Physics And Chemistry Of The Earth

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

Build a seismic velocity profile of Earth's mantle using the Herglotz-Wiechert method based on travel time observations. Analyze seismic wave travel times (P and S waves) to deduce velocity variations with depth, compare results with existing models, and interpret data considering the assumptions and limitations of the methods used.

Paper For Above instruction

Introduction

Understanding the Earth's internal structure, particularly the variation of seismic wave velocities with depth, is fundamental in geophysics. Seismology provides invaluable data through the analysis of seismic waves generated by earthquakes or artificial sources. By interpreting these data, scientists can infer the Earth's internal composition and layering. A prominent approach to this interpretation involves using travel time data of seismic waves, notably P (primary) and S (secondary) waves, and inverting these observations to develop velocity profiles of the Earth's mantle. This paper discusses the process of constructing such profiles using the Herglotz-Wiechert integral method, starting from simplified assumptions to more complex models that account for radial variations in seismic velocities.

Seismic Wave Propagation in a Homogeneous Earth

Initially, considering a simplified model where seismic velocities are constant throughout the Earth’s mantle provides foundational insight. The assumption of uniform velocity V simplifies the calculation of travel times and the derivation of the seismic ray path. In a spherical Earth, the seismic wave travels along a curved path characterized by the incident angle and the ray parameter p, which remains constant if the medium is homogeneous. The relation p = r sin i(r) / V(r) simplifies to p = R cos Δ / V in this case because the velocity V is constant, allowing us to express travel times proportionally to the angular distance Δ.

Specifically, the travel time T for a wave propagating along such a ray over an angular distance Δ can be derived from geometric considerations. The wave’s path length in terms of the angular distance relates directly to the velocity V and the incident angles involved. The homogeneous Earth model yields a straightforward formula relating travel time to Δ, which serves as a baseline for further comparison with more realistic heterogeneous models.

Variable Velocity Earth Model and the Herglotz-Wiechert Inversion

Realistic models must account for the fact that seismic wave velocities increase with depth due to changes in composition, temperature, and pressure. The variation V(r) influences the form of the travel time and the seismic ray path. The Herglotz-Wiechert method is particularly suited for regions where V(r) monotonically increases with depth, allowing the inversion of observed p(Δ) curves to deduce V(r).

The core concept is that the ray parameter p remains constant along the path, but the depth at which the wave bottoms out (radius rb) can be deduced from p and V(rb). The key integral relation (equation 4) connects the radius rb to the observed p(Δ) curve through the inverse hyperbolic cosine function integrated over the angular distance. This integral effectively accumulates the variations of p(Δ) and allows reconstruction of the velocity profile V(r) as a function of r, assuming V increases with depth.

The relation between p, rb, and V(rb) follows from the definition of the ray parameter, which indicates that the seismic wave’s azimuthal properties and velocity profile are interconnected. Using the observed p(Δ) curves, it is possible to numerically compute rb(Δ) and, consequently, the velocity profile V(rb) by differentiating and integrating the integral equation. This process involves discretizing the data and applying numerical integration techniques such as the trapezoidal rule.

Data Analysis and Velocity Profile Estimation

Employing observed travel time tables (from the ak135 model or similar), we can visualize travel time curves for P and S waves versus angular distance Δ. These curves often display deviations from the idealized constant velocity model, reflecting heterogeneities in Earth's mantle. From the observed travel times T(Δ), the ray parameter p(Δ) can be computed as p = dT/dΔ, expressed through numerical differentiation.

Graphic comparisons reveal that the constant velocity model approximates the data only in limited regions, emphasizing the need to incorporate velocity variations. The extracted p(Δ) curves serve as input for the Herglotz-Wiechert integral to estimate the bottoming radius rb for each Δ, further deriving V(rb). Plotting these V(rb) profiles against depth elucidates the Earth's velocity structure, showing how seismic velocities increase with increasing depth.

Constructing and Interpreting Velocity Profiles

Through numerical implementation—using programming languages like Python or MATLAB—one can compute rb(Δ) by evaluating the integral relations with observed p(Δ). Subsequently, the velocity at each rb can be calculated, generating a depth-dependent V(r) profile. These profiles are then plotted to visualize how seismic velocities vary with radius beneath the Earth's surface.

The resulting seismic velocity profiles are compared to standard models such as PREM (Preliminary Reference Earth Model). PREM provides a globally averaged velocity structure that incorporates many features of Earth's interior, including velocity discontinuities at mantle boundaries. By comparing computed profiles with PREM, we can assess the accuracy of the inversion process and the assumptions underlying the Herglotz-Wiechert method.

Conclusion

The process of constructing Earth’s mantle seismic velocity profiles involves a sequence of steps starting from travel time data, deriving p(Δ) curves, and applying the Herglotz-Wiechert integral. While assumptions about increasing velocity with depth simplify the inversion, real Earth heterogeneities pose challenges that require refined models and numerical techniques. When successfully implemented, this approach yields valuable insights into the Earth's internal structure, aligning well with global models like PREM. Such analyses enhance our understanding of mantle composition, thermal state, and dynamic processes driving plate tectonics and mantle convection.

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