Single-channel approach

If we have an estimate S₀ of the interfering signal the following subtraction can be applied to the traces (Jenkins & Watts,1968):

 

$$X_{ir}=X_i-{{\overline{X_iS_0^{\ast}}}\over{\overline{S_0S_0^{\ast}}}}\approx U_i,$$ (2)

where X_ir is the residual, $\overline{X_iS_0^{\ast}}$ is the smoothed cross-spectrum between the i-th trace and the estimate, and $\overline{S_0S_0^{\ast}}$ is the smoothed power spectrum of the estimate.

The problem with (2) is that often it is difficult to obtain a good estimate S₀. Now assume S_i>>U_i and substitute into (2) instead of S₀ X_k-just one trace. Remembering that X_k=U_k+S_k we get:  

$$X_{ir}\approx S_i-{{\overline{S_iS_k^{\ast}}}\over{\overline... ...ver{\overline{S_kS_k^{\ast}}}}U_k\approx U_i-{S_i\over S_k}U_k,$$ (3)

and, rewriting the result:  

$$X_{ir}\approx U_i-S_i{U_k\over S_k}.$$ (4)

Here U_k/S_k is a complex constant. The contribution of the interfering source to each trace has been reduced by the absolute value of this constant (which is <<1 under our assumption), but the phase shift between channels is preserved. To summarize, the magnitude of the interfering source was reduced significantly without adding major artifacts, and location of the weaker source is now more feasible.