A least-squares formalism

Consider a point ${\bf x}$ in the subsurface. I define the reflectivity matrix ${\bf {\cal R}}({\bf x};{\bf x}_s,{\bf x}_r)$ at the point ${\bf x}$ by the following relation between the incident displacement vector field ${\bf u}^s({\bf x},t;{\bf x}_s)$ due to a source at ${\bf x}_s$, and the reflected displacement vector field ${\bf u}^r({\bf x},t;{\bf x}_r)$ due to a receiver at the position ${\bf x}_r$:

 

$${\bf u}^r= {\bf {\cal R}}{\bf u}^s\;,$$ (1)

where all quantities are evaluated locally at the point ${\bf x}$.The reflectivity matrix ${\bf {\cal R}}$ is of order (3x3) and, in general, is a scattering matrix which relates ${\bf u}^s$ to ${\bf u}^r$ and contains refractions, diffractions and reflections. However, by anticipating that ${\bf u}^s$ will be simulated as a direct primary incident field, and ${\bf u}^r$ will be reconstructed from refraction-muted surface observations as a single-conversion primary reflected field, the interpretation of ${\bf {\cal R}}$ as a generalized reflectivity matrix is justified.

Now consider the local plane-wave unit polarization vectors $\{{\bf e}_1,{\bf e}_2,{\bf e}_3\}$, where ${\bf e}_1({\bf x},t)$ is the local P-wave propagation direction, ${\bf e}_2({\bf x},t)$ the S₁ direction, and ${\bf e}_3({\bf x},t)$ the S₂ direction. The displacement vectors $\u$ can be expressed in this same ray-centered coordinate system without loss of generality:

$$\u = u_1{\bf e}_1 + u_2{\bf e}_2 + u_3{\bf e}_3 \;,$$

where u₁ is the P-wave displacement amplitude, u₂ is the S₁ displacement amplitude, and u₃ is the S₂ displacement amplitude. In this local ray-coordinate frame, the elements of ${\bf {\cal R}}$ are associated with generalized non-specular elastic reflection coefficients:

 

$${\bf {\cal R}}= \left( \begin{array} {ccc} \grave{P}\!\acut... ...e{S_2}& \grave{S_2}\!\acute{S_2} \end{array} \right) \,\,\, .$$ (2)

The generalized non-specular reflection coefficients are related to the plane-wave specular Zoeppritz coefficients as discussed by Frazer and Sen (1985), and will be examined in more detail later in this paper. It is apparent that the generalized reflection coefficients within ${\bf {\cal R}}$ can be ``isolated'' by the appropriate vector dot products. For example, the $\grave{P}\!\acute{P}$ coefficient can be isolated as

 

$$\u^r{\bf \cdot}{\bf e}_1^r = \grave{P}\!\acute{P}\,\u^s{\bf \cdot}{\bf e}^s_1 \,\,\,,$$ (3)

where ${\bf e}_1^s$ is the local P-wave polarization vector of the incident displacement vector ${\bf u}^s$, evaluated at the subsurface point ${\bf x}$, and ${\bf e}_1^r$ is the local P-wave polarization vector of the reflected displacement vector $\u^r$, evaluated at the subsurface point ${\bf x}$. Please refer to Figure to see the appropriate polarization vector geometry. Similar dot product relations can be stated for the other reflection coefficients of ${\bf {\cal R}}$ using the appropriate vector directions ${\bf e}_k({\bf x},t)$.For example, the $\grave{P}\!\acute{S_1}$ coefficient can be isolated as

 

$$\u^r{\bf \cdot}{\bf e}_2^r = \grave{P}\!\acute{S_1}\,\u^s{\bf \cdot}{\bf e}_1^s \,\,\,.$$ (4)

I will define the reflected scalar function on the left of (4) as $\Phi^r_i$ and the incident scalar function on the right of (4) as $\Phi^s_j$:

$$\Phi^s_j({\bf x},t;{\bf x}_s) = {\bf u}^s{\bf \cdot}{\bf e}_j^s$$

and  

$$\Phi^r_i({\bf x},t;{\bf x}_r) = \u^r{\bf \cdot}{\bf e}_i^r \;.$$ (5)

 

geometry
Figure 1 The elastic polarization vector geometry. The source position is at ${\bf x}_s$, the receiver position is at ${\bf x}_r$ and the subsurface reflection point is at ${\bf x}$. The polarization vectors $\{{\bf e}_1,{\bf e}_2,{\bf e}_3\}$ are the local P, S₁ and S₂ plane-wave directions, and functions of $({\bf x},t)$. The ${\bf e}_2$ polarization vectors for S₂ waves point out of the plane of this page.

