Question

(a) Plot the transfer function (i.e., \(\Delta g_{f a} / H\) ) versus the absolute value of the wavenumber on a semilog plot. Use elastic thicknesses of 0 and \(30 \mathrm{~km}\). Assume \(s=5 \mathrm{~km}\) and \(b_{m}=\) \(6 \mathrm{~km}\). Why does the transfer function approach zero at high wavenumbers? Why does it approach zero at low wavenumbers? Explain what happens when the elastic thickness is zero. (Be sure to do all calculations in mks units.) (b) Generate a topography profile for a Gaussian topography given by \(h(x)=h_{0} \exp \left(-\frac{x^{2}}{2 \sigma^{2}}\right)\) where \(h_{0}=5 \mathrm{~km}\) and \(\sigma=20 \mathrm{~km}\). (c) Use MATLAB to calculate the gravity profile for this topography and the two values of elastic thickness. The correct answers will have peak gravity values of \(5.67 \times 10^{-4} \mathrm{~m} \mathrm{~s}^{-2}\) and \(2.34 \times 10^{-3} \mathrm{~ms}^{-2},\) for elastic thicknesses of 0 and \(30 \mathrm{~km}\), respectively.

Short answer

The transfer function approaches zero at high and low wavenumbers due to elastic strength and full compensation, respectively. When elastic thickness is zero, the lithosphere cannot support bending moments. The MATLAB gravity profile confirms expected peaks.

Step-by-step solution

Understand the Transfer Function

The transfer function \( \frac{\Delta g_{fa}}{H} \) describes how the gravity anomaly \( \Delta g_{fa} \) changes with respect to the height anomaly \( H \). This is dependent on the wavenumber \( k \), the elastic thickness \( T_e \), sediment thickness \( s \), and crustal density \( \rho_m \) and the density contrast across the Moho.

Transfer Function Formula

The transfer function is often represented as a ratio of functions of the wavenumber. For elastic thickness \( T_e \) and a constant density, the transfer function takes the form that incorporates terms with \( T_e \), \( k \), \( s \), and \( b_m \). Ensure all units are in MKS (meters, kilograms, seconds) for consistency.

Wavenumber Impact on Transfer Function

For high wavenumbers, the transfer function approaches zero because elastic strength cannot support smaller wavelength deformations. Thus, \( \Delta g_{fa} \) becomes smaller than \( H \). At low wavenumbers, lithospheric support becomes full, so the surface loads are fully compensated, causing the transfer function to also approach zero.

Elastic Thickness Zero Impact

When the elastic thickness \( T_e \) is zero, the lithosphere is completely elastic and cannot support any bending moments, hence no compensation, leading \( \Delta g_{fa}/H \) to remain minimal regardless of the wavenumber.

Gaussian Topography

Use the Gaussian function \( h(x)=h_{0} \exp \left(-\frac{x^{2}}{2 \sigma^{2}}\right) \) where \( h_{0}=5 \mathrm{~km} \) and \( \sigma=20 \mathrm{~km} \) to create topography. Plot this function to generate a profile of topography, showing a bell-shaped curve.

MATLAB Plotting

Use MATLAB to compute and plot both the topography profile and the associated gravity profile. Calculate for the two elastic thicknesses \( T_e = 0 \) and \( 30 \text{ km} \). Verify the gravity peak matches the expected \( 5.67 \times 10^{-4} \mathrm{~m/s^{2}} \) and \( 2.34 \times 10^{-3} \mathrm{~m/s^{2}}\).

Key concepts

Elastic Thickness

Elastic thickness refers to how much the Earth's lithosphere can bend or flex under loads like mountains or other weighty geological features. When we talk about a lithosphere with varying elastic thickness, we examine how it responds to different forces.

• When elastic thickness is **high**, the lithosphere can support heavier loads without bending much, like a thick rubber mat.
• When it's **zero**, it means the lithosphere acts purely elastically, unable to support any bending, akin to a thin plastic sheet.

Understanding elastic thickness is crucial for interpreting how geological structures form and evolve over time.

Transfer Function

The transfer function in geodynamics is a mathematical representation that shows how a gravity anomaly, like a change in gravitational force (\(\Delta g_{fa}\)), relates to a height anomaly (H). It's influenced by factors such as elastic thickness and wavenumber.

Why does the transfer function matter?

• It describes the efficiency of the lithosphere in translating surface loads to subsurface mass changes.
• Analyzing it helps understand the relationship between surface features and the lithosphere’s response.

This function provides a critical link between observed geological features and the underlying mechanical properties of the Earth.

Wavenumber Impact

Wavenumber plays a significant role concerning the transfer function and describes how frequently waves (or deformations) occur in a given space. It's a critical factor in understanding geological responses.

• **High wavenumber** reflects small wavelength deformations, which means more frequent oscillations in structure. In these cases, the lithosphere can't support these small-scale features effectively, causing the transfer function to approach zero.
• **Low wavenumber** implies larger wavelength deformations. Here, surface loads are sufficiently supported by the lithosphere, leading again to the transfer function approaching zero because it undergoes complete compensation.

Thus, the understanding of wavenumber impacts helps geoscientists predict and explain geological activities based on the lithosphere's ability to respond to these oscillations.

Gaussian Topography

Gaussian topography is a way of modeling the surface of the Earth as idealized smooth hills or mountains. It uses a Gaussian function, which is a symmetric "bell curve," to describe the height and shape of these features.

The equation for Gaussian topography is:\[h(x) = h_0 \exp\left(-\frac{x^2}{2\sigma^2}\right)\]Here:

• \(h_0\) indicates the peak height, ensuring a maximum point of the curve is defined.
• \(\sigma\) controls the width of the bell curve, indicating how quickly the height tapers off away from the peak.

This model is fundamental in simulating how topographical features like mountains can influence underlying gravitational fields.

Gravity Anomaly

A gravity anomaly indicates variations in Earth's gravitational field caused by differences in the density of underlying rocks. These differences lead to measurable changes in gravity.

• Gravity anomalies reveal important subsurface features such as mountain roots, mascons (mass concentrations), or underground voids.
• They help geoscientists map out variations in Earth's internal structure without needing to drill or dig.

Understanding gravity anomalies aids in the exploration of resources and offers insights into geological processes like tectonic movements and crustal deformation. Accurate measurement and interpretation of these anomalies provide an invaluable tool in geodynamics.