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.