Multisource Crosstalk

Crosstalk in the communications industry is defined in the online Merriam-Webster dictionary as ``unwanted signals in a communication channel (as in a telephone, radio, or computer) caused by transference of energy from another circuit (as by leakage or coupling).'' As we sometimes discover with analog phones, crosstalk can take the form of hearing someone else's conversation $s(t)_2$ instead of hearing the intended signal $s(t)_1$ . Mathematically the noisy signal $s(t)$ can be represented by

$$s(t) = s(t)_1+s(t)_2,$$ (1.1)

where additive random noise can also be considered.

Sometimes, $s(t)_2$ is also considered signal, and the efficient transmission of both signals requires that they both be simultaneously transmitted, but should be separated from one another after recording^1.1. To achieve this feat, the two signals are encoded and summed together to give $s(t)'=F[s(t)_1]+F[s(t)_2]$ prior to transmission, and then decoded to get the separate signals $F_1^{-1}[s(t)'] =s(t)_1$ and $F_2^{-1}[s(t)'] =s(t)_2$ . This efficient means of simultaneously transmitting two signals and their subsequent decoding is a monumental achievement because it greatly reduces both the cost and the number of channels required for today's multi-channel communication.

The decoder $F_i^{-1}$ is a device or operation^1.2 that does the reverse of the $i^{th}$ channel encoder, undoing the encoding so that the original information can be retrieved. An example in digital electronics is where a decoder can take the form of a multiple-input, multiple-output logic circuit that converts coded inputs into coded outputs. Decoding is also necessary in applications such as data multiplexing, which is seen in the oil industry with transmission of multichannel seismic data.

A more geophysically relevant model of crosstalk is to include correlation terms that must be eliminated, i.e.,

$$s(t) = s(t)_1+s(t)_2 + \alpha s(t)_1 \star s(t)_2,$$ (1.2)

where $\alpha$ is a scalar weighting term, and $\star$ denotes correlation. For our purposes, one might think of two shot gathers summed together and simultaneously migrated to give the desirable sum of the individual migration images $s(t)_1$ and $s(t)_2$ , and the undesirable crosstalk noise $\alpha s(t)_1 \star s(t)_2$ . Similar to the benefits in the communications industry, Morton and Ober (1998) and Romero et al. (2000) tested the possibility of simultaneously migrating a sum of encoded shot gathers (herewith known as a supergather) to tremendously reduce computation time and memory+I/O demands. Unfortunately, their early results did not show significant efficiencies because their method did not easily annihilate the crosstalk term $s(t)_1 \star s(t)_2$ .

In fact, decoding is mathematically simpler in the frequency domain where equation 1.2 at $\omega$ is given by

$$S(\omega) = S(\omega)_1 + S(\omega)_2 + \alpha S(\omega)_1^* \,S(\omega)_2,$$ (1.3)

where $S(\omega)_i$ is the Fourier transform of $s(t)_i$ . This frequency domain representation presents the opportunity for decoding encoded signals by fdm, where interfering signals are transmitted in several non-overlapping frequency ranges (Bates and Gregory, 2007). The distinct frequency band of each signal means that the spectral product in the above equation is zero. It also means that $S(\omega)_1$ and $S(\omega)_2$ can be recovered by appropriate bandpass filters applied to $S(\omega)$ . One of FDM's most common applications is cable television, where different TV channels are FDM encoded and sent over the cable simultaneously. The decoding box at home then separates each channel from one another by decoding. There are many other multiplexing schemes such as time-division, statistical, wavelength-division, orthogonal frequency-division, code-division multiplexing (Bates and Gregory, 2007).