NULL SPACE AND INTERVAL VELOCITY

A basic problem in seismology is building the velocity as a function of depth (or vertical travel time) starting from certain measurements. The measurements are described elsewhere (BEI for example). They amount to measuring the integral of the velocity squared from the surface down to the reflector. It is known as the RMS (root-mean-square) velocity. Although good quality echos may arrive often, they rarely arrive continuously for all depths. Good information is interspersed unpredictably with poor information. Luckily we can also estimate the data quality by the ``coherency'' or the ``stack energy''. In summary, what we get from observations and preprocessing are two functions of travel-time depth, (1) the integrated (from the surface) squared velocity, and (2) a measure of the quality of the integrated velocity measurement. Some definitions:

$\bold d$

is a data vector whose components range over the vertical travel time depth $\tau$,and whose component values contain the scaled RMS velocity squared $\tau v_{\rm RMS}^2/\Delta \tau$where $\tau /\Delta \tau$ is the index on the time axis.

$\bold W$

is a diagonal matrix along which we lay the given measure of data quality. We will use it as a weighting function.

$\bold C$

is the matrix of causal integration, a lower triangular matrix of ones.

$\bold D$

is the matrix of causal differentiation, namely, $\bold D=\bold C^{-1}$.

$\bold u$

is a vector whose components range over the vertical traveltime depth $\tau$,and whose component values contain the interval velocity squared $v_{\rm interval}^2$.

From these definitions, under the assumption of a stratified earth with horizontal reflectors (and no multiple reflections) the theoretical (squared) interval velocities enable us to define the theoretical (squared) RMS velocities by

$$\bold C\bold u \quad =\quad\bold d$$ (10)

With imperfect data, our data fitting goal is to minimize the residual

$$\bold 0 \quad\approx\quad \bold W \left[ \bold C\bold u - \bold d \right]$$ (11)

To find the interval velocity where there is no data (where the stack power theoretically vanishes) we have the ``model damping'' goal to minimize the wiggliness $\bold p$of the squared interval velocity $\bold u$.

$$\bold 0 \quad\approx\quad \bold D \bold u \quad =\quad\bold p$$ (12)

We ``precondition'' these two goals by changing the optimization variable from interval velocity squared $\bold u$ to its wiggliness $\bold p$.Substituting $\bold u=\bold C\bold p$ gives the two goals expressed as a function of wiggliness $\bold p$.

$$\begin{eqnarray} \bold 0 &\approx& \bold W \left[ \bold C^2\bold p - \bold d \right] \\ \bold 0 &\approx& \epsilon \; \bold p\end{eqnarray}$$ (13) (14)