z_o
• Approach is to use the wave equation to propagate s(x,z,t) backwards from depth z from 0 to z_o, which is called migration. We want to extrapolate from s(x, z₀, t) to s(x, z₀ + Dz, t).

• Define the 2-D Fourier transform of the wave field with respect to x and t at depth z

• Taking the Fourier transform of both sides of the wave equation

We have turned a partial differential equation in the space-time domain into an ordinary differential equation in the wavenumber-frequency domain.

Case 1:

• These solutions do not correspond to propagating waves

• They can be eliminated on physical grounds

• The positive exponent corresponds to an upwardly propagating wave.
• The negative exponent corresponds to a downwardly propagating wave. Set B=0.

• The necessary extrapolation can be performed using a linear, shift-invariant filter
• The desired frequency response is all-pass with a hyperbolic phase response.

• For sampled measurements s(n_1D x, lD z, n_2D t), the extrapolation can be performed by a digital filter. Let

)

• For |a w ₂| > |w ₁|, transfer function has unit magnitude and phase

• In the evanescent region, |a w ₂| < |w ₁|, waves do not propagate but are attenuated.

Filter design problem. We design either

• an all-pass filter with the proper phase characteristic

• a fan filter where the Propagating Region is the passband and the Evanescent Region is the stopband (pages 274-275 of the first edition of Dudgeon & Mersereau).

evanescent- tending to vanish or pass away like vapor