Geophysical data processing

Dynamic range: the ratio of the largest measurable amplitude Amax to the smallest
measurable amplitude Amin in a sampled function
= the ratio of two power values P1 and P2 is given by 10 log10(P1/P2) Db

Nyquist frequency ( fN):

.Convolution: if gi (i = 1, 2, . . . , m) is an input function and fj ( j =1, 2, . . . , n) is a

convolution operator, then the convolution output function yk is given by

.Cross correlation : The crosscorrelation operation is similar to convolution but does

not involve folding of one of the waveforms. Given two digital waveforms of finite
length, xi and yi (i = 1, 2, . . . , n), the cross-correlation function is given by

Ex. Convolve A={3,-1} with B={3,1,-1}

sum the elements in direction of arrows output={9,0,-7,2}

Ex. Crosscorrelate A={3,-1} with B={3,1,-1}

sum the elements in direction of arrows output={-3,8,5,-6}

Ex. . Crosscorrelate B={3,1,-1} with A={3,-1}

sum the elements in direction of arrows output={-6,5,8,-3

 Fourier series
Let g(t) be a periodic function with period T that is,
Fourier transform
The inverse Fourier transform

The discrete Fourier transform (DFT)

Inverse DFT

Hilbert transforms
The Hilbert transform of a function f(x) is defined by:

Computationally one can write the Hilbert transform as the

convolution:

ANALYTICAL SIGNAL
For a function f(t),with its Hilbert transform F(t),its analytical signal A(t)is
defined as 𝐴(𝑡) = 𝑓(𝑡) + 𝑖𝐹(𝑡)
Ex. For f(t)=sin t
𝐴(𝑡) = 𝑠𝑖𝑛 𝑡 − 𝑖𝑐𝑜𝑠 𝑡
WAVELET TRANSFORM
CONDITION FOR WAVELET:

Compactly supported

Admissibility condition

Translational Invariance
The continuous wavelet transformation (CWT)

Z-transform

Delay operator
𝑧 −1 (= 𝑒 −𝑖𝑤 ) Corresponds to a delay of one unit in the signal.
Region of convergence for
Right-handed Z-Transform (causal) |𝑧| > |𝑎|

Left-handed Z-Transform (acausal) |𝑧| < |𝑎|

Two sided z-transform |𝑏| > |𝑧| > |𝑎|

Theorems Related to vector analysis

 Earthquake seismology

Wadaati diagram
If D is the distance travelled by the seismic wave the travel-times of P- and S-waves are
respectively

Intercept to the horizontal axis (in Wadaati diagram) gives the time of occurrence of
Earthquake t0

Knowing the P-wave velocity𝛼, the distance to the earthquake is obtained from

Epicental and focal depth relationship distance

∆𝑘𝑚 is the epicentral distance and D the distance travelled by the wave, to a first
approximation The focal depth of the earthquake, d

Tsunami velocity

Earthquake magnitude

Surface-wave magnitude (Ms)

Body-wave magnitude (mb)

Moment magnitude (Mw)

Seismic moment M0

The area S of the fractured segment and the amount by which it slipped D can be
inferred.
𝜇 = 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝑟𝑖𝑔𝑖𝑑𝑖𝑡𝑦 of adjacent rocks

Relationship between magnitude and intensity(for depth h <50 km)

Earthquake frequency
8<a<9 and while b is approximately unity for regional and global
seismicity
Energy released in an earthquake

For two diff earthquakes of magnitudes with difference of surface wave

𝐸2
magnitude ∆𝑀𝑠 causes energy difference are related as
𝐸1
𝐸2
⁄𝐸 = 101.5(∆𝑀𝑠 )
1
AMPLITUDE RATIO OF BODY WAVES
Here two earthquakes of p-wave amplitude ratio 𝐴𝑃1 and 𝐴𝑃2 with body
wave magnitude 𝑚𝑏1 and 𝑚𝑏2 respectively.

