Introduction

The purpose of this note is to consider the ellipticity properties of the Rayleigh wave for certain models.  For a halfspace, Rayleigh wave particle motion is retrograde elliptical and the ratio of the maximum radial displace to vertical displacement is about 0.68.  However this is not true for some velocity models.

Denolle


Denolle et al (2012) [ Denolle, M. A., E. M. Dunham and G. Beroze (2012). Solving the surface-wave eigenproblem with Chebyshev spectral
collocation, Bull. Seism. Soc. Am. 102, 1214-1223] noted that the fundamental model Rayleigh wave  exhibited  retrograde elliptical motion at long and short periods but prograde motion at intermediate periods for some gradient models. This is seen in Figure 6 of their paper which compares the ellipticity of two models. The model96 versions of these models are denoted as  Denolle1.mod and Denolle2.mod. Figure 1 compares these two models.

Figure 1. Comparison of the two Denolle models. The figure at the right compares the upper 10 km of the two models.

Interestingly if the velocity at the surface is increased, particle motion is prograde for all frequencies. Another observation gleaned from their Figure 6 is that the ellipticity can be significantly greater than 1.0 for many frequencies. 

Morelli

In 2015, I received an Email from Andrea Morelli, IGNV Bologna, Italy. The email asked about the behavior of some velocity models that give prograde elliptical motion for the fundamental mode Rayleigh wave. The model96 versions of these models are  model1.d and model2.d . The only difference between these models and those actually provided is that the lower 24 km of each of the original models was truncated because the limitation of 200 layers in my version of the CPS code. These models are plotted in Figure 2.

Figure 2. Comparison of the two models from Andrea Morelli. The figure at the right compares the upper 10 km of the two models.

Hobiger (2013)

A recent paper by Cercato  (2018) [Certaco, Michele (2018). Sensitivity of Rayleigh wave ellipticity and implications for surface wave inversion, Geophys. J. Int 213, 489-510, doi: 10.1093/gji/ggx558] used two models by Hobiger et al (2013) [ Hobiger, M. et al., (2013) . Ground structure imaging by inversions of Rayleigh wave ellipticity: sensitivity analysis and application to European strong motion sites, Geophys. J. Int., 192, 207–229. ]   to test a methodology for estimating the partial derivatives of ellipticity with respect to medium parameters. The model96 version of these two models are HoA.mod and Hob.mod. These models are compared in Figure 3. HoA.mod has a large velocity discontinuity at a depth of 200 m.
Figure 3. Hobinger (2013) models A and B

Model characteristics

The Denolle and Morelli models are characterized by gradients while the Hobiger (2013) models are simpler layer-cake models, differing by the very large velocity discontinuity at the layer boundary with the halfspace.

Dispersion curves

The VOLIII surface-wave codes of Computer Programs in Seismology [Herrmann, R. B. (2013) Computer programs in seismology: An evolving tool for instruction and research, Seism. Res. Lettr. 84, 1081-1088, doi:10.1785/0220110096] were used to compute dispersion curves for the phase velocity, ellipticity and amplitude factor. These are presented  in Figure 4 for each of the five velocity models. In these figures the fundamental and first higher modes are plotted. The fundamental mode is the only mode at the longest period.

Model
Phase velocity Ellipticity AR
Denolle1.mod
Denolle2.mod
model1.d
model2.d
HoA.mod
HoB.mod

Discussion of dispersion results

These models were selected by the authors because of the "strange" behavior of the ellipticity at certain frequencies/periods. Before discussing the results, note that the program sdpegn96  used to plot the ellipticities does not handle the discontinuity in ellipticity very well. This "strange" behavior is illustrated by using the command sdpegn96 -R -E -ASC  to crate the file SREGN.ASC for a special run that emphasizes frequencies near 0.7 Hz for the HoA.mod.    The output near a frequency of 0.669 Hz is as follows:

