3.3 Reflection and refraction of body waves

When seismic waves hit boundaries between different rock layers, they bounce and bend. This reflection and refraction behavior follows specific laws, like Snell's law, and depends on the velocity differences between layers.

Understanding these principles is key to interpreting seismic data. By analyzing how waves reflect, refract, and convert between types, scientists can map out Earth's interior structure and composition.

Reflection and Refraction Principles

Fundamental Laws and Concepts

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• Snell's law governs the relationship between angles of incidence and refraction for waves passing through a boundary between two different media
• Formula for Snell's law expresses as $\frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2}$, where θ represents angles and v represents velocities
• Critical angle occurs when the angle of refraction equals 90 degrees, resulting in total internal reflection
• Calculation of critical angle uses the formula $\sin \theta_c = \frac{v_1}{v_2}$, where v₁ < v₂
• Huygens' principle states every point on a wavefront acts as a source of secondary wavelets
• Secondary wavelets combine to form a new wavefront in the direction of wave propagation

Velocity Contrasts and Wave Behavior

• Velocity contrasts refer to differences in seismic wave velocities between different rock layers or media
• Higher velocity contrasts lead to stronger reflections and more pronounced refractions
• P-wave velocity contrasts typically range from 2 km/s in unconsolidated sediments to 8 km/s in the upper mantle
• S-wave velocities generally about 60% of P-wave velocities in the same medium
• Velocity contrasts affect wave propagation paths, creating complex patterns of reflections and refractions in the Earth's interior
• Understanding velocity contrasts crucial for interpreting seismic data and creating accurate Earth models

Seismic Wave Coefficients

Reflection and Transmission Coefficients

• Reflection coefficient (R) quantifies the amplitude ratio of reflected wave to incident wave
• Calculation of reflection coefficient uses the formula $R = \frac{\rho_2v_2 - \rho_1v_1}{\rho_2v_2 + \rho_1v_1}$, where ρ represents density and v represents velocity
• Reflection coefficient values range from -1 to 1, with higher absolute values indicating stronger reflections
• Transmission coefficient (T) measures the amplitude ratio of transmitted wave to incident wave
• Formula for transmission coefficient expressed as $T = 1 - R$, ensuring energy conservation
• Coefficients depend on the acoustic impedance contrast between media, calculated as the product of density and velocity

Mode Conversion and Energy Partitioning

• Mode conversion occurs when an incident P-wave or S-wave generates both P and S waves at an interface
• P-to-S conversion happens when a P-wave strikes an interface at non-normal incidence
• S-to-P conversion occurs when an S-wave encounters an interface, generating both P and S waves
• Energy partitioning describes how the energy of an incident wave distributes among reflected and transmitted waves
• Partitioning depends on the angle of incidence, velocity contrast, and densities of the media
• Understanding mode conversion and energy partitioning crucial for interpreting complex seismic records and identifying different wave arrivals

Seismic Wave Propagation

Travel Time Curves and Seismic Interpretation

• Travel time curves graphically represent the relationship between arrival times of seismic waves and their distance from the source
• Curves plot travel time on the vertical axis and distance on the horizontal axis
• Different seismic phases (direct waves, reflected waves, refracted waves) produce distinct curves on the travel time plot
• Slope of a travel time curve indicates the apparent velocity of the wave
• Crossover distance on travel time curves marks the point where refracted waves arrive before direct waves
• Interpreting travel time curves helps determine layer velocities, thicknesses, and depths in the Earth's structure

Advanced Seismic Analysis Techniques

• Ray tracing technique uses travel time curves to model seismic wave paths through the Earth
• Involves solving the eikonal equation to determine wave propagation paths in heterogeneous media
• Tomography utilizes travel time data to create 3D velocity models of the Earth's interior
• Involves inverting large datasets of seismic arrival times to reconstruct subsurface structures
• Waveform modeling compares synthetic seismograms with observed data to refine Earth models
• Requires solving the wave equation to simulate seismic wave propagation through complex media