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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).

$\displaystyle r_t$ $\displaystyle =$ $\displaystyle {\rm FT}^{-1} \ D(Z)\ e^{\cdots + u_2 Z^2 + u_3 Z^3 + u_4 Z^4 +\cdots}$ (9)
$\displaystyle \frac{dr_t}{du_\tau}$ $\displaystyle =$ $\displaystyle {\rm FT}^{-1} \ D(Z)\ Z^\tau e^{\cdots + u_2 Z^2 + u_3 Z^3 + u_4 Z^4 +\cdots}$ (10)
$\displaystyle \frac{dr_t}{du_\tau}$ $\displaystyle =$ $\displaystyle r_{t+\tau}$ (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$ .

$\displaystyle q_t$ $\displaystyle =$ $\displaystyle g_t\ r_t \ =\ r_t\ g_t$ (12)
$\displaystyle \frac{dq_t}{du_\tau}$ $\displaystyle =$ $\displaystyle \frac{dr_t}{du_{\tau}}\ g_t
\ =\
r_{t+\tau}\ {g_t}$ (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$ .
$\displaystyle \Delta\mathbf{ u}$ $\displaystyle =$ $\displaystyle \sum_t
\quad \quad \quad \quad \tau\ne 0$ (14)
  $\displaystyle =$ $\displaystyle \sum_t
\frac{dH(q_t)}{dq_t}$ (15)
$\displaystyle \Delta\mathbf{ u}$ $\displaystyle =$ $\displaystyle \sum_t \
\ \
\left( g_t H'(q_t) \right)
\quad \quad \tau\ne 0$ (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:

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

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Next: TAKING THE STEP Up: Claerbout et al.: Log Previous: MINIMUM PHASE EXTENSION