Reciprocity Theorem

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

where is a surface completely enclosing an elastic body of arbitrary shape, heterogenity, and anisotropy. The are the vector components of displacement on due to surface traction (force per unit area of ) , and the are displacements on due to some other traction , separately applied. This is a fundamental theorem of elasticity, assuming only linear Hookean elasticity.

In the wave propagation context, it may be stated as

where are the vector components of force applied at source-point , and are the vector components of displacement recorded at , sourced from a different point . Similarly, are the force applied at source-point , and are the displacements recorded at , sourced from . 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 are mutually parallel, as are the two vectors at point . Hence the vector dot product above (the sum over components) reduces to a scalar product:

In s seismic survey, normally the source strength is the same at all points, including both and . Hence, the P-wave result above becomes:

which means that the data 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.