Blocky models

Sometimes we seek a velocity model that increases smoothly with depth through our scattered measurements of good-quality RMS velocities. Other times, we seek a blocky model. (Where seismic data is poor, a well log could tell us whether to choose smooth or blocky.) Here we see an estimation method that can choose the blocky alternative, or some combination of smooth and blocky.

Consider a five layer model, each layer having unit traveltime thickness (so integration is simply summation). Let the squared interval velocities be (a,b,c,d,e) with strong reliable reflections at the base of layer c and layer e, and weak, incoherent, ``bad'' reflections at bases of (a,b,d). Thus we measure V_c² the RMS velocity squared of the top three layers and V_d² that for all five layers. Since we have no reflection from at the base of the fourth layer, the velocity in the fourth layer is not measured but a matter for choice. In a smooth linear fit we would want d=(c+e)/2. In a blocky fit we would want d=e.

Our screen for good reflections looks like (0,0,1,0,1) and our screen for bad ones looks like the complement (1,1,0,1,0). We put these screens on the diagonals of diagonal matrices $\bold G$ and $\bold B$.Our fitting goals are:

$$\begin{eqnarray} 3V_c^2 &\approx& a+b+c \\ 5V_e^2 &\approx& a+b+c+d+e \\ u_0 &\a... ...& -a+b \\ 0 &\approx& -b+c \\ 0 &\approx& -c+d \\ 0 &\approx& -d+e\end{eqnarray}$$ (15) (16) (17) (18) (19) (20) (21)

For the blocky solution, we do not want the fitting goal (20). We can remove it by multiplying the model goals by a diagonal-matrix badpass screen $\bold B$ eliminating the goal $0 \approx -c+d$.In abstract, our fitting goals become

$$\begin{eqnarray} \bold 0 &\approx& \bold r \quad =\quad\bold C \bold u - \bold d... ... 0 &\approx& \bold p \quad =\quad\bold B\bold D\bold u - \bold u_0\end{eqnarray}$$ (22) (23)

where $\bold u_0$ is a zero vector with a top component of u₀. Since $\bold B$ is not invertable, we cannot backsolve the preconditioned variable $\bold p$ for the squared interval velocity $\bold u= \bold u_0 + \bold D^{-1}\bold B^{-1}\bold p$.Instead, we use $\bold G$ for $\bold B^{-1}$thus redefining the implicit relationship for $\bold u$. 

$$\bold u \quad =\quad\bold u_0 + \bold D^{-1}\bold G\bold p$$ (24)

where $\bold G$ is the goodpass screen. Since $\bold D^{-1}=\bold C$ the fitting goals become

 

$$\begin{array} {lll} \bold 0 &\approx& \bold r \quad =\quad\b... ...d C \bold u_0 -\bold d \\ \bold 0 &\approx& \bold p\end{array}$$ (25)

After fitting with $\bold p$,we define the squared interval-velocity $\bold u$ using (24).

The formulation (25) is so logical that we might have guessed it: The goal $\bold 0 \approx \bold p$ says that $\bold p$ is mostly zero. What emerges from $\bold G$ is a sprinkling of impulses. Then $\bold C^2$ converts the pulses to ramp functions (zero until a certain place, then growing linearly) which are used to fit the data (integrated velocity). Differentiating the data-fitting ramps converts them to the desired blocks of constant velocity. One iteration is required for each impulse.

Choosing $\bold G$ to be an identity $\bold I$ gives smooth velocity models, such as caused by the increasing consolidation of the rocks with depth. Choosing the screen $\bold G$ to have a sprinkling of pass locations picks the boundaries of blocks of constant velocity. The choice can be made by people with subjective criteria (like geologists) or we can assist by using the data itself in various ways to select our degree of preference between the blocky and smooth models. For example, we could put seismic coherency or amplitude on the goodpass diagonal matrix $\bold G$.Clearly, much remains to be gained from experience.