Theory

The development of the solution of the wave propagation problem using the Generalized Reflection/Transmission matrices  is given in Reflection.pdf. The notation follows the author's Seismic Wave in Layered Media text. The reason for this effort was the need to test wave propagation in a model consisting of and ice layer  overlying water which in  turn sat on top of an elastic solid. Given current seismological monitoring in  the Arctic  and Antarctic it was necessary to overcome limitations in the hspec96 and hspec96p wavenumber integration codes.  The other case of interest is wave propagation in the Earth that includes the effect of the outer core when modeling reflections at the core-mantle boundary.  When generating Green's functions for body-wave moment tensor inversion, the current codes were used and the S-wave velocity in the outer core was set to a small non-zero value.


In its current form the rspec96 and rspec96p compute all Green's functions, including the stress field in a fluid when the shear velocity at the receiver depth is zero. Depending on the source, the PDD, PDS, PSS, PEX, PVF and PHF

Thus one objective of implementing the Generalized Reflection/Transmission method is to test the limitations on the assumption that a small non-zero S-velocity can be for a fluid.  The rspec96  permits one to  compute the response of a medium consisting of a sequence of solid/liquid/solid/liquid, etc., layering even though such a general model would be unstable. Practically, the most extreme model that I can conceive of is atmosphere/ice/water/solid/outer core. 



Summary of tests

The tests demonstrate that the rspec96 gives the same results as the hspec96 code for the following media:  all liquid layers, all solid layers, a sequence of liquid layers above solid layers and a sequence of solid layers above liquid ones. 

To test the more difficult solid/liquid/solid problem, rspec96   is run with the solid/liquid/solid model and  hspec96   is run with the liquid layer approximated as one with the same P-velocity but with a small, but non-zero S-velocity.  It is found that the Z component Green's functions, e.g., ZDD, ZDS, ZSS, ZEX, ZVF and ZHF, are identical throughout the medium. Surprisingly the radial Green's functions, RDD, RDS, RSS, REX, RVF, and RHF agree in the solid layers and also in many of the liquid layers if the very slow S-wave arrival moves away from the initial signal.  Because the S-wave velocity is so slow in the approximate medium, the hspec96 results differ from those from rspec96 only near the solid-liquid boundary.

As a result it seems that hspec96 and hspec96p could be further modified by doing the following:

a) if the medium is fluid/solid/fluid or solid/fluid/solid, etc., set the S-wave velocity for the fluid to be very small. Then
b) run the code but also apply filter for a receiver in fluid of  -RPUP -RPDN to filter out the S-wave in the pseudo-fluid.
Admittedly this is a hack, but could useful.

Test cases

The test cases exercise the boundary conditions that there is a free surface at the top and a halfspece at the bottom. There are the default conditions of hprep96, e.g, -TF -BH.

Download and install

To run the test cases, do the following:

The codes rspec96 and rspec96p were tested with isotropic and completely fluid models. The results agreed with those of hspec96 and hspec96p. The critical test is a model consisting of a solid - fluid - solid layer sequence. 

W - Model is a fluid

This model is as follows: 
  H(KM) VP(KM/S) VS(KM/S) RHO(GM/CC)   QP   QS  ETAP  ETAS  FREFP  FREFS
  4.0000  6.0000  0.0000  2.7000 0 0 0 0 1 1
  4.0000  6.0000  0.0000  2.7000 0 0 0 0 1 1
  7.0000  6.0000  0.0000  2.7000 0 0 0 0 1 1
 26.0000  6.0000  0.0000  2.7000 0 0 0 0 1 1
   .0000  8.0000  0.0000  3.3000 0 0 0 0 1 1
For this model only the ZEX, REX and PEX Green's functions are created. The source is at a depth of 20 km and the receiver depths vary from the surface to 30 km. The rspec96 and hspec96 Green's functions are plotted in red and blue, respectively. The plots are a true amplitude plot, meaning that the relative amplitudes between rceiver depths is preserved. Click on the links to see the comparison.

The display shows only subtle differences between the results of the two methods.

