# Reciprocity Theorem

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

$\int _{S}{}\sum _{i=1}^{3}\delta u_{i}({\vec {x}})\delta \tau '_{i}({\vec {x}})dS=\int _{S}{}\sum _{i=1}^{3}\delta u'_{i}({\vec {x}})\delta \tau _{i}({\vec {x}})dS$ where $S$ is a surface completely enclosing an elastic body of arbitrary shape, heterogenity, and anisotropy. The $\delta u_{i}$ are the vector components of displacement on $S$ due to surface traction (force per unit area of $S$ ) $\delta \tau _{i}$ , and the $\delta u'_{i}$ are displacements on $S$ due to some other traction $\delta \tau '_{i}$ , separately applied. This is a fundamental theorem of elasticity, assuming only linear Hookean elasticity.

In the wave propagation context, it may be stated as

$\sum _{i=1}^{3}f_{i}(A)u_{i}(A|B)=\sum _{i=1}^{3}f_{i}(B)u_{i}(B|A)$ where $f_{i}(A)$ are the vector components of force applied at source-point $A$ , and $u_{i}(A|B)$ are the vector components of displacement recorded at $A$ , sourced from a different point $B$ . Similarly, $f_{i}(B)$ are the force applied at source-point $B$ , and $u_{i}(B|A)$ are the displacements recorded at $B$ , sourced from $A$ . 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 $A$ are mutually parallel, as are the two vectors at point $B$ . Hence the vector dot product above (the sum over components) reduces to a scalar product:

$f(A)u(A|B)=f(B)u(B|A)$ In s seismic survey, normally the source strength $f$ is the same at all points, including both $A$ and $B$ . Hence, the P-wave result above becomes:

$u(A|B)=u(B|A)$ which means that the data $u$ 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.