If I can forward model numerous source wavefields $\u^s$ and I have numerous reconstructions of the reflected wavefield $\u^r$ from multiple surface shot gather observations, then (5) (and all of its nine converted-wave counterparts) can be rearranged into an optimization problem to estimate the isolated reflection coefficient, hereafter called ${\bf {\cal R}}_{ij}$. The objective function J for this problem has the general l_p form

 

$$J = \Vert W_d (\Phi^r_i - {\bf {\cal R}}_{ij}\Phi^s_j) \Vert^2_p + \Vert W_m {\bf {\cal R}}_{ij}\Vert^2_p$$ (6)

where W_d and W_m are arbitrary weighting functions in data and model space respectively. The weighting functions in this form are nonconvolutional, and therefore correspond to diagonal covariance functions ${\bf W}^T{\bf W}$ in the usual least-squares theory sense. Note that adding the second term to (6) will add stability to the final least-squares solution, and will ensure that ${\bf {\cal R}}_{ij}$ will be a minimum energy model in the l_p sense, which is appropriate for a desired spiky reflectivity solution. Choosing the l₂ norm and integrating over the implicit functional dependencies, one obtains:

$$\begin{eqnarray} J({\bf {\cal R}}_{ij}) & = & \int_t \int_{{\bf x}} W_d^2({\bf... ...{\bf x}} W_m^2({\bf x}){\bf {\cal R}}_{ij}^2({\bf x})\;d{\bf x}\;.\end{eqnarray}$$ (7)

To extremize J and thus find an ``optimal'' solution for ${\bf {\cal R}}_{ij}$, I perform the following stationary point analysis. Perturb J by an arbitrary small amount $\d{\bf {\cal R}}_{ij}({\bf x})$:

$$\begin{eqnarray} J({\bf {\cal R}}_{ij}+\d{\bf {\cal R}}_{ij}) & = & \int_t \int... ...}} W_m^2({\bf {\cal R}}_{ij}+\d{\bf {\cal R}}_{ij})^2\;d{\bf x}\;.\end{eqnarray}$$ (8)

A stationary point of J with respect to ${\bf {\cal R}}_{ij}$ occurs when

$$\lim_{\d{\bf {\cal R}}_{ij}\rightarrow 0} \left\{ J({\bf {\c... ...d{\bf {\cal R}}_{ij}) - J({\bf {\cal R}}_{ij})\right\} = 0 \;.$$

Neglecting terms of order $\d{\bf {\cal R}}_{ij}^2$,

 

$$\int_{{\bf x}}\; \left\{ \int_t W_d^2({\bf {\cal R}}_{ij}\Ph... ...}\right\} \; \d{\bf {\cal R}}_{ij}({\bf x})\;d{\bf x}\;=\; 0\;.$$ (9)

Since $\d{\bf {\cal R}}_{ij}({\bf x})$ is arbitrary, then the integrand kernel $\{\cdots\}$ must vanish at the stationary point. This results in a least-squares reflectivity solution:

 

$${\bf {\cal R}}_{ij}({\bf x}) = \frac {\int_t W_d^2({\bf x},t... ...2 \left(\Phi^s_j({\bf x},t)\right)^2\,dt + W_m^2({\bf x})} \;.$$ (10)

Equation (10) is a weighted zero-lag correlation of the incident and reflected wavefield scalar products, normalized by the weighted minimum-threshold energy of the incident wavefield scalar product. Equation (10) is valid for all of the generalized elastic reflection coefficients of (2) by simply making the appropriate redefinitions of $\Phi^s_j$ and $\Phi^r_i$ as in (5).