S - Model is a solid

This model is as follows: 
  H(KM) VP(KM/S) VS(KM/S) RHO(GM/CC)   QP   QS  ETAP  ETAS  FREFP  FREFS
  4.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
  4.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
  7.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
 26.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
   .0000  8.0000  4.7000  3.3000 0 0 0 0 1 1
For this model the ZDD, RDD, ZDS, RDS, TDS, ZSS, RSS, TSS, ZEX, REX, ZVF, RVF, ZHT, RHF and THF Green's functions are created. The source is at a depth of 20 km and the receiver depths vary from the surface to 30 km. The rspec96 and hspec96 Green's functions are plotted in red and blue, respectively. The plots are a true amplitude plot, meaning that the relative amplitudes between rceiver depths is preserved. Click on the links to see the comparison.

The display shows only subtle differences between the results of the two methods.

SW - Upper 8 km is a solid and the rest of the model is a fluid

This model is as follows: 
  H(KM) VP(KM/S) VS(KM/S) RHO(GM/CC)   QP   QS  ETAP  ETAS  FREFP  FREFS
  4.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
  4.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
  7.0000  6.0000  0.0000  2.7000 0 0 0 0 1 1
 26.0000  6.0000  0.0000  2.7000 0 0 0 0 1 1
   .0000  8.0000  0.0000  3.3000 0 0 0 0 1 1
For this model only the ZEX, REX and PEX Green's functions are created since the source is in the fluid. The PEX Green's function is created only for the receiver in the fluid. The source is at a depth of 20 km and the receiver depths vary from the surface to 30 km. The rspec96 and hspec96 Green's functions are plotted in red and blue, respectively. The plots are a true amplitude plot, meaning that the relative amplitudes between rceiver depths is preserved. Click on the links to see the comparison.

The display shows only subtle differences between the results of the two methods.

WS - Upper 8 km is a fluid and the rest of the model is a solid

This model is as follows: 
  H(KM) VP(KM/S) VS(KM/S) RHO(GM/CC)   QP   QS  ETAP  ETAS  FREFP  FREFS
  4.0000  6.0000  0.0000  2.7000 0 0 0 0 1 1
  4.0000  6.0000  0.0000  2.7000 0 0 0 0 1 1
  7.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
 26.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
   .0000  8.0000  4.7000  3.3000 0 0 0 0 1 1
For this model the ZDD, RDD, ZDS, RDS, TDS, ZSS, RSS, TSS, ZEX, REX, ZVF, RVF, ZHT, RHF, THF, PDD, PDS, PSS, PEX, PVF and PHF Green's functions are created. The source is at a depth of 20 km and the receiver depths vary from the surface to 30 km. The rspec96 and hspec96 Green's functions are plotted in red and blue, respectively. The plots are a true amplitude plot, meaning that the relative amplitudes between rceiver depths is preserved. Click on the links to see the comparison.

The display shows only subtle differences between the results of the two methods. Note that the P?? are not computed at a receiver depth of 8 km since the code computes the response on the solid side of tat boundary, e.g., at 8+ km.

T - solid/fluid/solid model

Although the hspec96 and hspec96p codes cannot handle such a model, it was instructive to run those codes with a near-zero S-wave velocity. The models considered are shown below. The difference between the models is highlighted in red.  The S.mod was used with rspec96 while Sa.mod and Sb.mod were used with hspec96.

S.mod
 Sa.mod
 Sb.mod
 
MODEL.01
CUS Model with Q from simple gamma values
ISOTROPIC
KGS
FLAT EARTH
1-D
CONSTANT VELOCITY
LINE08
LINE09
LINE10
LINE11
  H(KM) VP(KM/S) VS(KM/S) RHO(GM/CC) QP QS ETAP ETAS FREFP FREFS
  4.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
  4.0000  6.0000  0.0000  2.7000 0 0 0 0 1 1
  7.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
 26.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
   .0000  8.0000  4.7000  3.3000 0 0 0 0 1 1
 
