Solving the particular LSJIMP problem  

Now that we have derived appropriate imaging and amplitude correction operators, we are ready to translate the general LSJIMP modeling equation (2) to my particular implementation. The primary image, $\bold m_0$, is mapped into data space primary events by NMO, $\bold N_0$. Similarly, a given pegleg image, $\bold m_{i,k,m}$, is mapped into data space by sequentially applying the differential geometric spreading correction ($\bold G_{i,m}$), Snell resampling ($\bold S_{i,m}$), HEMNO ($\bold N_{i,k,m}$), and finally, a reflection coefficient ($\bold R_{i,k,m}$). Let us rewrite equation (2) accordingly:  

$$\bold d_{\rm mod} = \bold N_0 \bold m_0 + \sum_{i=1}^p \sum... ...m} \bold N_{i,k,m} \bold S_{i,m} \bold G_{i,m} \bold m_{i,k,m}.$$ (28)

We see that in equation (28), the analog to $\bold L_{i,k,m}$ in equation (2) is $\bold R_{i,k,m} \bold N_{i,k,m} \bold S_{i,m} \bold G_{i,m}$.

The data residual weight in equation (5), $\bf W_d$, can often strongly influence the success of the inversion. Technically, $\bf W_d$ carries a heavy burden: it must decorrelate and balance the residual. However, I have found that a simpler form for $\bf W_d$ nontheless pays dividends. I set $\bf W_d$, which has the same dimension as a CMP gather, to zero where the data, $\bf d$, has an empty trace, and also above the onset of the seabed reflection.