Eliminating Marine Acquisition Crosstalk

Once sources have been assigned non-overlapping sets of frequencies, marine acquisition crosstalk can be eliminated. The key idea is, after completing multisource forward modeling by computer simulation, at each receiver $h$ any extraneous frequency component $j$ is pruned; $j$ is considered extraneous if $j$ is assigned to a source, to which, according to the marine geometry, $h$ is not associated. I illustrate the proposed algorithm with an example shown in Figure 2.1. Figure 2.1(a) depicts $S$ =3 sources, $n_\omega$ =5 frequencies, and a specific way of frequency selection described by the frequency encoders $N_s(j)$ 's identified as, from left to right, $N_3=[1,0,0,0,1]^T$ , $N_2=[0,1,0,0,0]^T$ , and $N_1=[0,0,1,1,0]^T$ . Figure 2.1(b) depicts a towed marine geometry, where each source is associated with $\gls{nh}=5$ receivers. For instance, source $s_3$ is associated with receivers $h_5$ -$h_8$ , but not with receivers $h_1$ -$h_4$ . Consider for example $j$ =5 at receiver $h_4$ . Because, according to Figure 2.1(a), $j$ =5 is assigned to source $s_3$ , $j$ =5 is considered extraneous at receivers $h_4$ and should be pruned. The rationale is as follows. When sources are blended, all frequency components are present (see equation 2.26) in forward modeling and consequently at every receiver. Receiver $h_4$ would have detected frequency component 5, which comes from source $s_3$ , but $h_4$ lies outside the aperture associated to $s_3$ , and therefore $h_4$ should not pick up any signal stemming from $s_3$ . This explains the pruning of the extraneous frequency component 5 at $h_4$ . This is indicated by the absence of a bar corresponding to $j$ =5 at $h_4$ in Figure 2.1(c). Other unoccupied frequency slots in Figure 2.1(c) are likewise inferred.

Figure 2.1: Frequency division of sources for one supergather of towed-marine data. Sources and receivers are identified with their indices. (a) Unique spectra assigned to, and hence will be emitted by, the sources. The three spectra patterns are non-overlapping. (b) The association, signified with the same line width and fill style, between sources and their respective receiver groups. $f$ denotes near offset; $l$ denotes line length. (c) Frequencies listened to at each receiver.

The pruning operation is equivalent to selective filling in as follows. Let F_blen(frequency, receiver) of size $n_\omega \times n_{htot}$ be the outcome^2.1in the frequency domain detected by receivers generated by forward modeling with blended sources prior to pruning, and let F_prun(frequency, receiver) of the same size be the result of pruning applied to F_blen. Here, $n_{htot}$ is the total number of receivers covered by the supergather, and $n_{htot}=8$ in this example. F_prun is obtained by first initialization with 0 and subsequently filling in with valid entries in F_blen; an entry F_blen($j$ , $h$ ) is valid if frequency component $j$ is not extraneous at receiver $h$ . For instance for $j=5$ , we have

$$\ce{F_{prun}}(j=5,[h_5,h_6,h_7,h_8]) \gets \ce{F_{blen}}(j=5,[h_5,h_6,h_7,h_8])$$ (2.28)

Similarly, the encoded supergather ${\textrm{CSG}_\textrm{enc}}$ , of size $n_\omega \times n_{htot}$ , can be formed as follows, assuming that the observed csg have been transformed to the frequency domain and are indexed as CSG(frequency, receiver, source), of size

$$M_{CSG} = n_\omega \times \gls{nh} \times S.$$ (2.29)

Here, nh is the number of receivers associated with each source in acquisition, and $\gls{nh}=4$ in this example. First, ${\textrm{CSG}_\textrm{enc}}\gets 0$ . Next, fill in ${\textrm{CSG}_\textrm{enc}}$ with the corresponding entries in CSG according to the current frequency encoders. Specifically, loop j over nomega, and for a given $j$ , find to which source $s$ it belongs, and subsequently find which receivers $h$ 's are associated to this $s$ . Then execute ${\textrm{CSG}_\textrm{enc}}(j$ , $h$ 's) $\gets$ CSG($j$ , :, $s$ ). An example for $j$ =3 is given as

$${\textrm{CSG}_\textrm{enc}}(j=3,[h_1,h_2,h_3,h_4]) \gets \textrm{CSG}(j=3, ~:, ~s_1).$$ (2.30)

Finally, the misfit function is computed by $\ce{F_{prun}}-{\textrm{CSG}_\textrm{enc}}$ . By pruning or equivalently selective filling in, the mismatch problem between the limited number, $n_h$ , of live traces/shot in observed CSG, and the pervasive number, $n_{htot}$ , of traces in simulation-generated F_blen is now resolved.

Note that since there are $n_\omega$ equations similar to equation 2.30, each reading $n_h$ entries, the total number of entries read from CSG by selective filling in is

$$M_{CSG_{enc}} = n_\omega \times n_h,$$ (2.31)

$$= \frac{M_{CSG}}{S}.$$ (2.32)

In this example $M_{CSG_{enc}} = 5\times 4 = 20$ , coinciding with the number of bars in Figure 2.1(c).