INTEGRAL EXPRESSION OF THE OPERATOR ${\bf H^TH}$

The aim of this section is to derive the expression for the operator $\bold H^{\bold T}\bold H$.We know that

$$(\bold u_1,\bold H \bold d_2)=(\bold H^{\bold T}\bold u_1,\b... ...bold u_1(\sqrt{t^2-x^2m},m){t \over \sqrt{t^2-x^2m}}dm)dt\, dx$$ (36)

for any $\bold u_1$ from $(\tau,m)$-space and $\bold d_2$ from (t,x)-space. If we define

$$\bold u_1(\tau,m)=\int \bold d_1(\sqrt{\tau^2+y^2m},y)dy,$$ (37)

then

$$({\bf H^THd}_1,\bold d_2)=\int\!\! \int \bold d_2(t,x)(\int\... ...old d (\sqrt{t^2-(x^2-y^2)m},y){t \over \sqrt{t^2-x^2m}}dy) dm.$$ (38)

From this equation it follows  

$${\bf H^THd}(t,x)=\int\!\! \int \bold d(\sqrt{t^2-(x^2-y^2)m},y){t \over \sqrt{t^2-x^2m}} dy\, dm,$$ (39)

which is the expression for the operator ${\bf H^TH}$.

Equation $T=\sqrt{t^2-(x^2-y^2)m}$ represents a hyperbola in (T,y)-space passing through the point (t,x). With changing m we obtain a set of hyperbolas passing through the point (t,x) (Fig ). A weighting factor is assigned to each hyperbola. The higher the slope of a hyperbola is, the higher weighting factor is assigned. The value of the operator ${\bf H^TH}$ at the point (t,x) is equal to the weighted sum of all values lying on the hyperbolas passing through the point (t,x).

This result may also be explained in the following way. The sums along hyperbolas passing through the point (t,x) form a parabola in $(\tau,m)$-space (Jedlicka, 1989). ${\bf H^TH}(t,x)$ is obtained by weighted summation along this parabola. Each point of the parabola represents a sum along a corresponding hyperbola, from which the previous result follows immediately.

If d(t,x)=1 for all t and x then from equation (39) it follows

$${\bf H^THd}(t,x)=C\int {t \over \sqrt{t^2-x^2m}} dm.$$ (40)

This means that count(t,x) for velocity analysis is equal to the sum of NMO counts over the range of velocities used in velocity analysis. An expression for a count for NMO is derived in Appendix C.