MODEL.01
CUS Model with Q from simple gamma values
ISOTROPIC
KGS
FLAT EARTH
1-D
CONSTANT VELOCITY
LINE08
LINE09
LINE10
LINE11
  H(KM) VP(KM/S) VS(KM/S) RHO(GM/CC) QP QS ETAP ETAS FREFP FREFS
  4.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
  4.0000  6.0000  0.0100  2.7000 0 0 0 0 1 1
  7.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
 26.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
   .0000  8.0000  4.7000  3.3000 0 0 0 0 1 1
 
MODEL.01
CUS Model with Q from simple gamma values
ISOTROPIC
KGS
FLAT EARTH
1-D
CONSTANT VELOCITY
LINE08
LINE09
LINE10
LINE11
  H(KM) VP(KM/S) VS(KM/S) RHO(GM/CC) QP QS ETAP ETAS FREFP FREFS
  4.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
  4.0000  6.0000  0.0050  2.7000 0 0 0 0 1 1
  7.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
 26.0000  6.0000  3.5000  2.7000 0 0 0 0 1 1
   .0000  8.0000  4.7000  3.3000 0 0 0 0 1 1

The next figure plots the Shear velocities of these models. Since the S-velocities in the second layer are very small, their differences are not seen in the figure.

SHWMOD96.png

To compare results of the different codes, the computations make using a single ray parameter with hspec96p or rspec96p are the fastest, but for waveform modeling the complete synthetics computed using hspec96 or rspec96 must be computed. There is not too much difference in the execution times of these codes.   

For these models the ZDD, RDD, ZDS, RDS, TDS, ZSS, RSS, TSS, ZEX, REX, ZVF, RVF, ZHT, RHF, THF, PDD, PDS, PSS, PEX, PVF and PHF Green's functions are created. The source is at a depth of 20 km and the receiver depths vary from the surface to 30 km. The rspec96 and hspec96 Green's functions are plotted in red and blue, respectively. The plots are a true amplitude plot, meaning that the relative amplitudes between rceiver depths is preserved.

For this comparison the S.mod is used with rspec96 and Sa.mod with hspec96.

Receiver depth sections 0 - 30 km

In the trace comparisons the only real differences are in the R?? components. For example look at the REX Grens/ function at 5 and 7 km at late times. These would be a problem is a shorter time window were used because of the periodicity of the FFT used. Note that these later arrivals are about 100 sec after the first P arrival.

Click on the links to see the comparison. Note the P?? are componented only for the rspec96 since the Sa.mod is a solid model even though the S velocity is very small.

Receiver depth sections 3.5 - 8.5 km

These plot focus on the arrivals in the water layer. Remembering that the color blue is used for the hspec96 synthetics. A set of arrivals is genereated at the sharp velocity boundary at 4 and 8 km. The moveout indicates that these are the S-wave propagating at 0.01 km/sec. In genereal the initial part of the traces agree very well. It is only at location near the boundary that these arrivals interfer with the expected motion.

One cannot make the S velocity too small to simulate a fluid when using hspec96 if the computations are done in single precision.

The other conclusion is that the Z?? displacements in the fluid comapre well. Although the hspec96 will not give the P?? in the fluid, the displacements in the overlying solid are correctly computed. This one could use hspec96 for the ice/water/solid problem.

Click on the links to see the comparison.

One subtle difference in the implementation occurs when the source or receiver depth is at the solid/liquid boundary, e.g., at a depth of 4.0 or 8.0 km in  the S.mod above. Actually in the code, the decision is whether the observation point is at the bottom of the layer above or at the top of the layer below. One could follow the code, but one could also experiment  by computing synthetics at just above and below the boundary and then comparing those to the synthetics computed at the boundary,  This distinction is pertinent since the vertical displacement and stress are continuous across a fluid/solid boundary, but the radial displacement is not.

I mention this because of the use of OBS platforms. Thus to make synthetics for the particle velocities, I would perform the computations at a depth of about 1 meter or so below the ocean/solid boundary. To get the pressure field, I would run the codes for a receiver in the fluid at a height of about 1 meter above the boundary.