Seismic source model - stochastic kinematic model (kappa-inverse-square)

🔍 Overview of the κ⁻² Kinematic Model

The κ⁻² model is a kinematic seismic source model used to describe the heterogeneous spatial distribution of slip on an earthquake fault. It was originally proposed to explain the observed high-frequency behavior of seismic waves, particularly the ω⁻² (omega-inverse-square) decay of displacement spectra.

In this model:

The fault slip is described as a random field.
The spatial Fourier transform of slip amplitude decreases with the inverse square of wavenumber k.
The model is kinematic: it prescribes slip and rupture propagation, rather than deriving it from stress and strength (as in dynamic models).

📚 Background

Observations show that:

Far-field displacement spectra of earthquakes decay as $\omega^{-2}$ at high frequencies.
This implies that the source spectrum must also decay in a similar fashion.
To produce such spectra, the spatial slip distribution on the fault must contain both long-wavelength and short-wavelength components, with decreasing energy at higher wavenumbers.

To reproduce this behavior, the κ⁻² model imposes a slip distribution with a Power Spectral Density (PSD) that decays as $1/k^2$ , where k is the spatial wavenumber.

🧮 Derivation Process of the κ⁻² Model

🔹 Step 1: Define the Power Spectral Density (PSD)

We assume that the slip distribution s(x, y) over the fault is a random spatial field with the following 2D PSD:

$P(k) = |S(k)|^2 \propto \frac{1}{k^2 + k_c^2}$

P(k) is the power spectral density.
S(k) is the 2D spatial Fourier transform of the slip distribution.
$k = \sqrt{k_x^2 + k_y^2}$ is the radial spatial wavenumber.
k_c is a corner wavenumber that smooths the singularity at k = 0 and limits the total slip energy.

✅ This is the key assumption of the κ⁻² model: the spatial slip PSD decays with the square of the wavenumber.

This form is directly inspired by the Brune ω⁻² spectral model, where the amplitude spectrum decays like:

$|U(\omega)|^2 \propto \frac{1}{\omega^2 + \omega_c^2}$

By analogy, the spatial domain uses k instead of angular frequency \omega.

🔹 Step 2: Generate Slip in Frequency Domain

To create a synthetic slip distribution:

Define a 2D grid of wavenumbers (k_x, k_y).
Compute $k = \sqrt{k_x^2 + k_y^2}.$
Assign random complex values to S(k_x, k_y) with magnitudes scaled by $1/\sqrt{k^2 + k_c^2}$ , and random phases.

That is:

$S(k_x, k_y) = \frac{A(k_x, k_y)}{\sqrt{k^2 + k_c^2}}$

where A(k_x, k_y) is a complex random number with unit variance (often generated as Gaussian noise with random phase).

🔹 Step 3: Inverse Fourier Transform

Apply the 2D inverse Fourier transform to obtain the spatial slip distribution:

$s(x, y) = \iint S(k_x, k_y) e^{i(k_x x + k_y y)} \, dk_x \, dk_y$

This yields a spatially heterogeneous slip distribution consistent with the assumed PSD.

🔹 Step 4: Rupture Propagation and Ground Motion Simulation

To complete the kinematic model:

Assign a rupture velocity and rise time for each point on the fault.
Use the slip field and rupture timing to compute synthetic ground motions (via Green’s functions or numerical methods).

This slip field will naturally lead to broadband seismic radiation, including realistic high-frequency content.

🧠 Why the 1/k^2 PSD?

The PSD controls how much energy is in large vs. small-scale features.
A 1/k^2 fall-off means that:
- Large-scale features dominate (e.g. main patches of slip),
- Small-scale variations are present but weaker.
This reflects real observations from earthquakes.
It leads to a far-field displacement spectrum that decays like \omega^{-2}, consistent with data.

📌 Summary

Component	Description
Model Type	Kinematic (prescribed slip)
Key Assumption	Spatial slip PSD decays as 1/(k^2 + k_c^2)
Wavenumber k	Spatial frequency on the fault plane
PSD Interpretation	Controls roughness and energy of slip heterogeneity
Output	Heterogeneous slip → broadband ground motion
Benefit	Matches observed ω⁻² spectral decay in seismic waves

Here’s a detailed explanation of Step 1 of the κ⁻² kinematic source model, especially why the Power Spectral Density (PSD) is modeled as proportional to 1/k², in English:

🔍 Step 1: Slip Distribution and the Origin of the 1/k² Power Spectrum

🎯 Goal

We aim to model the spatial heterogeneity of slip on a fault during an earthquake, which is known to be irregular rather than uniform. In the κ⁻² kinematic source model, the spatial slip spectrum follows:

$|S(k)|^2 \propto \frac{1}{k^2}$

Where:

S(k) is the 2D Fourier transform of the slip distribution.
|S(k)|^2 is the Power Spectral Density (PSD).
k is the wavenumber, i.e., spatial frequency.

1. 🔬 Why Use a 1/k^2 PSD?

📌 Empirical Motivation

Strong motion observations show that acceleration spectra often decay as f^{-2} at high frequencies (the omega-square model).
This spectral decay can’t be explained by smooth or uniform slip. It suggests that slip is spatially rough or heterogeneous.
Inversion results of real earthquake slip distributions show clear heterogeneities—slip varies across the fault, forming “asperities” or “patches.”

These empirical features motivate the use of a random field with a specific spectral decay, and 1/k² is found to match real data well.

2. 🧠 From Spatial Autocorrelation to 1/k² Spectrum

Step A: Slip as a Random Field

We model the slip s(x, y) as a stationary random field on the fault plane.

By the Wiener–Khinchin theorem, the PSD P(k) is the 2D Fourier transform of the autocorrelation function R(\rho):

$P(k) = \int R(\rho) e^{-i k \cdot \rho} d\rho$

Step B: Exponential Correlation Assumption

A typical spatial autocorrelation function is:

$R(\rho) = \sigma^2 e^{-\rho/a}$

Where:

$\sigma^2$ is the variance,
a is the correlation length.

The 2D Fourier transform of this yields:

$P(k) = \frac{C}{1 + (k a)^2}$

So when $k \gg 1/a$ , we get:

$P(k) \approx \frac{1}{k^2}$

This gives us the desired κ⁻² spectrum.

3. 🔧 Physical Interpretation

Low wavenumbers (small k) capture large-scale slip variation.
High wavenumbers (large k) represent small-scale roughness.
A spectrum with $P(k) \propto 1/k^2$ implies:
- Moderate spatial variability,
- Realistic representation of observed heterogeneous slip,
- Physically reasonable energy decay at small scales.

4. 📐 Constructing the Random Slip Field

In practice, the slip field is built in the frequency domain as:

$\tilde{S}(k) = \sqrt{P(k)} \cdot e^{i \phi(k)} \quad \text{where} \quad P(k) = \frac{C}{1 + (k/k_c)^2}$

$\phi(k) \sim \text{Uniform}[0, 2\pi]$ is a random phase,
$k_c = 1/a$ is a characteristic wavenumber (corresponds to the correlation length),
The field s(x, y) is obtained by inverse Fourier transform (IFFT).