The Herglotz-Wiechert method and Earth’s mantle seismic velocities profiles

The goal of this problem is to build a model of the P and S wave velocity profiles in the Mantle, from travel times tables build from observations. To do this, we will use the Herglotz-Wiechert method, a method developed by Gustav Herglotz and Emil Wiechert at the beginning of the twentieth century.

We consider a seismic ray going from point S to point A, as depicted on figure 1. We denote by ∆ the angular distance of travel (i.e. the angle ŜCA), and by T (∆) the travel time of the seismic wave as a function of angular distance. We recall that in spherical geometry the ray parameter is defined as

p = r sin i(r) V (r) , (1)

and is constant along a given ray. Here r is the distance from the center of the Earth, i(r) is the incidence angle (i.e. the angle between the ray and the vertical direction at a given r), and V (r) is the wave velocity. We denote by R = 6371 km the radius of the Earth.

∆ d∆

R

p

p + dp

i

A

A’B

S

C

rb

Figure 1 – Two rays coming from the same source S with infinitesimally different ray parameters p and p + dp. Their angular distances of travel are ∆ and ∆+ d∆, and their travel-times are T and T + dT . The line going through points A and B is perpendicular to both rays.

1 Constant velocity model

Let us first assume that the wave velocity V does not vary with depth. 1. Draw on a figure the ray going from a source S to a point A of the surface, without any reflexion.

This ray could represent either the P or S phase. 2. Find the expression of the travel time T along this ray as a function of ∆. 3. Find the expression of the incidence angle i of the ray at point A as a function of the epicentral distance ∆, and then show that the ray parameter is given by

p = R V

cos ∆

2 . (2)

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2 Linking p to T and ∆

We now turn to a more realistic model and allow for radial variations of the waves velocities.

4. By considering two rays coming from the same source with infinitesimally different ray parameters p and p + dp, and travel times T and T + dT (Figure 1a), demonstrate that

p = dT d∆

. (3)

Hints : (1) Since the two rays are very close, the arcs connecting A to A’, A to B, and B to A’ can be approximated as straight lines. (2) Show first that the segment AB is part of a wavefront. What does it imply for the times of arrivals at points A and B ?

5. Check that the expressions of p and T found for the constant velocity model are consistent with eq. (3).

3 Travel time curves and estimate of the p(∆) curves You will find on Chamillo a file containing travel time tables obtained from the global Earth’s seis-mological model ak135 (either a text file, AK135tables.txt, or an Excel spreadsheet, AK135tables.xlsx). The files contain three columns :

— the first gives the angular distance of travel ∆ (in °) ;

— the second column gives the travel time (in seconds) of the P phase (i.e. a P -wave travelling in the mantle without any reflexion) ;

— the third column gives the travel time (in seconds) of the S phase (i.e. a S-wave travelling in the mantle without any reflexion).

6. Travel time curves :

(a) Using the programming language of your choice (Python, Matlab/Octave, Excel, …), make a plot showing the travel times of the P and S waves as a function of ∆.

(b) Compare with the prediction of the constant velocity model. Can you find a P -wave velocity that allows for a good agreement between the constant velocity model and the observed travel times ? Comment.

7. p(∆) curves :

(a) From the travel time tables, compute the ray parameter p for each value of ∆, for the P and S phases.

(b) Make a plot of p as a function of ∆, for the P and S phases.

(c) Compare with the prediction of the constant velocity model. Comment.

4 Estimating the Mantle’s VP and VS profiles using the Herglotz- Wiechert method

The Herglotz-Wiechert method is an ’inversion’ method allowing to determine a vertical seismic velocity profile from a p(∆) curve obtained from observations. The method only works in regions where the velocity increases with depth, and its use is therefore restricted to regions without low-velocity zones.

We denote by rb the radius of the bottoming point of the ray (figure 1a), and by V (rb) the wave velocity at r = rb.

8. From the definition of the ray parameter p (eq. (1)), find a relation between p, rb, and V (rb).

The Herglotz-Wiechert method is based on the following formula, which links the radius rb of the bottoming point of a ray of angular distance ∆ to an integral involving the ray parameter p :

rb(∆) = R exp(− 1

π ∫

∆

0 arcosh(p(∆

′) p(∆) )d∆′) . (4)

(Note that arcosh(x) = ln (x + √ x2 − 1).)

9. Explain qualitatively how one can use this formula together with the results from the previous questions to estimate the radial profiles VP (r) and VS(r).

10. Given the p v.s. ∆ tables you have obtained on question 7, write a program allowing you to

(a) compute rb as a function of ∆ , using equation (4),

(b) and then compute the seismic velocity VP (rb) at each rb. (Please hand back your program with your homework.) Hint : To compute the integral, you can either use a built-in integration function from you pro- gramming language, or write a simple integration program (either the rectangular or trapezoidal rule can be used).

11. Use your program to compute VP and VS as functions of r, and make a plot of the resulting velocity profiles.

12. Compare your results with P and S velocity models you can find on the internet (for example from the PREM model).

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