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Combined Geo-Scientst (Main)
Examination-2025 BGSP-A-GPH
GEOPHYSICS
PAPER—I
Time Allowed : Three Hours Maximum Marks : 200
QUESTION PAPER SPECIFIC INSTRUCTIONS
Please read each of the following instructions carefully
before attempting questions
There are TEN questions divided under TWO Sections.
Candidate has to attempt SIX questions in all.
Question Nos. 1 and 6 are compulsory.
Out of the remaining EIGHT questions, FOUR questions are to be attempted choosing
TWO from each Section.
The number of marks carried by a question/part is indicated against it.
Neat sketches may be drawn to illustrate answers, wherever required. These shall be
drawn in the space provided for answering the question itself.
Unless otherwise mentioned, symbols and notations have their usual standard meanings.
Assume suitable data, if necessary, and indicate the same clearly.
Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a
question shall be counted even if attempted partly.
Any page or portion of the page left blank in the Question-cum-Answer Booklet must be
clearly struck off.
Answers must be written in ENGLISH only.
BGSP-A-GPH/63 1 [PIMo:
He ain?’ 1
1. (a) Formulate a linear inverse problem to determine p, (average density of core)
(b)
()
(@)
and p,, (average density of mantle) from the two data points d, and dy as per
the details given below :
d, : Mass of the Earth
dz : Moment of inertia
c: Radius of core
\ / a: Radius of mantle
Also write the expression for least square solution of the inverse problem.
(Assume the Earth as a spherical body) 10
() Explain, briefly, seismic body waves using suitable sketches. 5
(ii) Differentiate between compressional waves and transverse waves. 5
What is natural period of seismometer? At critical damping, the response of
seismometer to a periodic disturbing signal with frequency is given by
Aw?
u=
(wo +)
cos(wt — A)
Using this equation, define how long-period seismometer becomes
displacement meter and short-period seismometer becomes accelerometer.
Also give the frequency range of seismic signals recorded by displacement
seismometer and accelerometer. 10
(i) Explain divergence theorem. 4
(ii) Solve the following using divergence theorem :
bag a 2.% ae > its : 5
If F = xyi+y*zj+z~k, evaluate Ih (F -n)dS, where S is the unit cube
defined by O< x <1, O<y<1,0<ZzS1. 6
BGSP-A-GPH/63 2
(b)
()
(b)
(¢)
4. (a)
What is ‘a priori’ information? Assume that the average value of three model
parameters (m, m2, m3) of a linear inverse problem is given ‘a priori’
information. Write this information in the form F m=h (m is model parameter
vector). Write an expression for least square solution of a linear inverse
problem without and with ‘a priori’ information. 10
What do you understand by the reference spheroid and the geoid? Explain with
the help of suitable sketch for the situations—
(i) as wrapping of the geoid for local mass;
(i) large-scale wrapping. 10
(i) Give the geometric interpretation of Snell’s law. How can Fermat’s
principle be used in deriving Snell’s law? 8
(i) Give ray curvature due to increasing and decreasing velocity with depth. 2
() Describe, briefly, Biot-Savart law with the help of suitable diagram. 5
(i) Explain vector potential of a current element as drawn for above (i). 5
Describe the orbit of a planet or comet in the solar system as an ellipse with
the Sun at one of its focal points. Write down the condition under which the
object will escape from the solar system. 8+2=10
What is generalized inverse G9? Write the expressions for generalized inverse
(GY) for least square solution, minimum length solution and damped least
square solution of a linear inverse problem d= Gm (d : data vector, G: data
kernel and m : model parameter vector). 10
() Show that the Laplace equation V?V =0 satisfies for current flow in a
homogeneous medium, where V is electrical potential.
(ti) Solve the Laplace equation when single current electrode is kept at a depth
in the homogeneous and isotropic media. 10
BGSP-A-GPH/63 3 [P:T:0.
(b) What is ray parameter in seismology? Derive the travel time equation for
spherical Earth
T=Pa+2{? o> ar
.
dr
where — = ba, 7, is radius to the turning point of the ray, P=
v
A=2P)”
rsini
u(r)
radius of the sphere. 10
and ro is the
(c) Write an expression for Singular Value Decomposition (SVD) of a matrix.
