Wave migration and the matched-filter Diffraction summation

components become zero resulting in the iden- tity matrix. For a monostatic radar configura- tion, r Y sr 4 Ž . and a sa . 5 Ž . ˆ ˆ i r Ž . Through reciprocity, Eq. 3 is simplified to H r X ,r, v s G 2 r X , r , v , 6 Ž . Ž . Ž . Ý a r a where the subscript r corresponds to the direc- tion of the receiving antenna and a corresponds to three orthogonal spatial directions. Finally, through the use of the exploding reflector model Ž . Loewenthal et al., 1976 , the squaring opera- Ž . tion in Eq. 6 is expressed as H r X ,r, v s G 2r X ,2r, v . 7 Ž . Ž . Ž . Ý a r a Converting back to the time-domain, the general form of the image is expressed as X X S r s g 2r ,2r,t mu r,y t d r . Ž . Ž . Ž . Ý H ts0 a r a 8 Ž . At this point, we disregard the mathematics and interpret the physical processes implied by this equation. The result is a forward scattering problem. First, the received signals become time-reversed sources. Next, these sources are placed at the receiver locations in the exact environment of the survey with all dimensions increased by a factor of two. The sources gener- ate waves and these waves are propagated until time equals zero. At this time, the spatial distri- bution of the scalar fields or the sum of the vector field components represents the image. This process can be accomplished using any forward scattering technique as will be shown later in this paper, but for now some properties Ž X . of the filter, h r ,r,t , are addressed with their implications on the image. Ž X . Ž X First, h r ,r,t is not orthogonal with h r , r, 1 2 . X X t when r is not equal to r . This can be 1 2 expressed mathematically as h r X ,r,t h r X , r ,t d rd t 0 9 Ž . Ž . Ž . HH 1 2 and implies that a non-zero output will occur at r X when a scatterer exists at r X . The result is 2 1 range sidelobes, causing ambiguities of the ob- ject’s actual location. However, we can assume that this will only occur when r X is in close 2 X X X proximity to r and as the distance r yr 1 1 2 increases the output at r X will become insignifi- 2 cant. An example of this phenomenon is illus- trated by the finite bandwidth of the transmitted signal. Reflections in the x–t domain due to a scatterer at r X do not exist as a single event but 1 are dispersed over time and tend to ring as they Ž X . decay. Since h r ,r,t is not orthogonal to 1 Ž X . X h r ,r,t , values r can be chosen such that a 2 2 significant energy will occur at the output of Ž . Eq. 9 , and this ringing will be translated into the x–z domain. Ž X . Second, the inner product of h r ,r,t is not normal or even constant for all values of r X . This is expressed as h r X ,r,t h r X ,r,t d rd t sf r X constant Ž . Ž . Ž . HH 10 Ž . Ž X . and implies that an amplitude variation, f r , will exist over the image. Usually, this function is laterally independent and decreases with Ž X . depth, and if f r can be approximated, its effects can be eliminated by applying it as an inverse gain to the image.

3. Wave migration and the matched-filter

As stated earlier, migration, in a seismic or ground-penetrating radar sense, can be inter- preted as a redistribution of energy that was initially dispersed along a receiving array during the collection process to the actual or initial locations within the ground. In seismic tech- niques, migration is used to approximate the reflectivity at a point in space by estimating what the response would be given that both the transmitter and receiver were collocated at this point during the survey. Using this model, the reflectivity would be the received waveform at time equal to zero. Physically, this task is im- possible without digging into the ground, and, therefore, the fields that were collected at the surface are extrapolated back into the ground. These extrapolation processes that have been based on many ideas can usually be reduced to Ž . a forward scattering problem similar to Eq. 8 with the main differences lying in the physical interpretation of wave propagation. In the next sections, four different solutions of the wave equation are applied to the matched-filter image Ž . shown by Eq. 8 and resulting equations are compared to current migration techniques.

4. Diffraction summation

The simplest form of wave migration is diffraction summation or wavefront interference Ž migration Schneider, 1978; Gazdag and . Sguazzero, 1984 . This method uses a geometric interpretation of wave propagation rather than a solution of the wave equation. In this method, waves are assumed to propagate spherically with no dispersion, distortion, or attenuation. The function describing this process is expressed as X 2 r yr X g 2r,2r ,t sd ty . 11 Ž . Ž . ž Õ Ž . Substituting the delta function into Eq. 8 and integrating over time is equivalent to sampling the scattered-field response at times correspond- ing to the two-way travel distance divided by the velocity of the medium, Õ. X 2 r yr X S r s u r,ts d r 12 Ž . Ž . H ž Õ Ž . The output in Eq. 12 is equivalent to the conventional process of summing over a hyper- bola described by the two-way travel times from the receiver array to a pixel in space.

5. f–k migration