Question

A sedimentary basin has a thickness of \(7 \mathrm{~km}\). Assuming that the crustal stretching model is applicable and that \(h_{c c}=35 \mathrm{~km}, \rho_{m}=3300 \mathrm{~kg} \mathrm{~m}^{-3}\), \(\rho_{c c}=2700 \mathrm{~kg} \mathrm{~m}^{-3},\) and \(\rho_{s}=2450 \mathrm{~kg} \mathrm{~m}^{3},\) determine the stretching factor. A MATLAB code for solving this problem is given in Appendix \(D\).

Short answer

Use MATLAB to numerically solve for the stretching factor \( \beta \). This involves substituting given values into an isostatic balance equation.

Step-by-step solution

Understand the Problem and Parameters

The problem involves finding the stretching factor in a sedimentary basin using the parameters given. Critical parameters include the thickness of the sedimentary basin, crustal thickness, and densities of different layers (mantle, crust, and sediments). These parameters are typically used in isostasy and lithospheric stretching models.

Define the Stretching Factor (β) in Crustal Stretching Model

In the context of crustal stretching, the stretching factor \( \beta \) is defined as the ratio of the original thickness of the crust \( (h_{cc}) \) to the final thickness of the crust after stretching. The equation for stretching factor \( \beta \) is expressed as follows:\[\beta = \frac{L_{new}}{L_{original}}\]where \( L_{new} \) is the new thickness including the stretched crust and sediment, and \( L_{original} = h_{cc} \) is the original thickness.

Use Isostatic Balance Equation

The isostatic condition for the sediments overlaying a stretched crust is given by:\[\rho_{cc} \cdot (h_{cc} - \frac{L_{new}}{\beta}) + \rho_s \cdot S = \rho_m \cdot \left( L_{new} - \frac{L_{new}}{\beta} \right)\]where \( S \) is the thickness of sediments (7 km), \( \rho_{cc} \), \( \rho_s \), and \( \rho_m \) are the given densities of the crust, sediments, and mantle respectively.

Solve the Equation for β

We need to substitute all given parameters into the isostatic balance equation and solve for \( \beta \). The equation modifies to:\[2700 \times (35 - \frac{L_{new}}{\beta}) + 2450 \times 7 = 3300 \times (L_{new} - \frac{L_{new}}{\beta})\]This complex equation involves some algebra involving \( \beta \). Solving it typically will require numerical methods or software like MATLAB as suggested. You can use iterative methods or built-in function solvers in MATLAB.

Obtain Numerical Solution using MATLAB

Given the complexity of the equation, MATLAB provides the environment to solve it numerically. Use the provided MATLAB code or create one using functions such as `fsolve` to find the value of \( \beta \). Ensure all parameters are correctly inserted and check the code for proper implementation of the equation.

Key concepts

Sedimentary Basin Thickness

Understanding sedimentary basin thickness is crucial in geophysics, especially when analyzing crustal stretching. Sedimentary basins are areas where sediment collects over time, often resulting in thick layers that can range over kilometers. In our exercise, we have a sedimentary basin thickness of 7 km. This measurement is significant when calculating the stretching factor of the crust, as it provides information about the amount of sediment that has accumulated and therefore must be balanced in the modeling equations.

Crucially, the sedimentary thickness plays a role in the isostatic balance of the crust, as it affects how the weight of accumulated materials is distributed over the underlying crust and mantle. This can influence the mechanics of crustal stretching and subsequently the formation of the basin itself. Understanding its thickness is, therefore, a primary step in any geophysical analysis of a sedimentary basin.

In a modeling scenario, such as the described exercise, sedimentary basin thickness data feeds into equations that describe the balance of forces along the lithosphere. This thickness, along with initial crustal thickness and densities, allows us to derive formulations that predict changes in crustal structure, such as the stretching factor (β), using crustal stretching models and isostatic principles.

Isostatic Balance Equation

The isostatic balance equation is key to understanding how the Earth's lithosphere naturally maintains equilibrium. Isostasy governs the buoyancy of the Earth's crust floating on the denser, semi-fluid mantle, much like an iceberg on the ocean.

In our scenario, this equation accounts for different layers' densities: the crust, sediments, and mantle. It's visually affirmed in the equation:
\[\rho_{cc} \cdot (h_{cc} - \frac{L_{new}}{\beta}) + \rho_s \cdot S = \rho_m \cdot \left( L_{new} - \frac{L_{new}}{\beta} \right)\]
where:

• \(\rho_{cc}\) is the density of the crustal column.


• \(h_{cc}\) is the original crustal thickness.


• \(\rho_s\) is the density of sediments.


• S is the sediment thickness (in our case, 7 km).


• \(\rho_m\) is the density of the mantle.

The balance equation signifies a dynamic equilibrium between the added sediment weight and displacement within the crust and mantle. By solving the equation for the stretching factor \(\beta\), we can better understand the degree to which the crust has undergone extension during basin development. This calculation is intrinsic to determining how geological features develop over geological time scales.

Numerical Solution using MATLAB

Numerical solutions are often the go-to when dealing with complex equations like the isostatic balance equation. MATLAB is a powerful computational tool that can handle the calculations required for our exercise efficiently.

In this exercise, the primary task is to solve for the stretching factor \(\beta\) using MATLAB. MATLAB's numerical solvers, such as `fsolve`, accommodate non-linear equations and can iterate to find solutions that represent physical reality. The process typically involves defining the equation within MATLAB's script environment, inputting the known parameters (densities, crustal, and sediment thicknesses), and executing the solver.

The iterative nature of `fsolve` or similar functions allows it to test multiple values, refining estimates, and converging towards an accurate \(\beta\). This approach is particularly essential here, as analytic solutions might not be feasible due to the equation's complexity. Hence, mastering MATLAB for these purposes is pivotal for geoscientists and engineers who need to bridge the gap between theory and real-world applications in crustal stretching models.