𝑨𝑷𝟐
= 𝟏𝟎(𝒎𝒃𝟐 −𝒎𝒃𝟏 )
𝑨𝑷𝟏
Earth’s Heat

Volume expansion coefficient

 The adiabatic temperature gradient

involving entropy (S), pressure (p), temperature (T) and volume (V). and
adiabatic temp. grad. is

Approximate estimates of adiabatic temperatures inside the Earth can also

be obtained with the aid of the Grüneisen thermodynamic parameter,
𝜸

where Ks is the adiabatic incompressibility or bulk modulus.

Heat flux 𝒒𝒛

Where k is thermal conductivity.

The amounts of heat generated per second by these elements (in 𝜇W /kg)
are: natural uranium, 95.2; thorium, 25.6; and natural potassium,
0.00348.
 The heat Qr produced by radioactivity in a rock that has concentrations
CU, CTh and CK, respectively, of these elements is

Heat-flow province
Each province is characterized by a different linear relation between q and A,
such that

The parameters qr and D typify the heat-flow province. The intercept of the
straight line with the heat-flow axis, qr, is called the reduced heat flow
Equation of heat conduction

With unit of m2/s.

Comparison of the decay depths for daily and annual temperature variations in
the same ground

Penetration of external heat into the Earth

Variation of oceanic heat flow and depth with lithospheric age

The optimum relationships between depth (d, m) and age (t, Ma) can be written

Relation in Permeability,velocity of flowand hydraulic conductivity

Velocity of the flow = v,
The gradient of the hydraulic pressure head (dp/dx).
Thermal convection
Prandtl number
The ratio of the other forces(driving forces exerted by pressure gradient and buoyancy) to the
inertial forces

The Rayleigh number (RaT)

The Rayleigh number, which is proportional to the ratio of the buoyancy force to the
diffusive–viscous force
For convection due to the super adiabatic temperature gradient in a fluid layer of thickness D
is

Second Rayleigh number (RaQ)

For convection driven by radiogenic heat:

Reynolds number, Re
The conditions favoring turbulence are high momentum𝜌𝑣, at large scale (D) whereas it is
inhibited by high viscosity 𝜂,so the ratio of two .
Defined as

Nusselt number this is defined as the ratio of the heat transport in the presence of
convection to the heat transport without convection(the non-convective heat transport)
Ratio of Rayleigh number RaT and the critical Rayleigh number Rac

Super adiabatic temperature

The difference between the real and adiabatic temperature gradients is the
superadiabatic temperature gradient 𝜃
Thermal diffusivity κ
the heat that would contribute to the buoyancy is removed by thermal
conduction; the efficacy of this process is expressed by the thermal diffusivity

Seismic
Nyquist frequency

Nyquist Wave no.

Frequency aliasing
To compute the alias frequency fa, use the following
Relation

Bin size

Foldage

NMO Stretching

NMO velocity for dipping reflector

Where φ is the dip angle of the reflector

Radius of Freznel Zone

IN-LINE FOLD

CROSS-LINE FOLD

TOTAL FOLD

Aperture width A for depth Z and dip𝜽

MAX. OFFSET

Max. offset should be compatible with fold requirements at shallow and deep
horizons of intrest .If Fs and Fd be fold at shallow and deeper horizon of intrest.V1,T1
be velo.and time corresponding to shallow reflecter,then
𝑋𝑚𝑎𝑥 = 𝑉1𝑇1𝐹𝑑⁄2𝐹𝑠

Relation of dip moveout and dip of reflector

Faust relation of velocity for depth of burial(Z) and resistivity(R)

Here V in ft/s, R in ohm-ft, Z in ft.

