Trouble validating Greens function reciprocity in Specfem3d and Specfem2d.

[padesh]

I am working on reciprocity in coupled acoustic-elastic systems and found that SPECFEM3D is not producing the expected results. To investigate further, I revisited the Green's function reciprocity for acoustic and elastic systems using a homogeneous half-space model. These reciprocity relations are discussed in Wapenaar's 2006 paper Green’s Function Representations for Seismic Interferometry.

The specific reciprocity relations are as follows:

1. Acoustic reciprocity ->G^{p,q} (x_A,x_B,t) = G^{p,q} (x_B,x_A,t)
2. Elastic reciprocity 1 --> G^{v,f}_{m,n} (x_A,x_B,t) = G^{v,f}_{n,m} (x_B,x_A, t)
3. Elastic reciprocity 2 --> G^{tau,f}_{kl,n} (x_A,x_B,t) = G^{v,h}_{n,kl} (x_B,x_A, t)

I was able to validate the first two reciprocity relations, but the third one always results in a π/2 phase difference. I checked this using SPECFEM2D, assuming its pressure output as stress = pressure /3, and the results were the same, showing a π/2 phase difference.

There are 2 reciprocity that I worked out for a coupled system
4. G^{tau,q}_{kl} (x_A,x_B,t) = - G^{p,h}_{,kl} (x_B,x_A, t)
5. G^{p,f}_{,m} (x_A,x_B,t) = - G^{v,q}_{m} (x_B,x_A, t)

The first case I could validate successfully, but the second shows a π/2 phase difference.

Notice that all combinations that could be validated involve same or equivalent physical quantity measured and source applied ( force/velocity, pressure/pressure, or pressure/stress). However, whenever cross-combinations occur—such as when a force source is paired with a pressure/stress measurement, or a pressure/stress source is paired with a velocity measurement—a π/2 phase discrepancy is observed in the SPECFEM simulations.

What actually matches in those equations (3 and 5 items above) is displacement, not velocity, but this is inconsistent with what source-receiver reciprocity states. Other equations (1,2 and 4) would not give this error because of same physical quantities on both the sides. I am unable to determine the cause of this issue in SPECFEM. It could be something to do which where and how the force source in implemented in equation of motion? This phase discrepancy is introducing errors in ambient seismic simulations for a coupled system (e.g., ocean-bottom node cases), which, according to the theory and equations, should not exist.

Hope the Greens function notions are understandable above. Measurements are pressure->p, velocity->v, stress->tau. Sources are q->pressure-type, f-> force, h-> stress type with m,n,k,l as direction indices.

Does anyone have suggestions on why this might be happening?