Asymptotic pseudounitary stacking operators

Datuming

Let $x$ denote a point on the surface at which the propagating wavefield is recorded. Let $y$ denote a point on another surface, to which the wavefield is propagating. Then the summation path of the stacking operator for the forward wavefield continuation is

$$\theta(x;t,y) = t - T(x,y)\;,$$ (32)

where $t$ is the time recorded at the $y$ -surface, and $T(x,y)$ is the traveltime along the ray connecting $x$ and $y$ . The backward propagation reverses the sign in (32), as follows:

$$\widehat{\theta}(y;z,x) = z + T(x,y)\;.$$ (33)

Substituting the summation path formulas (32) and (33) into the general weighting function formulas (28) and (29), we immediately obtain

$$w^{(+)} = w^{(-)} = {1\over{\left(2 \pi\right)^{m/2}}} \left\vert{{\partial^2 T}\over{\partial x \partial y}}\right\vert^{1/2}\;.$$ (34)

Gritsenko's formula (Goldin, 1986; Gritsenko, 1984) states that the second mixed traveltime derivative ${{\partial^2 T}\over{\partial x \partial y}}$ is connected with the geometric spreading $R$ along the $x$ -$y$ ray by the equality

$$R(x,y) = {\sqrt{\cos{\alpha(x)} \cos{\alpha(y)}}\over v(x)} \left\vert{{\partial^2 T}\over{\partial x \partial y}}\right\vert^{-1/2}\;,$$ (35)

where $v(x)$ is the velocity at the point $x$ , and $\alpha(x)$ and $\alpha(y)$ are the angles formed by the ray with the $x$ and $y$ surfaces, respectively. In a constant-velocity medium,

$$R(x,y) = v^{m-1} T(x,y)^{m/2}\;.$$ (36)

Gritsenko's formula (35) allows us to rewrite equation (34) in the form (Goldin, 1988)

$$w^{(+)}(x;t,y) = {1\over{\left(2 \pi\right)^{m/2}}} {\sqrt{\cos{\alpha(x)} \cos{\alpha(y)}}\over {v(x) R(x,y)}}\;,$$ (37)

$$w^{(-)}(y;z,x) = {1\over{\left(2 \pi\right)^{m/2}}} {\sqrt{\cos{\alpha(x)} \cos{\alpha(y)}}\over {v(y) R(y,x)}}\;.$$ (38)

The weighting functions commonly used in Kirchhoff datuming (Wiggins, 1984; Berryhill, 1979; Goldin, 1985) are defined as

$$w(x;t,y) = {1\over{\left(2 \pi\right)^{m/2}}} {{\cos{\alpha(x)}}\over {v(x) R(x,y)}}\;,$$ (39)

$$\widehat{w}(y;z,x) = {1\over{\left(2 \pi\right)^{m/2}}} {{\cos{\alpha(y)}}\over {v(y) R(y,x)}}\;.$$ (40)

These two operators appear to be asymptotically inverse according to formula (10). They coincide with the asymptotic pseudo-unitary operators if the velocity $v$ is constant ($v(x)=v(y)$ ), and the two datum surfaces are parallel ( $\alpha(x) = \alpha(y)$ ).

Asymptotic pseudounitary stacking operators