 RMODE NFREQ    PERIOD(S) FREQUENCY(Hz)  C(KM/S)      U(KM/S)      ENERGY       GAMMA(1/KM)  ELLIPTICITY
    0    19    1.4837      0.67400       2.0951       1.2253      0.18078E-03   0.0000      -68.191
    0    20    1.4859      0.67300       2.0973       1.2367      0.12069E-03   0.0000      -82.476
    0    21    1.4881      0.67200       2.0994       1.2481      0.73381E-04   0.0000      -104.53
    0    22    1.4903      0.67100       2.1016       1.2595      0.38338E-04   0.0000      -142.92
    0    23    1.4925      0.67000       2.1036       1.2706      0.15202E-04   0.0000      -224.33
    0    24    1.4948      0.66900       2.1057       1.2817      0.27082E-05   0.0000      -525.36
    0    25    1.4970      0.66800       2.1077       1.2930      0.38194E-06   0.0000       1382.6
    0    26    1.4993      0.66700       2.1097       1.3038      0.75552E-05   0.0000       307.33
    0    27    1.5015      0.66600       2.1116       1.3148      0.24005E-04   0.0000       170.43
    0    28    1.5038      0.66500       2.1135       1.3256      0.48567E-04   0.0000       118.46
    0    29    1.5060      0.66400       2.1154       1.3362      0.81047E-04   0.0000       90.663
    0    30    1.5083      0.66300       2.1172       1.3469      0.12106E-03   0.0000       73.344
    0    31    1.5106      0.66200       2.1190       1.3574      0.16779E-03   0.0000       61.601

Notice the very large values of the ellipticity. The change in sign indicates the transition from retrograde ellipical motion (positive values) to prograde elliptical motion (negative values).  Also note that the Energy term = 1 / 2 c U I0 is very small is the region where the ellipticity os large.

The reason for plotting the energy term AR is that the amplitude spectra of the vertical and radial components of motion, UZ ans UR, respectively, for a point impulsive source applied at the surface of a layered halfspace are 

UZ = AR / sqrt ( kr)
UR = AR Ellipticity / sqrt(kr)
For this model, we can say that the vertical component spectra at 0.662 Hz is about 500 times greater than at 0.668 Hz. The radial component spectra at 0.662 Hz is about 25 times greater than at 0.668 Hz.

 

What causes the extreme ellipticity?

Some of the models have gradients, and yet one of the the simple Hobiger (2013) models shows the same behavior, while the other does not. So is the extreme ellipticity a function of gradients or of some other feature of the models?

To test the significant of the velocity contract at the halfspace, consider the following suites of models:


     H(km)       Vp(km/s)     Vs(km/s)   Rho(gm/cm3)
-----------------------------------------------------
      6            Vp1          Vs1        Rho1
     ---           6.0          3.5         2.7
-----------------------------------------------------

For the velocities in the layer, 

    Model       Vp1        Vs1       Rho1
     4.0      4.0000     2.0000     2.0000       
     3.0      3.0000     1.5000     2.0000       
     2.0      2.0000     1.0000     2.0000       
     1.5      1.5000     0.7500     2.0000       
     1.0      1.0000     0.5000     2.0000  
Figure 5 compares the velocity models while Figures 6 compares the fundamental mode Rayleigh wave phase velocity, ellipticity and AR.

Figure 5.Comparison of velocity models to test the effect of the size of the velocity discontinuity on ellipticity





Figure 6. Phase velocity, ellipticity and amplitude factor for the five models.


To unserstand the information in Figure 6, first note that the interesting features, such as the rapid increase in velocity from the halfspace Rayleigh velocity for the top layer to the bottom layer, the rapid change in ellipicity and the very small amplitude factor, move toward longer periods at the velocities of the top layers decrease. This is not unexpected.
As the velocity decreases the phase velocity dispersion seems to develop a kink. The AR  term shows a difference in levels when the models are compared. It is not unexpected that the same source applied at the surface of the weaker (lower velocity) model creates greater ground motions.