Describe the SVD analysis of a least square solution of a linear inverse
problem. How does SVD analysis help to identify an ill-posed inverse problem? 10
5. (a) (i Try to draw a suitable diagram illustrating the difference between the
Earth’s geographic, geomagnetic and magnetic poles and equator. 5
(i) The palaeomagnetic latitude from the magnetic measurement was made
on basalt flow at present at 47° N, 20° E. The angle of inclination of the
remanent magnetization of this basalt is 30°. Compute the magnetic
latitude at this site at the time the basalt was magnetized. 5
(b) (ij For normal incidence P wave, give the expression of reflection and
transmission coefficient. Compute the reflection coefficient (RC) and
transmission coefficient (TC), energy reflected (E,) and energy transmitted
(E,). M, and M, are medium one and medium two : 6
Incident Reflected
P wave P wave
M, a, = 40 km/s, p, = 1 gm/cm$
M, 2 = 5:0 km/s, py = 2 gm/cm3
Transmitted P wave
(ii) Describe b value in seismology and the equation used to calculate b value
from earthquake data. 4
(c) Explain the gravity effect of a sphere at a point with the help of suitable sketch.
Compute depth to the centre of the spherical body, z and mass of the
anomalous body, M from the given data as x,;2 =25 mand gmax =0-5 mgal. 10
BGSP-A-GPH/63 4
SECTION—B
6. (a) Using Cayley-Hamilton theorem matrix, find the eigenvalues of the matrix
B=A*-3A° +3A? -2A+8]
here A=(+ —}
WAETE El i - |
10
(b) What is Cherenkov radiation? Derive the condition for emission of Cherenkov
radiation from a charged particle passing through a medium. 10
(c) Derive the condition for static equilibrium for plasma as a conducting fluid. 10
(d) When radio waves propagate through the ionosphere, show that the induced
current developed due to propagation of wave is inductive in nature. 10
7. (a) Given, z(x, 0)=x? and z(l, y) =cosy. Find the solution of
2
0*2(% Y _ ay
ox dy 10
11 . ; : 5 : A
(b) IfA= 11 and I is a2 x2 identity matrix, then what will be the value of e4?
10
2
(c) Show that the Fourier transform of a Gaussian function f(x) = Ae ®*" is also a
Gaussian function, where both A and 8 are constants. 10
8. (a) Prove that for a moving conductor loop in a time-varying magnetic field
VxE =~ 28. vxGt nd)
ot 10
(b) Show that the radiation emitted by a charged particle (Bremsstrahlung) is
independent of acceleration or deceleration. 10
(c) For an aluminium airplane, using Gauss’ law, show that no electric field exists
inside the plane. 10
9. (a) Describe the physical significance of Maxwell’s equations in differential and
integral forms. Explain Faraday’s law and its consequences using differential
form of equations. 10
(b) Prove that for an electromagnetic wave, the sum of reflection coefficient and
transmission coefficient at normal incidence is equal to unity. 10
(c) If two plane waves travelling with speed c at angles 6, and @,, compare their
net velocities in a rectangular waveguide. 10
BGSP-A-GPH/63 5 [ P.T.O.
10. (a) (i) Explain the term fading’ in radio wave propagation and state the factors
responsible for fading. 5
(ii) Discuss in brief how fading can be minimized for radio wave propagation. 5
(b) (i) Explain the terms Virtual height’, ‘skip distance’ and ‘maximum usable
frequency’ as applied to sky wave propagation. 6
(ii) For the Earth, the atmosphere consists of 22% Oz and 78% Ny. What is
the scale height of the Earth atmosphere?
(Given, g=9-81m/ s?, T=290K = and Boltzmann constant
k =1-38x107%5 joule per kelvin) 4
(c) (i) What is satellite communication? Distinguish between active and passive
satellite communication systems. 4
(i) Calculate the value of frequency at which an electromagnetic wave must
propagate through the D-region having refractive index 0:5 and electron
density of 3-24x10* electrons/m°. 6
kkk
BGSP-A-GPH/63 6 25SS—288
