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Seismic source model -- Brune model ( ω^2 model)

news 来源：原创 2025/5/24 6:22:14

The Brune model, proposed by James N. Brune in his seminal 1970 paper “Tectonic stress and the spectra of seismic shear waves from earthquakes”, was developed to explain the observed spectral characteristics of seismic waves generated by earthquakes. Here’s a detailed look at the background and the derivation process of the Brune model:

🔷 1. Background

📌 Motivation

In the 1960s and early 1970s, seismic observations showed that:

Displacement spectra of earthquakes typically had a flat level at low frequencies and decayed at high frequencies.
This spectral shape seemed consistent across many events, suggesting a universal behavior tied to the source of the earthquake rather than path or site effects.

📌 Objective

Brune aimed to derive a simple analytical model for the earthquake source that could:

Match observed spectral shapes,
Be used to estimate source parameters like seismic moment, stress drop, and source radius,
Be suitable for engineering and seismological applications.

🔷 2. Physical Assumptions

Brune made the following idealizations:

Point Shear Dislocation Source: A small circular fault area experiences sudden slip.
Isotropic Medium: Homogeneous, elastic medium.
Radiation of S-Waves: Focus on far-field S-wave displacement.
Instantaneous Stress Drop: The stress changes instantly, and slip occurs rapidly.
No Attenuation or Scattering: Only source effects are considered.

🔷 3. Derivation Process

🔹 A. Time-Domain Representation

Brune assumed that the slip on the fault increases over time and then stops, producing a time-dependent moment release M(t).

A simple model of the source time function $\dot{u}(t)$ is:

$\dot{u}(t) = u_0 \cdot e^{-t/\tau}$

Where:

u_0: final slip,
\tau: rise time (duration of slip).

This slip function gives a displacement u(t) and associated ground motion that can be Fourier transformed.

🔹 B. Frequency-Domain Spectrum

Taking the Fourier transform of the above slip rate function yields:

$U(f) \propto \frac{1}{1 + (2\pi f \tau)^2}$

This is a low-pass filter shape, where:

At low frequencies $(f \ll 1/\tau)$ , $U(f) \approx \text{constant}$ ,
At high frequencies $(f \gg 1/\tau)$ , $U(f) \propto f^{-2}$ .

This matches the observed spectra: flat at low frequencies, decaying at high frequencies.

🔹 C. Defining the Corner Frequency

The corner frequency f_c is defined as:

$f_c = \frac{1}{2\pi\tau}$

It separates the flat (low-frequency) and decaying (high-frequency) portions of the spectrum.

🔹 D. Final Form of the Brune Spectrum

$U(f)=\frac{\Omega_0}{1 + (f / f_c)^2}$

Where:

Ω_0 (low-frequency plateau) is related to the seismic moment M_0:

$\Omega_0 \propto \frac{M_0}{\rho R \beta^3}$
- \rho: density,
- R: hypocentral distance,
- \beta: shear-wave velocity.

🔷 4. Stress Drop and Source Radius

Brune further related f_c to source radius r:

$r = \frac{k \cdot \beta}{2\pi f_c}, \quad \text{with } k \approx 0.32$

And the stress drop \Delta\sigma is estimated as:

$\Delta\sigma = \frac{7}{16} \cdot \frac{M_0}{r^3}$

This connects spectral observations to physically meaningful earthquake parameters.

🔷 5. Importance of the Brune Model

The Brune model was revolutionary because it:

Provided the first practical way to estimate source parameters from seismic data,
Explained the universality of earthquake spectra,
Became the foundation for many developments in engineering seismology, seismic hazard analysis, and strong-motion modeling.

The Brune model laid the groundwork for earthquake source spectrum modeling, but its simplifying assumptions—such as a point source, isotropy, and instantaneous rupture—limit its realism. Since Brune’s 1970 work, many researchers have developed extensions and improvements to address these limitations and to better capture the complexity of real earthquake sources.

🔷 6. Major Developments and Improvements of the Brune Model

1. Extended Source Models

Brune’s model is a point source model. Real earthquakes rupture over finite fault areas.

✅ Improvements:

Haskell’s finite fault model (1964–1972): Introduced multiple subfaults and rupture propagation.
Boatwright (1978): Modeled the source as a circular rupture with finite area and a more complex slip distribution.
Somerville et al. (1999): Developed stochastic finite fault models that account for heterogeneous slip, asperities, and rupture directivity.

2. Kinematic vs. Dynamic Models

✅ Kinematic models:

Assume a prescribed slip function and rupture velocity.
Popular for ground motion simulation (e.g., Graves and Pitarka models).

✅ Dynamic models:

Solve elastodynamic equations based on friction laws and stress conditions.
Can simulate self-consistent rupture processes (e.g., Kaneko and Lapusta, 2010s).

3. Spectral Modifications and Two-Corner Models

To better match observed spectra, some models include more complex spectral shapes.

✅ Two-corner frequency models:

Proposed by Atkinson and Silva (2000s) for large earthquakes.
Introduce a second corner frequency to account for large-scale rupture characteristics or multiple stages of rupture.

$U(f) \propto \frac{1}{(1 + (f/f_{c1})^2)(1 + (f/f_{c2})^2)}$

4. Stochastic Source Models

✅ Anderson and Hough (1984)** and Boore (2003):

Added random variability to account for scattering, heterogeneous rupture, and site effects.
Form the basis of stochastic ground motion simulations used in hazard analysis.

5. Stress Drop Scaling and Depth Dependence

Brune assumed a fixed stress drop, but:

Observations show that stress drop varies with depth, tectonic setting, and earthquake size.
Studies (e.g., Allmann and Shearer, 2009) have explored these dependencies.
Self-similarity vs. breakdown of self-similarity remains a debated topic.

6. Anisotropic and Non-Circular Source Models

To account for fault geometry, researchers have modeled:

Elliptical rupture areas (e.g., Sato and Hirasawa, 1973),
Ruptures on dipping faults or multi-segment faults,
Rake variation, fault roughness, and off-fault plasticity (e.g., Dunham et al., 2011).

7. Frequency-Dependent Radiation Pattern and Directivity

Brune’s model assumes isotropic radiation, but real earthquakes exhibit directivity—higher amplitudes in the rupture direction.
Spudich and Frazer (1984) and later studies introduced models that incorporate frequency-dependent directivity effects.

8. Spectral Decomposition and Empirical Green’s Functions (EGFs)

EGF method allows empirical removal of path and site effects, isolating the source spectrum.
Helps refine estimates of M_0, f_c, and stress drop without relying on simplified assumptions of Brune’s model.

🔷 Summary Table of Key Improvements

Aspect	Brune Model	Improvement
Source size	Point source	Finite-fault models
Slip	Uniform	Heterogeneous (asperities)
Geometry	Circular	Elliptical, segmented
Rupture	Instantaneous	Finite rise time, directivity
Spectrum	One-corner	Two-corner, broadband models
Variability	Deterministic	Stochastic models
Radiation	Isotropic	Anisotropic, directionally dependent