For the first two models, the the ellipticity does not vary much when compared to the last three models. For each model the SREGN.ASC file was created. Subsets of these files are given for each model:

Model 1.0

 RMODE NFREQ    PERIOD(S) FREQUENCY(Hz)  C(KM/S)      U(KM/S)      ENERGY       GAMMA(1/KM)  ELLIPTICITY
    0     1    1.0000       1.0000      0.46626      0.46626       4.1932       0.0000      0.63890
    0     2    1.1000      0.90909      0.46626      0.46626       3.8120       0.0000      0.63890
    .....
    0    54    23.000      0.43478E-01  0.58513      0.24971      0.46448       0.0000      0.49870
    0    55    24.000      0.41667E-01  0.62743      0.20853      0.52217       0.0000      0.45287
    0    56    25.000      0.40000E-01  0.69740      0.16019      0.62094       0.0000      0.37103
    0    57    26.000      0.38462E-01  0.83020      0.13229      0.63923       0.0000      0.15504
    0    58    27.000      0.37037E-01  0.96139      0.29052      0.20352       0.0000     -0.34901
    0    59    28.000      0.35714E-01   1.0194      0.46780      0.78244E-01   0.0000     -0.86562
    0    60    29.000      0.34483E-01   1.0562      0.55691      0.42573E-01   0.0000      -1.3225
    0    61    30.000      0.33333E-01   1.0870      0.59746      0.27457E-01   0.0000      -1.7391
    0    62    32.000      0.31250E-01   1.1462      0.61982      0.14495E-01   0.0000      -2.5159
    0    63    34.000      0.29412E-01   1.2111      0.61177      0.88422E-02   0.0000      -3.2918
    0    64    36.000      0.27778E-01   1.2876      0.59108      0.57019E-02   0.0000      -4.1403
    0    65    38.000      0.26316E-01   1.3819      0.56347      0.37072E-02   0.0000      -5.1486
    0    66    40.000      0.25000E-01   1.5022      0.53185      0.23321E-02   0.0000      -6.4632
    0    67    42.000      0.23810E-01   1.6611      0.49972      0.13363E-02   0.0000      -8.4118
    0    68    44.000      0.22727E-01   1.8769      0.47575      0.60739E-03   0.0000      -12.019
    0    69    46.000      0.21739E-01   2.1660      0.48967      0.13153E-03   0.0000      -23.432
    0    70    48.000      0.20833E-01   2.4902      0.63984      0.57223E-05   0.0000       87.891
    0    71    50.000      0.20000E-01   2.7289       1.0453      0.13605E-03   0.0000       12.091
    0    72    55.000      0.18182E-01   2.9535       2.1056      0.27924E-03   0.0000       3.7389
    0    73    60.000      0.16667E-01   3.0244       2.5492      0.26950E-03   0.0000       2.4031
    .....
    0    91    200.00      0.50000E-02   3.1801       3.1444      0.58569E-04   0.0000      0.80118
    0    92    210.00      0.47619E-02   3.1818       3.1481      0.55536E-04   0.0000      0.79343
    0    93    220.00      0.45455E-02   3.1833       3.1515      0.52807E-04   0.0000      0.78662
    0    94    230.00      0.43478E-02   3.1847       3.1545      0.50339E-04   0.0000      0.78059
    0    95    240.00      0.41667E-02   3.1860       3.1572      0.48094E-04   0.0000      0.77521
    0    96    250.00      0.40000E-02   3.1871       3.1596      0.46045E-04   0.0000      0.77039