Array-designing-Spacing for cancelling the wavelength 𝝀 using n geophones

∆𝑥 = 𝝀/n
Thumb rule of array length L
0.44𝑉𝑥 0.44
𝐿= =
𝑓𝑚𝑎𝑥 𝑘𝑚𝑎𝑥

Geophone response
For a given magnetic field B, Number of turns N, relative velocity V,
geometry factor c, emf 𝑒 = 𝑐𝑁𝐵𝑉

Rule of thumb of signal to noise ratio in vibrosies is

𝑆
𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑚𝑒𝑛𝑡 ∝ √𝐹𝐿𝑊
𝑁
Where
F=force,L=sweep length,W=band-width

Travel time of harmonic ghost using down-sweep

𝐿𝑓1
𝜏=
𝑓2 − 𝑓1
Where𝑓2 , 𝑓1 are max. and min. frequencies.L= sweep length.
Down sweep gives correlation of pilot trace at positive times so harmonic
ghosts appear on record. But upsweep solves this problem by giving
correlation at negative time (not realizable) so harmonics not seen on
record.

Source and receiver depth (ghost effect in marine)

Charge depth is adjusted if destructive notch frequencies are there ,
 given by 2𝑑 = 𝑛𝜆
𝑛𝑉
Or 2𝑑 = where n = 0,1,2, ….
𝑓

Elements of seismic surveying

Body wave velocity:

Poisson’s ratio:

Seismic wave velocities of rocks

Reflection and transmission Coefficients

Snell’s Law of Refraction:

The critical angle is given by

Time of
 a) Reflected ray

b) Refracted ray

c) Intercept time

 At the crossover distance Xcros the travel times of direct

and refracted rays are equal.
d) Direct wave

Unit of geophone sensitivity-Volt/m/s.

`hydrophone sensitivity-mV/Pa
Absorption coefficient-dB𝜆−1

Depth = d
Xcr = crossover distance
Critical distance
𝟐𝒛𝒗𝟏
𝑿𝒄 = 𝟐𝒛𝒕𝒂𝒏𝜽𝒄 = 𝟐
(𝒗𝟐 − 𝒗𝟏 𝟐 )𝟏/𝟐
Seismic reflection surveying
The interval velocity is given by, If zi is the thickness of such an interval
and ti is the
One-way travel time of a ray through it

Time-average velocity or, simply, average velocity

Normal move out (NMO) at an offset distance x is the difference in

travel time DT between reflected arrivals at x and at zero offset

The root-mean-square velocity

of the section of ground down to the nth interface is given by
the NMO for the nth reflector is given by

The Dix formula, To compute the interval velocity 𝑣𝑛 for the nth interval

Time velocity, depth dip relationship

Dip move out ∆Td

For small angles

Diffractions

Diffraction travel time curve

Move out for diffractions

Path difference b/w ghost and direct wave
𝜆
= + 2𝑑
2
𝑑 𝑖𝑠 𝑠ℎ𝑜𝑡 𝑑𝑒𝑝𝑡ℎ and 𝝀 is wavelength.

The width w of the Fresnel zone is related to the dominant wavelength

𝝀 of the source and the reflector depth z by

Apparent velocity

If 𝜶𝒔 is the dip of
the record surface
and is the true dip
of the reflector :-
  Crossover distance is always greater than twice the depth to the
Refractor

Seismic refraction surveying

Three-layer case with horizontal interface
Multilayer case with horizontal interfaces
Faulted planar interfaces the throw of the fault ∆z is given by

Delay time
The plus–minus interpretation method

The travel time of a refracted ray travelling from one end of the line to the other
is given by

Profile.

The generalized reciprocal method

The method uses a velocity analysis function tv given by

The values being referred to the mid-point between each pair of detector
positions D1 and D2. The overall interpretation method is more complex than
the plus–minus method, but can deliver better velocity discrimination, greater
lateral resolution and better depth estimates to boundaries.

Gravity surveying
Unit of the micrometre per second per second is referred to as the gravity unit
(gu).

Absolute measurement of gravity

A simple pendulum

where I is the moment of inertia of the pendulum about a pivot, h is the distance
of the center of mass from the pivot, and m is the mass of the pendulum. The
distance L between the pivots is then measured accurately.
Free-fall method
a starting position z0 with initial velocity u the equation of motion gives the
position z at time t as

Rise-and-fall method
Let the time spent by the sphere above the first timing level be T1 and the time
above the second level be T2. The distance h between the two timing levels
(around 1m) was measured accurately by optical interferometry
Relative measurement of gravity: the gravimeter
A mass m suspended from a spring of length s0 causes it to stretch to a new
length s.

