Consider a two-dimensional reflection experiment, where a horizontal boundary between two layers is at a depth of 500 m below the free surface. A source and geophone are placed on the free surface with an offset of 2000 m. The P-wave velocity of the top layer is 3000 m/s. The travel time of a recorded free-surface reflection multiple, which got reflected twice at the free surface, is ________ s.
Show Hint
Solution and Explanation
In this two-dimensional reflection experiment, the source and geophone are placed on the free surface, with the boundary at a depth of 500 m. The recorded wave is a multiple reflection, where the wave is reflected twice at the free surface.
The total travel time for a wave that is reflected twice at the free surface consists of:
- The time for the wave to travel from the source to the boundary at depth \( h = 500 \, {m} \),
- The time for the wave to travel back to the free surface after reflection,
- The time for the wave to travel from the free surface to the receiver, which has an offset of \( 2000 \, {m} \).
The total travel time for the free-surface reflection multiple is the sum of the round-trip travel times, taking into account the path the wave travels and the P-wave velocity of the top layer.
The travel time to the boundary is calculated as: \[ t_1 = \frac{h}{v_p} = \frac{500}{3000} = 0.1667 \, \text{seconds}. \] where \( v_p = 3000 \, \text{m/s} \) is the P-wave velocity.
The travel time for the wave to travel from the source to the receiver with an offset of 2000 m is calculated using the straight-line distance between the source and the receiver. The distance traveled is the hypotenuse of a right triangle with legs \( 2000 \, {m} \) (offset) and \( 500 \, {m} \) (depth). The distance traveled is: \[ d = \sqrt{2000^2 + 500^2} = \sqrt{4000000 + 250000} = \sqrt{4250000} \approx 2061.55 \, \text{m}. \] The travel time for this distance is: \[ t_2 = \frac{d}{v_p} = \frac{2061.55}{3000} \approx 0.6872 \, \text{seconds}. \]
The total travel time for the recorded free-surface reflection multiple is the sum of the following:
1. The travel time to the boundary: \( t_1 = 0.1667 \, \text{s} \),
2. The travel time for the offset: \( t_2 = 0.6872 \, \text{s} \),
3. The wave is reflected twice, so the total time is the sum of the round-trip travel times for both reflections: \[ \text{Total travel time} = 2 \times (t_1 + t_2) = 2 \times (0.1667 + 0.6872) = 2 \times 0.8539 \approx 1.7078 \, \text{seconds}. \] Thus, the travel time of the recorded free-surface reflection multiple is approximately: \[ \boxed{1.71 \, \text{seconds}}. \]
Top Questions on Seismology
In seismology, Born approximation of the scattered (perturbed) wavefield is given by \[ \delta u(\mathbf{r}, \mathbf{s}; t) \approx \int_V \delta r(\mathbf{x}) \left(u_0(\mathbf{x}, \mathbf{s}; t) _t u_0(\mathbf{r}, \mathbf{x}; t)\right) \, d\mathbf{x}. \] Here, \( _t \) denotes temporal convolution, \( \delta r(\mathbf{x}) \) is the strength of the scatterer at \( \mathbf{x} \) in volume \( V \), \( \delta u(\mathbf{r}, \mathbf{s}; t) \) is the scattered wavefield measured at the receiver \( \mathbf{r} \) from the source \( \mathbf{s} \), \( u_0(\mathbf{x}, \mathbf{s}; t) \) is the downgoing wavefield (to the scatterer at \( \mathbf{x} \) from the source \( \mathbf{s} \)) in the unperturbed medium, \( u_0(\mathbf{r}, \mathbf{x}; t) \) is the upgoing wavefield (to the receiver \( \mathbf{r} \) from the scatterer at \( \mathbf{x} \)) in the unperturbed medium.
Select the correct statement(s).- What factor(s) determine(s) the magnitude of peak ground acceleration measured at a particular station due to an earthquake?
For a horizontal liquid-solid interface as shown, which one of the following ray diagrams with an incident P wave is correct? SH and SV denote shear-horizontal and shear-vertical waves, respectively.
A geophysicist is analyzing the elastic-wave radiation to infer the body force equivalent of a seismic source. She has plotted the horizontal component of the displacement field, denoted as \( u(m, t) \), for time \( t \) and location \( m \). The measured field at two locations \( m \) and \( n \) is plotted in the figure. Note that \( S \) and \( P \) waves have negligible amplitudes at locations \( m \) and \( n \), respectively. Assuming a homogeneous medium, select the most probable direction (specified by the angle \( \alpha \)) along which the force \( \vec{F} \) is applied.
Questions Asked in GATE GG exam
While doing Bayesian inference, consider estimating the posterior distribution of the model parameter (m), given data (d). Assume that Prior and Likelihood are proportional to Gaussian functions given by \[ {Prior} \propto \exp(-0.5(m - 1)^2) \] \[ {Likelihood} \propto \exp(-0.5(m - 3)^2) \]
The mean of the posterior distribution is (Answer in integer)
- GATE GG - 2025
- Bayesian construction of posterior probabilities
Suppose a mountain at location A is in isostatic equilibrium with a column at location B, which is at sea-level, as shown in the figure. The height of the mountain is 4 km and the thickness of the crust at B is 1 km. Given that the densities of crust and mantle are 2700 kg/m\(^3\) and 3300 kg/m\(^3\), respectively, the thickness of the mountain root (r1) is km. (Answer in integer)
- GATE GG - 2025
- Geophysics
- GATE GG - 2025
- Geophysics
- A primary electromagnetic field (\( H_P \)) is being generated from current \( I_P \) flowing in a coil A with negligible capacitance, such that \( H_P = K I_P \sin \omega t \), where \( \omega \) is the angular frequency, \( K \) is a constant and \( t \) is time. A secondary electromagnetic field is being produced by induction in a conducting coil B. The phase difference between the primary and the secondary electromagnetic fields depends on which of the following factors?
- GATE GG - 2025
- Electromagnetic
- A loop of radius \( R \) carries a current \( I \) and produces a magnetic field \( B \). Which of the following statements is/are correct about \( B \)?
- GATE GG - 2025
- Electromagnetic