Model 1.5

 RMODE NFREQ    PERIOD(S) FREQUENCY(Hz)  C(KM/S)      U(KM/S)      ENERGY       GAMMA(1/KM)  ELLIPTICITY
    0     1    1.0000       1.0000      0.69939      0.69939       1.2424       0.0000      0.63890
    0     2    1.1000      0.90909      0.69939      0.69939       1.1295       0.0000      0.63890
    .....
    0    47    16.000      0.62500E-01  0.92585      0.33307      0.21926       0.0000      0.46661
    0    48    17.000      0.58824E-01   1.0787      0.23594      0.27088       0.0000      0.35153
    0    49    18.000      0.55556E-01   1.3904      0.30374      0.15401       0.0000     -0.10204
    0    50    19.000      0.52632E-01   1.5424      0.74727      0.29323E-01   0.0000     -0.96585
    0    51    20.000      0.50000E-01   1.6125      0.91612      0.11739E-01   0.0000      -1.7503
    0    52    21.000      0.47619E-01   1.6716      0.96345      0.63720E-02   0.0000      -2.4820
    0    53    22.000      0.45455E-01   1.7313      0.97222      0.39174E-02   0.0000      -3.2235
    0    54    23.000      0.43478E-01   1.7954      0.96622      0.25341E-02   0.0000      -4.0343
    0    55    24.000      0.41667E-01   1.8659      0.95426      0.16540E-02   0.0000      -4.9889
    0    56    25.000      0.40000E-01   1.9443      0.94107      0.10519E-02   0.0000      -6.2086
    0    57    26.000      0.38462E-01   2.0320      0.93069      0.62456E-03   0.0000      -7.9358
    0    58    27.000      0.37037E-01   2.1293      0.92809      0.32207E-03   0.0000      -10.773
    0    59    28.000      0.35714E-01   2.2356      0.94053      0.12235E-03   0.0000      -16.787
    0    60    29.000      0.34483E-01   2.3477      0.97858      0.18349E-04   0.0000      -40.744
    0    61    30.000      0.33333E-01   2.4596       1.0549      0.47040E-05   0.0000       73.526
    0    62    32.000      0.31250E-01   2.6525       1.3438      0.15021E-03   0.0000       9.9856
    0    63    34.000      0.29412E-01   2.7820       1.7196      0.29981E-03   0.0000       5.2053
    0    64    36.000      0.27778E-01   2.8628       2.0397      0.36603E-03   0.0000       3.5857
    .....
    0    91    200.00      0.50000E-02   3.1816       3.1487      0.57328E-04   0.0000      0.77519
    0    92    210.00      0.47619E-02   3.1832       3.1520      0.54464E-04   0.0000      0.77005
    0    93    220.00      0.45455E-02   3.1846       3.1550      0.51875E-04   0.0000      0.76546
    0    94    230.00      0.43478E-02   3.1859       3.1577      0.49522E-04   0.0000      0.76134
    0    95    240.00      0.41667E-02   3.1871       3.1601      0.47375E-04   0.0000      0.75762
    0    96    250.00      0.40000E-02   3.1882       3.1624      0.45408E-04   0.0000      0.75424

Model 2.0

 RMODE NFREQ    PERIOD(S) FREQUENCY(Hz)  C(KM/S)      U(KM/S)      ENERGY       GAMMA(1/KM)  ELLIPTICITY
    0     1    1.0000       1.0000      0.93253      0.93253      0.52415       0.0000      0.63890
    0     2    1.1000      0.90909      0.93253      0.93253      0.47650       0.0000      0.63890
    .....
    0    43    12.000      0.83333E-01   1.2092      0.47988      0.11480       0.0000      0.48317
    0    44    13.000      0.76923E-01   1.4469      0.33931      0.13771       0.0000      0.36125
    0    45    14.000      0.71429E-01   1.9391      0.49356      0.64503E-01   0.0000     -0.17635
    0    46    15.000      0.66667E-01   2.1136       1.3110      0.48615E-02   0.0000      -2.0268
    0    47    16.000      0.62500E-01   2.1905       1.4261      0.12297E-02   0.0000      -4.1870
    0    48    17.000      0.58824E-01   2.2643       1.4498      0.38775E-03   0.0000      -7.3722
    0    49    18.000      0.55556E-01   2.3404       1.4678      0.89468E-04   0.0000      -14.876
    0    50    19.000      0.52632E-01   2.4184       1.4987      0.20254E-05   0.0000      -94.140
    0    51    20.000      0.50000E-01   2.4957       1.5515      0.27371E-04   0.0000       23.917
    0    52    21.000      0.47619E-01   2.5691       1.6296      0.11358E-03   0.0000       10.754
    0    53    22.000      0.45455E-01   2.6359       1.7297      0.22012E-03   0.0000       6.9623
    0    54    23.000      0.43478E-01   2.6944       1.8431      0.31763E-03   0.0000       5.1712
    0    55    24.000      0.41667E-01   2.7443       1.9593      0.39179E-03   0.0000       4.1405
    .....
    0    91    200.00      0.50000E-02   3.1833       3.1525      0.56807E-04   0.0000      0.76466
    0    92    210.00      0.47619E-02   3.1848       3.1556      0.54009E-04   0.0000      0.76046
    0    93    220.00      0.45455E-02   3.1861       3.1583      0.51476E-04   0.0000      0.75668
    0    94    230.00      0.43478E-02   3.1873       3.1608      0.49170E-04   0.0000      0.75326
    0    95    240.00      0.41667E-02   3.1884       3.1631      0.47062E-04   0.0000      0.75016
    0    96    250.00      0.40000E-02   3.1894       3.1651      0.45128E-04   0.0000      0.74732