Correction of gravity measurements

Latitude correction = 0.8140 sin 2𝜃 mGal per kilometer of north–south

displacement correction is 0 at equator and pole.
Terrain corrections =

The Bouguer plate correction =

=
Where the density 𝜌 is in kg/m3
Free-air correction

Combined elevation correction=

The free-air and Bouguer plate corrections are often combined into a single
elevation correction

Eötvös correction
1 km/hr=0.539 knots

: ranges up to 0.3mGal.

MASS OF SPHERE IN TERMS OF X

½ AND gmax
 Here w = 2𝑥1/2

Calculation of the gravity anomaly of geometric

model by an infinite horizontal cylinder the “half-
height width” w of the anomaly is again dependent
on the depth z to the axis of the cylinder; in this
case the depth is given by Z=0.5w
 OVERBURDEN

This is for 3-D body. For 2-D body factor will be will be 0.65.

ACTUAL MASS AND EXCESS MASS RELATION

mGal
∆𝒛 is sme as h here.

Relative density: If weight of rock mass in air and

water be 𝑾𝒂 and 𝑾𝒘 respectively.
ISOSTACY

The Airy–Heiskanen model

The Pratt–Hayford isostatic model

Isostatic gravity anomalies

Airy
Pratt

Magnetic surveying
The force F between two magnetic poles of strengths m1 and m2
separated by a distance r is given by
ELEMENTS OF EARTHS MAGNETIC FIELD

LARMOUR FREQUENCY

SO IF,

𝛾𝑝 = Gyromagnetic ratio of proton

Magnetic minerals
Ferromagnetism

some metals (e.g., iron, nickel, cobalt)

Ferrimagnetism maghemite and Pyrrhotite geothite.

Paramagnetic Platinum, Hematite and Franklinite

Many clay minerals and other rock-forming minerals (e.g., chlorite, amphibole, pyroxene, olivine) are
paramagnetic at room temperature
Diamagnetism, Bismuth Quartz and Calcite, Gold, Silver, Gypsum
Antiferromagnetic
A common example of an antiferromagnetic mineral is Ilmenite (FeTiO3)
Parasitic ferromagnetism
Iron mineral hematite (𝛼-Fe2O3)

CURIE’S LAW
At temperatures 𝑇 > 𝜃 the paramagnetic susceptibility k is given by the
Curie–Weiss law

𝜃 = paramagnetic Curie temperature or Weiss constant of the material

RELATION B/W INCINATION AND MAGNETIC LATITUDE

Magnetic latitude 𝜆 = 90-polar angle 𝜃

Radial component
Tangential component
Königsberger ratio (Qn)
𝑴𝒓
𝑸𝒏 = , 𝑴𝒓 = remnant magnetization,𝑴𝒊 = induced magnetization
𝑴𝒊

1. The altitude correction is about 0.015 nT /m near equator; near the

magnetic poles (Bt_60,000 nT) it is about 0.030 nT /m.
2. The latitude variation is rarely greater than 6 nT/km.
3.
4. Diurnal correction
Magnetization contrast
If k represents the susceptibility of an orebody, k0 the susceptibility of the host
rocks and F the strength of the inducing magnetic field then magnetization.
contrast is given by:

Magnetic anomaly of a surface distribution of magnetic poles

𝛺 = solid angle

Magnetic anomalies of simple geometric bodies

WERNER DECONVOLUTION
To isolate an anomaly from interference of nearby anomalies
Ex.
Rearranging

DEPTH ESTIMATION
Smith rule for maximum depth of estimations
If magnetisation is parallel throughout the body, but not necessarily uniform and
if:

EULER DECONVOLUTION
More rigorous method of determining the depth to magnetic sources derives
from a technique known as Euler deconvolution
Euler’s homogeneity relation can be written

N is also called structural index.

