Teleseismic P- and S-wave Geometrical Spreading

## Introduction

Langston and Helmberger (1975) proposed a simple technique to determine the earthquake size, depth and faulting parameters from an analysis of the contribution of P, pP and sP to the observed teleseismic signal immediately following P.  They used a simple halfspace  model for the source and receiver crusts and required a geometrical spreading and T* attenuation parameter to propagate the signal from the source region to the receiver.

```       We follow Bullen and the
Herrmann class notes http://www.eas.slu.edu/People/RBHerrmann/Courses/EASA462/
for Ass14.pdf
The geometrical spreading is dimensionless and gives the decrease in amplitude from
a distance of 1 km from the source to the receiver
2                            2
(   a sin Is Vs                  d T      )
sqrt |  ------------------------     -----     |
|                              2      2   |
(  sin DEG  cos Ir Rs Cos Is Rr  d DEL    )
a = radius of sphere about source - we use 1.0 km
Is= incident angle at source
Rs= distance from center of Earth to source
Rr= distance from center of Earth to receiver
DEG=epicental distance in degrees
note that the second derivative divided by Rr^2 is sec/km^2 - which we can get
from dp/dx  where dp is in sec/km and dx is the distance in km, so we compute
2                            2
(   a sin Is Vs                  d T      )
sqrt |  ------------------------     -----     |
|                                  2      |
(  sin DEG  cos Ir Rs Cos Is     dx       )
The only imperfection is the numerical computation of dp/dD```

This description describes is a function of the ratio of ray tube areas at the receiver to the source.  Aki and Richards (199?) and others note that a consideration of wave dynamics requires this relation to be modified by the following term

( Density (source)     Wave velocity (source)   )
sqrt |  ----------------------   ------------------------------  |

When considering the composite effect of P, pP and sP,  the wave velocities are both that of the P wave.

## Script

The script DOIT
requires the use of AK135-f in the Computer Programs in Seismology model format ( tak135sph.mod )

## Sample Run

Assuming that the program time96 in in your PATH, entering the command DOIT yields the following:

```--------------------------------------------------
Phase P
DEG    HS    T(s)    p(s/deg)    Geom     T*(sec)
--------------------------------------------------
30.0  10.0   368.48   8.787    7.115e-05  1.15
32.5  10.0   390.45   8.787    6.531e-05  1.18
35.0  10.0   412.18   8.644    6.056e-05  1.19
37.5  10.0   433.56   8.502    7.408e-05  1.20
40.0  10.0   454.56   8.363    7.045e-05  1.22
42.5  10.0   475.11   8.096    6.622e-05  1.22
45.0  10.0   495.22   7.967    6.889e-05  1.24
50.0  10.0   534.11   7.595    6.187e-05  1.27
55.0  10.0   571.18   7.247    5.624e-05  1.31
60.0  10.0   606.44   6.916    5.552e-05  1.35
65.0  10.0   639.85   6.501    5.422e-05  1.39
70.0  10.0   671.47   6.108    4.986e-05  1.43
75.0  10.0   701.28   5.733    4.625e-05  1.48
80.0  10.0   729.24   5.373    4.325e-05  1.53
85.0  10.0   755.32   5.024    4.072e-05  1.58
90.0  10.0   779.45   4.685    3.255e-05  1.63
--------------------------------------------------
Phase pP
DEG    HS    T(s)    p(s/deg)    Geom     T*(sec)
--------------------------------------------------
30.0  10.0   371.54   8.952    7.225e-05  1.18
32.5  10.0   393.51   8.787    6.531e-05  1.19
35.0  10.0   415.26   8.644    7.135e-05  1.20
37.5  10.0   436.65   8.502    7.408e-05  1.21
40.0  10.0   457.66   8.363    7.045e-05  1.23
42.5  10.0   478.23   8.096    6.622e-05  1.24
45.0  10.0   498.35   7.967    6.889e-05  1.25
50.0  10.0   537.28   7.595    6.187e-05  1.29
55.0  10.0   574.38   7.247    5.624e-05  1.32
60.0  10.0   609.65   6.916    5.552e-05  1.36
65.0  10.0   643.09   6.501    5.422e-05  1.40
70.0  10.0   674.74   6.108    4.676e-05  1.45
75.0  10.0   704.57   5.825    4.671e-05  1.49
80.0  10.0   732.55   5.373    4.598e-05  1.54
85.0  10.0   758.65   5.024    4.072e-05  1.59
90.0  10.0   782.80   4.685    3.255e-05  1.64
--------------------------------------------------
Phase sP
DEG    HS    T(s)    p(s/deg)    Geom     T*(sec)
--------------------------------------------------
30.0  10.0   372.79   8.952    7.225e-05  1.20
32.5  10.0   394.76   8.787    6.531e-05  1.20
35.0  10.0   416.51   8.644    7.135e-05  1.21
37.5  10.0   437.90   8.502    7.408e-05  1.23
40.0  10.0   458.90   8.363    7.045e-05  1.24
42.5  10.0   479.47   8.096    6.622e-05  1.25
45.0  10.0   499.59   7.967    6.889e-05  1.27
50.0  10.0   538.50   7.595    6.187e-05  1.30
55.0  10.0   575.60   7.247    5.624e-05  1.34
60.0  10.0   610.87   6.916    5.552e-05  1.38
65.0  10.0   644.30   6.501    5.422e-05  1.41
70.0  10.0   675.94   6.108    4.676e-05  1.46
75.0  10.0   705.77   5.825    4.671e-05  1.51
80.0  10.0   733.74   5.373    4.598e-05  1.55
85.0  10.0   759.84   5.024    4.072e-05  1.60
90.0  10.0   783.98   4.685    3.255e-05  1.65```