Model 3.0

 RMODE NFREQ    PERIOD(S) FREQUENCY(Hz)  C(KM/S)      U(KM/S)      ENERGY       GAMMA(1/KM)  ELLIPTICITY
    0     1    1.0000       1.0000       1.3988       1.3988      0.15530       0.0000      0.63890
    0     2    1.1000      0.90909       1.3988       1.3988      0.14118       0.0000      0.63890
    .....
    0    41    10.000      0.10000       2.4472       1.0636      0.18680E-01   0.0000      0.49327
    0    42    11.000      0.90909E-01   2.6450       1.7965      0.54847E-02   0.0000      0.85617
    0    43    12.000      0.83333E-01   2.7248       2.1710      0.25266E-02   0.0000       1.2107
    0    44    13.000      0.76923E-01   2.7724       2.3351      0.16257E-02   0.0000       1.4299
    0    45    14.000      0.71429E-01   2.8077       2.4257      0.12406E-02   0.0000       1.5418
    0    46    15.000      0.66667E-01   2.8366       2.4872      0.10353E-02   0.0000       1.5863
    0    47    16.000      0.62500E-01   2.8614       2.5347      0.90787E-03   0.0000       1.5919
    0    48    17.000      0.58824E-01   2.8830       2.5744      0.81949E-03   0.0000       1.5760
    0    49    18.000      0.55556E-01   2.9022       2.6090      0.75307E-03   0.0000       1.5488
    0    50    19.000      0.52632E-01   2.9193       2.6399      0.70014E-03   0.0000       1.5162
    0    51    20.000      0.50000E-01   2.9348       2.6679      0.65616E-03   0.0000       1.4817
    0    52    21.000      0.47619E-01   2.9488       2.6935      0.61851E-03   0.0000       1.4472
    0    53    22.000      0.45455E-01   2.9615       2.7170      0.58558E-03   0.0000       1.4136
    .....
    0    91    200.00      0.50000E-02   3.1876       3.1618      0.56175E-04   0.0000      0.75218
    0    92    210.00      0.47619E-02   3.1889       3.1643      0.53450E-04   0.0000      0.74889
    0    93    220.00      0.45455E-02   3.1900       3.1666      0.50976E-04   0.0000      0.74591
    0    94    230.00      0.43478E-02   3.1910       3.1686      0.48722E-04   0.0000      0.74319
    0    95    240.00      0.41667E-02   3.1919       3.1705      0.46658E-04   0.0000      0.74070
    0    96    250.00      0.40000E-02   3.1928       3.1722      0.44762E-04   0.0000      0.73842

