Question

A sedimentary basin has a thickness of \(4 \mathrm{~km}\). Assuming that the crustal stretching model is applicable and that \(h_{c c}=35 \mathrm{~km}, \rho_{m}=3300 \mathrm{~kg} \mathrm{~m}^{-3}\), \(\rho_{c c}=2750 \mathrm{~kg} \mathrm{~m}^{-3},\) and \(\rho_{s}=2550 \mathrm{~kg} \mathrm{~m}^{-3},\) determine the stretching factor. A MATLAB code for solving this problem is given in Appendix \(D\).

Short answer

The stretching factor \( \beta \) is approximately 1.1567.

Step-by-step solution

Understanding the given values

We are provided with several constants:- Thickness of the basin: \( h_s = 4 \text{ km} \)- Thickness of the continental crust: \( h_{cc} = 35 \text{ km} \)- Mantle density: \( \rho_m = 3300 \text{ kg/m}^3 \)- Continental crust density: \( \rho_{cc} = 2750 \text{ kg/m}^3 \)- Sediment density: \( \rho_s = 2550 \text{ kg/m}^3 \).

Write the equation for stretching factor

The stretching factor, \( \beta \), can be obtained from the crustal thinning equation:\[ \beta = \frac{h_{cc} + \frac{h_s(\rho_s - \rho_m)}{\rho_{cc} - \rho_m}}{h_{cc}} \].

Calculate the thinning due to sediment loading

Substitute the known values into the thinning equation:\[ \beta = \frac{35 + \frac{4(2550 - 3300)}{2750 - 3300}}{35} \].

Perform the calculation

Perform the arithmetic operation inside the fraction:\[ \beta = \frac{35 + 4 \times \frac{-750}{-550}}{35} = \frac{35 + 4 \times \frac{750}{550}}{35} \].Complete the multiplication and division to simplify: \[ \beta = \frac{35 + 4 \times 1.3636}{35} \].

Final computation

Compute the final value for \( \beta \):\[ \beta = \frac{35 + 5.4544}{35} = \frac{40.4544}{35} \approx 1.1567 \].

Key concepts

Sedimentary Basin

A sedimentary basin is a low-lying geographical area where sediments accumulate over time. These regions are crucial for understanding geological and environmental processes.
Sediments in basins are often derived from weathering and erosion of surrounding rocks, then transported by water, wind, or ice. The accumulation creates layers, storing historical records about the Earth's past climate and biological activity.
Understanding sedimentary basins is essential for natural resource exploration. They can be rich in oil, natural gas, and minerals. In geophysics, analyzing sediment accumulation helps us understand crustal movements and the effects of crustal stretching, as discussed in the given exercise.

Thinning Equation

The thinning equation describes the relationship between the crust's thickness before and after sediment deposition. It's vital for calculating the stretching factor, denoted by \( \beta \), to understand crustal deformation.
The equation given is:

• \( \beta = \frac{h_{cc} + \frac{h_s(\rho_s - \rho_m)}{\rho_{cc} - \rho_m}}{h_{cc}} \)

Here:

• \( h_{cc} \) is the original crust thickness.
• \( h_s \) denotes sediment thickness.
• \( \rho_m \), \( \rho_{cc} \), and \( \rho_s \) are densities of the mantle, continental crust, and sediments respectively.

By understanding this equation, you can determine how the weight and distribution of sediments affect the Earth's crust, leading to changes such as subsidence or uplift.

Density Calculations

Density calculations are fundamental in geophysics, especially for modeling how various layers of the Earth interact under stress.
Each geological layer—sediments, crust, and mantle—has a different density, impacting crustal movement. High density means more mass in a given volume, influencing gravitational pull and pressure.
In the exercise, you perform specific calculations to substitute into the thinning equation. By calculating density differences between sediments and mantle rocks, you understand their impact on stretching. This informs interpretations of basin formation, tectonic activity, and potential resource locations.

MATLAB for Geophysics

MATLAB is a powerful tool for solving complex mathematical and physical problems, making it invaluable in geophysics. It allows for robust data analysis, modeling, and visualization.
In the exercise, MATLAB is used to implement the thinning equation computationally. This program handles large datasets and performs precise calculations quickly, minimizing human error.
Geophysicists often rely on MATLAB for simulations, predicting crustal behavior under various conditions. It also aids in visualizing geological processes, such as subsidence in sedimentary basins, providing clearer insights into Earth's dynamic systems.