Physique Et Chimie De La Terre: Physics And Chemistry Of The

Physique Et Chimie De La Terre Physics And Chemistry Of The Earth 20

Build a model of the P and S wave velocity profiles in the Earth's mantle using travel times from observations and the Herglotz-Wiechert method. Analyze seismic ray data, compare models with observations, and estimate radial velocity profiles accordingly.

Paper For Above instruction

Understanding the Earth's internal structure through seismic wave analysis has been a cornerstone of geophysics. The method described in this assignment—centered on the Herglotz-Wiechert technique—provides a systematic approach to deriving the Earth's mantle velocity profiles for P-waves and S-waves. This paper details the steps involved, from theoretical derivations in simplified models to practical data analysis and interpretation, illustrating how seismic observations inform models of Earth's interior.

Initially, the simplified constant velocity model serves as a foundation for understanding seismic wave travel times and their relationships to Earth's geometry. Assuming uniform wave velocity V within the mantle, the travel time T for seismic waves traveling along a great-circle path from a source S to a surface point A can be mathematically expressed. In this simplified context, the travel time T is proportional to the angular distance ∆ divided by the velocity, i.e., T(∆) = (R ∆) / V, where R is Earth's radius. This relation assumes the wave travels along a straight path through a homogenous medium, and the incidence angle i aligns with the geometric considerations straightforwardly, leading to a simple expression for the ray parameter p, as p = R V * cos(∆) / 2. Such a model provides an initial understanding but does not account for radial variations of velocities, which are more realistic within Earth's mantle.

Moving beyond this simplification necessitates considering radially varying velocities V(r). The Herglotz-Wiechert method hinges on the concept that the ray parameter p remains constant along a seismic ray, enabling the inversion from observed travel times to the Earth's velocity structure. The key relation in this context links the bottoming radius rb (where the ray reaches its maximum depth) to the integral involving the variation of the ray parameter p with the angular distance ∆, particularly through the integral formula: rb(∆) = R exp(- (1/π) ∫ from 0 to ∆ of arcosh(p(•) / p(•)) d∆'). This integral, computed numerically, provides a means to infer the depth at which the wave bottoms out, ultimately leading to estimates of V(rb).

The practical application involves analyzing travel time data from the ak135 global model, which offers observed travel times as a function of angular distance for P and S waves. By plotting these travel times against ∆, one can compare observed data with predictions of the constant velocity model. Subsequently, calculating the ray parameters p from these data (p = dT/d∆) yields p(∆) curves for both phases. These curves are instrumental in applying the Herglotz-Wiechert integral formula to infer depth-dependent velocities. Computationally, numerical integration techniques such as trapezoidal or Simpson’s rule are employed to evaluate the integral, facilitating the derivation of the velocity profiles V(r).

The ultimate goal is to reconstruct the Earth's mantle velocity profiles for P and S waves by transforming the computed rb(∆) into radial velocity functions V P(r) and V S(r). The process involves translating the bottoming radius rb into depth r (since r = rb / R * 6371 km) and associating each depth with the corresponding seismic velocity. These results are then plotted to visualize the Earth's radial velocity structure, which can be compared with established models like PREM. Such comparisons validate the inversion procedure, highlight the velocity variations with depth, and contribute to understanding the Earth’s composition and thermal state.

References

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