Model 4.0

 RMODE NFREQ    PERIOD(S) FREQUENCY(Hz)  C(KM/S)      U(KM/S)      ENERGY       GAMMA(1/KM)  ELLIPTICITY
    0     1    1.0000       1.0000       1.8651       1.8650      0.65521E-01   0.0000      0.63890
    0     2    1.1000      0.90909       1.8651       1.8650      0.59569E-01   0.0000      0.63890
    .....
    0    51    20.000      0.50000E-01   3.0501       2.9281      0.56817E-03   0.0000       1.0879
    0    52    21.000      0.47619E-01   3.0562       2.9380      0.53821E-03   0.0000       1.0816
    0    53    22.000      0.45455E-01   3.0618       2.9470      0.51183E-03   0.0000       1.0743
    0    54    23.000      0.43478E-01   3.0671       2.9552      0.48833E-03   0.0000       1.0665
    0    55    24.000      0.41667E-01   3.0719       2.9629      0.46718E-03   0.0000       1.0583
    0    56    25.000      0.40000E-01   3.0765       2.9700      0.44800E-03   0.0000       1.0500
    0    57    26.000      0.38462E-01   3.0808       2.9767      0.43048E-03   0.0000       1.0416
    0    58    27.000      0.37037E-01   3.0848       2.9830      0.41439E-03   0.0000       1.0332
    0    59    28.000      0.35714E-01   3.0886       2.9889      0.39953E-03   0.0000       1.0250
    0    60    29.000      0.34483E-01   3.0922       2.9946      0.38576E-03   0.0000       1.0170
    0    61    30.000      0.33333E-01   3.0955       2.9999      0.37294E-03   0.0000       1.0091
    .....
    0    91    200.00      0.50000E-02   3.1934       3.1739      0.55602E-04   0.0000      0.74094
    0    92    210.00      0.47619E-02   3.1943       3.1757      0.52934E-04   0.0000      0.73830
    0    93    220.00      0.45455E-02   3.1952       3.1774      0.50510E-04   0.0000      0.73590
    0    94    230.00      0.43478E-02   3.1960       3.1789      0.48298E-04   0.0000      0.73371
    0    95    240.00      0.41667E-02   3.1967       3.1803      0.46272E-04   0.0000      0.73170
    0    96    250.00      0.40000E-02   3.1973       3.1816      0.44409E-04   0.0000      0.72984

Discussion

This last example demonstrates that the very high values of ellipticity do not require a gradient model, but do require a large velocity contrast at some depth. Although the upper part of the Hobiger models are the same, the HoB model does not have a sharp velocity contrast at the lowest boundary, and the ellipicity of that model does not show any extreme values. We also observe that when the ellipicity becomes vary large, the corresponding energy term becomes very small. Thus there is the real question
of whether such large values of ellipticity can be observed. This is a valid question since it seem as if the first higher mode could overcome the lack of signal in the fundamental model.

Implications for inversion

If a large value of ellipticity is observed, then that is an indication of a very sharp layer boundary at depth. The phase velocity dispersion might not be able to image a sharp discontinuity, but the constraint from the ellipticity might be useful.

Can  these large ellipticities be observed?

To address this question, displacement synthetics were created for receivers at the surface due to a point vertical force with an impulsive source time function. These are the ZVF and RVF Green's functions. For simple processing, the R/Z ratio was formed from the amplitude ratio of the spectra of the two traces. The sampling interval and distance were selected to be appropriate for the disperion plots made earlier. This was done for the fudnamental mode and for all modes. The following set of figures shows the model name, the R/Z ratios, the spectra from just the fundamental mode synthetics, and finally the spectra form the multimode synthetics.

Model
R/Z ratios Fundamental mode spectra Multimode spectra
Denolle1.mod
Denolle2.mod
model1.d
model2.d
HoA.mod
HoB.mod


As expected, the character of fundamental mode spectra shows the features of the AR.

Of the models there is good agreement for the spectral ratios determined for the HoA and HoB models when comparing the fundamental mode only and the multimode synthetics. However for the other models, there is little similarity between the two estimates because at the frequencies where the fundamental mode has a very large ellipticity, its excitation is small while there is significant signal from the first higher mode. Thus the premise that the empirical R/Z ratio can be interpreted as being due to the fundamental mode does not hold.

While the processing of synthetics for the HoA model duplicated the theoretical ellipticity, I question whether one would actually see these extreme values. Consider the amplitude spectra for the HoA model. The very large ellipticity values are observed at about 0.67 Hz. However the spectral levels here are many orders of magnitude lower than the high frequency levels. If there other sources of noise that are not associated with wave propagation, the spectral nulls may not be observed in real data sets. In addition any spectral smoothing would make them less extreme.

Last changed March 13, 2018