Decon in the log domain with variable gain

THE GRADIENT

Having data $d_t$ , having chosen gain $g_t$ , and having a starting log filter, say $u_t=0$ , let us see how to update $u_t$ to find a gained output $q_t = g_t r_t$ with better hyperbolicity. Our forward modeling operation with model parameters $u_t$ acting upon data $d_t$ (in the Fourier domain $D(Z)$ where $Z=e^{i\omega})$ produces deconvolved data $r_t$ (the residual).

$$r_t = {\rm FT}^{-1} \ D(Z)\ e^{\cdots + u_2 Z^2 + u_3 Z^3 + u_4 Z^4 +\cdots}$$ (9)

$$\frac{dr_t}{du_\tau} = {\rm FT}^{-1} \ D(Z)\ Z^\tau e^{\cdots + u_2 Z^2 + u_3 Z^3 + u_4 Z^4 +\cdots}$$ (10)

$$\frac{dr_t}{du_\tau} = r_{t+\tau}$$ (11)

This follows because $Z^\tau$ shifts the data $D(Z)$ by $\tau$ units which shifts the residual the same amount. Output formerly at time $t$ moves to time $t+\tau$ . This is not the familiar result that the derivative of an output with respect to a filter coefficient at lag $\tau$ is the shifted input $d_{t+\tau}$ . Here we have the output $r_{t+\tau}$ . This difference leads to remarkable consequences below.

It is the gained residual $q_t = g_t r_t$ that we are trying to sparsify. So we need its derivative by the model parameters $u_\tau$ .

$$q_t = g_t\ r_t \ =\ r_t\ g_t$$ (12)

$$\frac{dq_t}{du_\tau} = \frac{dr_t}{du_{\tau}}\ g_t \ =\ r_{t+\tau}\ {g_t}$$ (13)

Recall $u_0=0$ and hence $\Delta u_0=0$ . To find the update direction at nonzero lags $\Delta \mathbf{ u} = (\Delta u_t)$ take the derivative of the hyperbolic penalty function $\sum_t H(q_t)$ by $u_\tau$ .

$$\Delta\mathbf{ u} = \sum_t \frac{dH(q_t)}{du_\tau} \quad \quad \quad \quad \tau\ne 0$$ (14)

$$= \sum_t \frac{dq_t}{du_\tau} \ \frac{dH(q_t)}{dq_t}$$ (15)

$$\Delta\mathbf{ u} = \sum_t \ (r_{t+\tau}) \ \ \left( g_t H'(q_t) \right) \quad \quad \tau\ne 0$$ (16)

This says to crosscorrelate the physical residual $r_t$ with the statistical residual $g_t H'(q_t)$ . Notice in reflection seismology the physical residual $r_t$ generally decreases with time while the gain $g_t$ generally increases to keep the statistical variable $q_t$ roughly constant, so $g_t H'(q_t)$ grows in time(!)

In the frequency domain the crosscorrelation (16) is:

$$\Delta U = \ \overline{{\rm FT}(r_t)} \ \ {{\rm FT}(g_t \, {\rm softclip}(q_t))}$$ (17)

Equation (17) is wrong at $t=0$ . It should be brought into the time domain and have $\Delta u_0$ set to zero. More simply, the mean can be removed in the Fourier domain.

Causal least squares theory in a stationary world says the signal output $r_t$ is white (Claerbout, 2009); the autocorrelation of the signal output is a delta function. Noncausal sparseness theory (other penalty functions) in a world of echoes (nonstationary gain) says the crosscorrelation of the signal output with its gained softclip is also a delta function (equation (16), upon convergence).

Decon in the log domain with variable gain