Geophysical Inversion - Advanced topics in Geophysics - Geology & Geophysics Previous Year Questions

1

2008 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2008

In a linear inverse problem having rectangular system matrix that is rank deficient, the inverse solution is

A

unique solution

B

least square solution

C

minimum norm solution

D

minimum norm least square solution

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2

2008 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2008

In a linear inverse problem having eigenvalues 100, 10, 1, 0.1, 0.01, 0.001, the highest condition number of the system matrix is

A

100000

B

10000

C

1000

D

100

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3

2009 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2009

Geophysical inverse problems are described by

A

Fredholm's integral equation of first kind

B

Fredholm's integral equation of second kind

C

Volterra's equation of second kind

D

Legendre equation

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4

2009 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2009

Spot the ANN method from the following :

A

Singular value decomposition

B

Monte-Carlo technique

C

Ridge regression procedure

D

Back propagation technique

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5

2009 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2009

The concept of resolving kernel is used in

A

Tikhonov's regularization method

B

Ridge regression method

C

Backus-Gilbert method

D

Simulated annealing method

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6

2010 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2010

Unguided random-walk inversion technique signifies

A

Genetic algorithm

B

Simulated annealing

C

Monte Carlo inversion

D

Metropolis algorithm

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7

2010 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2010

If \(m\) represents the number of model parameters, \(d\) the number of data points and \(p\) the rank of matrix to be inverted, then which of the following defines an underdetermined system?

A

\(m < d\) and \(p = d\)

B

\(m > d\) and \(p = d\)

C

\(m = d\) and \(p = d\)

D

\(m < d\) and \(p \neq d\)

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8

2011 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2011

The least squares generalized inverse of an overdetermined problem is expressed as

A

\((G^T G)^{-1} G^T\)

B

\((G^T G)^{-1}\)

C

\(G^T (G G^T)^{-1}\)

D

\((G G^T)^{-1}\)

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9

2012 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2012

The solution to the purely under-determined problem Gm = d is given by

A

(GᵀG)⁻¹Gᵀd

B

(GᵀG)⁻¹Gdᵀ

C

Gᵀ(GGᵀ)⁻¹d

D

Gᵀd(GGᵀ)⁻¹

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10

2013 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2013

A singular value of an m x n matrix, A, is defined as

A

positive square root of eigenvalue of AA^T

B

modulus of eigenvalue of A

C

eigenvalue of A^TA

D

square of eigenvalue of A

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11

2013 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2013

In an ill-posed geophysical inverse problem, stated as non-singular matrix equation, the magnitude of determinant of the coefficient matrix is

A

large

B

zero

C

near zero

D

very large

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12

2013 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2013

The eigenvalues of A are

A

(4.01, -0.01)

B

(-4.01, -0.01)

C

(4.01, 0.01)

D

(-4.01, 0.01)

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13

2013 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2013

If the elements of A are expressed up to first decimal place only, then the number of possible solution(s) of the resulting inverse problem is

A

1

B

2

C

3

D

∞

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14

2014 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2014

In general, geophysical inverse problems dealing with real data obtained from field measurements are

A

grossly over determined

B

even determined

C

over determined

D

grossly underdetermined

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15

2014 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2014

Which one of the following functions is used as a source-term to obtain the Green’s function of a boundary value problem?

A

Heaviside unit step function

B

Exponential function

C

Rectangular function

D

Dirac delta function

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16

2014 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2014

The eigenvalue ($\Lambda$) and eigenvector (U) matrices for singular value decomposition of the matrix $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ respectively are

A

$\Lambda = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}$ and $U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$

B

$\Lambda = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}$ and $U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -1 \\ 1 & -1 \end{pmatrix}$

C

$\Lambda = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$ and $U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$

D

$\Lambda = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$ and $U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$

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17

2016 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2016

If J is the Jacobian matrix in a geophysical inverse problem, then the addition of the regularization parameter, \(\lambda\), as \((J^T J + \lambda I)\), in finding the inverse leads to

A

unstable solution with increased parameter resolution

B

stable solution with increased parameter resolution

C

unstable solution with decreased parameters resolution

D

stable solution with decreased parameter resolution

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18

2016 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2016

The Singular Value Decomposition of a square nonsingular matrix J is given by \(J = U \lambda V^T\). The inverse of matrix J will be

A

\(J^{-1} = U \lambda^{-1} V^T\)

B

\(J^{-1} = V \lambda^{-1} U^T\)

C

\(J^{-1} = U^T \lambda^{-1} V^T\)

D

\(J^{-1} = U \lambda^{-1} V\)

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19

2022 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2022

Choose the CORRECT statement(s) from the following on the solution of systems of linear equations without the application of regularization.

A

An under-determined system of linearly independent equations has either a trivial solution or an infinite number of solutions.

B

An ill-conditioned system of linear equations can yield stable solutions in the presence of noise.

C

An over-determined system of linearly independent equations does not have an exact solution.

D

A system of linearly independent equations with the number of equations equal to the number of unknowns is a mixed-determined system.

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20

2024 · Geology & Geophysics · Advanced topics in Geophysics · Geophysical Inversion

Geology & Geophysics (GG) 2024 [Session 2]

An overdetermined linear inverse problem is expressed as \(\mathbf{G} \mathbf{m} = \mathbf{d}\), where \(\mathbf{G}\) is the data kernel, \(\mathbf{m}\) is the vector of model parameters and \(\mathbf{d}\) is the vector of observed data. If damping is applied to the inverse problem and the resultant generalized inverse is represented by \(\mathbf{G}^{-g}\), the model resolution matrix can be expressed as

A

\(\mathbf{G}^T \mathbf{G}^{-g}\)

B

\(\mathbf{G}^{-g} \mathbf{G}^T\)

C

\(\mathbf{G}^{-g} \mathbf{G}\)

D

\(\mathbf{G} \mathbf{G}^{-g}\)

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