The task described in the problem involves generating samples from a statistical model of the Earth's subsurface, often represented with horizontal structures, using specific software. Here’s a breakdown of the problem and hints:

Problem Breakdown:

• Domain Setup:
  The unit square $\Omega = [0,1] \times [0,1]$ is discretized into a 100x100 grid. This domain is split into two regions, $\Omega_1$ and $\Omega_2$, based on the equation of a straight line $y = mx + c$.

• Random Variables:

  1. Slope $m$ and intercept $c$ are random variables: $m \sim \mathcal{N}\left(0, \frac{1}{2}\right), \quad c \sim \mathcal{N}\left(\frac{1}{2}, \frac{1}{8}\right)$
  2. Subdomain variables $\theta_1$ and $\theta_2$: $\theta_1 \sim \mathcal{N}(0, \Gamma_1), \quad \theta_2 \sim \mathcal{N}(0, \Gamma_2)$ where $\Gamma_1$ and $\Gamma_2$ are exponential squared covariance matrices, with variance $\sigma = 1$ and correlation length $\ell = 2$.

Task:

• Using a suitable software (Python is commonly used for these types of problems with libraries like numpy and matplotlib), you need to generate and plot 6 samples from the distribution $\pi(\theta_1, \theta_2, m, c)$.

Hints:

1. The samples should look similar to the images provided in the figure.
2. Factorizing the distribution using conditional and marginal distributions might help.
3. You could start by using a coarser discretization (e.g., a smaller grid) for debugging purposes.

How to Approach:

1. Generate $m$ and $c$:

   • These are normally distributed, so you can use a random number generator from a normal distribution to sample from these values.

2. Define $\Omega_1$ and $\Omega_2$:

   • Once you have $m$ and $c$, the regions $\Omega_1$ and $\Omega_2$ are defined as two parts of the unit square divided by the line $y = mx + c$.

3. Sample $\theta_1$ and $\theta_2$:

   • Use a multivariate normal distribution with covariance matrices $\Gamma_1$ and $\Gamma_2$, which can be modeled using exponential kernels. The exponential squared covariance matrix can be created using: $\Gamma_{ij} = \exp\left(-\frac{\|x_i - x_j\|^2}{2\ell^2}\right)$ This requires constructing the grid and calculating pairwise distances for all grid points.

4. Plotting:

   • Use a plotting library (such as matplotlib in Python) to create the plots of the sampled fields over the domain.

Would you like me to walk you through specific code examples for any of these steps?

Here are some related questions to expand on this topic:

1. How can we construct the covariance matrix for $\theta_1$ and $\theta_2$ using an exponential kernel?
2. What software options are available for generating random samples from multivariate normal distributions?
3. How do correlation length $\ell$ and variance $\sigma$ affect the sample generation?
4. How can conditional and marginal distributions simplify the factorization of $\pi(\theta_1, \theta_2, m, c)$?
5. What would be the effect of using different grid sizes in the discretization?

Tip: When working with large covariance matrices, you might want to consider using Cholesky decomposition to efficiently sample from multivariate normal distributions.