Reciprocity Theorem

The Theorem of Reciprocity is a fundamental result of linear responses. In the case of elasticity, it may be stated as

${\displaystyle \int _{S}{}\sum _{i=1}^{3}\delta u_{i}({\vec {x}})\delta \tau '_{i}({\vec {x}})dS=\int _{S}{}\sum _{i=1}^{3}\delta u'_{i}({\vec {x}})\delta \tau _{i}({\vec {x}})dS}$

where ${\displaystyle S}$ is a surface completely enclosing an elastic body of arbitrary shape, heterogenity, and anisotropy. The ${\displaystyle \delta u_{i}}$ are the vector components of displacement on ${\displaystyle S}$ due to surface traction (force per unit area of ${\displaystyle S}$) ${\displaystyle \delta \tau _{i}}$ , and the ${\displaystyle \delta u'_{i}}$ are displacements on ${\displaystyle S}$ due to some other traction ${\displaystyle \delta \tau '_{i}}$, separately applied. This is a fundamental theorem of elasticity, assuming only linear Hookean elasticity.

In the wave propagation context, it may be stated as

${\displaystyle \sum _{i=1}^{3}f_{i}(A)u_{i}(A|B)=\sum _{i=1}^{3}f_{i}(B)u_{i}(B|A)}$

where ${\displaystyle f_{i}(A)}$ are the vector components of force applied at source-point ${\displaystyle A}$, and ${\displaystyle u_{i}(A|B)}$ are the vector components of displacement recorded at ${\displaystyle A}$, sourced from a different point ${\displaystyle B}$. Similarly, ${\displaystyle f_{i}(B)}$ are the force applied at source-point ${\displaystyle B}$, and ${\displaystyle u_{i}(B|A)}$ are the displacements recorded at ${\displaystyle B}$, sourced from ${\displaystyle A}$. This is the general form, the Vector Reciprocity Theorem. It includes mixed-mode wave propagation, like converted waves.

When this is specialized to pure-mode P-wave propagation, the raypath is symmetrical, so the two vectors at point ${\displaystyle A}$ are mutually parallel, as are the two vectors at point ${\displaystyle B}$. Hence the vector dot product above (the sum over components) reduces to a scalar product:

${\displaystyle f(A)u(A|B)=f(B)u(B|A)}$

In s seismic survey, normally the source strength ${\displaystyle f}$ is the same at all points, including both ${\displaystyle A}$ and ${\displaystyle B}$. Hence, the P-wave result above becomes:

${\displaystyle u(A|B)=u(B|A)}$

which means that the data ${\displaystyle u}$ are the same, when the source and receiver positions are interchanged. This is the Scalar Reciprocity Theorem, valid only for pure-mode propagation (like P-waves). It is casually referred to as the "Reciprocity Theorem", but it is just the special case for pure-mode propagation. Split-spread P-wave gathers are necessarily symmetric; this is why off-end towed-streamer acquisition is successful.

For C-waves, we need the Vector Reciprocity Theorem. The force-vector above (generating a P-wave) is parallel to the downgoing raypath , but the upcoming data-vector is (almost) transverse to this, since the energy arrives as an S-wave. So, the vector dot product only says that 0=0; the Vector Reciprocity Theorem only constrains the source-parallel component of the data, which is minimal. Split-spread C-wave gathers are not necessarily symmetric.