 RESISTIVITY METHODS

Ohm’s law

Single Current Electrode at Depth

Single Current Electrode at Surface

Two Current Electrodes at Surface

ELECTRODE LAYOUTS

∆𝑉
p =geometrical factor. is resistance and factor left after this is called
𝐼
geometrical factor.
Wenner array

Schlumberger (gradient) array

Pole-dipole (three-point) array

Double-dipole (dipole-dipole) system

CONDITION OF :

Half- Schlumberger
𝒓𝟏 = 𝒂 − 𝜹𝒂/𝟐
𝒓𝟑 = 𝒂 + 𝜹𝒂/𝟐
Transverse unit resistance

Longitudinal unit conductance

Where 𝒛𝒆 = ∑𝒏𝒊=𝟎 𝒛𝒊
Reduction of layer to a single layer of resistivity 𝝆𝒎 and thickness 𝒛𝒆𝒒

𝑻
𝝆𝒎 = √ 𝒂𝒏𝒅 𝒛𝒆𝒒 = √𝑻𝑺
𝑺

Current Distribution
The fraction of total current through a long strip (Z2 - Z1) wide will be

Fraction of current will be maximum when

L=2(𝒛𝟏 𝒛𝟐 )1/2
Distortion of Current Flow at a Plane Interface

The potential V at the surface

over a series of horizontal
layers, the uppermost of
which has a resistivity r1, at a distance r from a current source of strength I
is given by-
Koefoed relation

RESISTIVITY TRANSFORM
A useful additional parameter is the resistivity transform T (λ) defined by

Coefficient of Anisotropy
Medium in which the resistivity is uniform in the horizontal direction and has
the value 𝜌ℎ in the vertical direction it is also constant and has a different
magnitude 𝜌𝑣 , 𝜌𝑣 almost invariably being larger than 𝜌ℎ

The quantities T and S are known as the Dar Zarrouk parameter.

An average square resistivity ρm and pseudoanisotropy λ are given by

Common relation in Wenner, Schlumberger and Double-dipole is

Factor c=1.39 wenner

=1 slumberger
=0.5 dipole dipole

Principle of equivalence
It is impossible to distinguish between two highly resistive beds of different z
and 𝜌 values if the product z𝜌(for K-type curve) is the same, or between two
highly conductive beds if the ratio z/ 𝜌(for H-type cuve) is the same.

Principle of Suppression
For A-and Q-type curves.

Resistivity relation for the Schlumberger array

Where the product 𝐾(𝜆)𝐽1 (𝜆) called Stefanescu function.

 INDUCED POLARISATION METHOD
INDUCED POLARIZATION MEASUREMENTS
Millivolts per volt (lP percent)

=V(t)/Vc
Decay-time integral
V(t)

Chargeability

Chargeability M is in milliseconds.
Frequency-Domain Measurements
Frequency-effect

Percent frequency effect

Metal factor

We can write for the chargeability

ELECTROMAGNETIC MATHODS
Maxwell’s equations

For poor conductors these reduces to

For good conductors,

Wavelength of EM Wave
Skin depth

Primary field

Phase difference

Figure of merit
In airborne EM coil configurations
For coaxial mounting

For coplanar mounting

GPR
VELOCITY OF RADAR WAVE

The velocity of a radar wave (V) is given by:

𝜅 𝑖𝑠 𝑑𝑖𝑒𝑙𝑐𝑟𝑖𝑐 𝑐𝑜𝑛𝑠𝑎𝑛𝑡.
The reflection coefficient K

The magnetotelluric (MT) method

The impedance phase

Kramers–Kroenig relationship
The E-polarisation

The B-polarisation

PENETRATION DEPTH AND

CONDUCTIVITY
Where E is in mV/km and H is in nT.
TENSOR IMPEDANCE RELATIONSHIP

IMPEDANCE FOR

Sensors in the Earth’s magnetic field transforms seismic noise into a

perturbation of the magnetic field, B according to:

RADIOACTIVE METHODS

Radioactive decay
The decay is described by an exponential curve.If the decay rate is equal to l,
then in a short time interval dt the probability that a given nucleus will decay is l
dt; if at any time we have P parent nuclei the number that decay in the following
interval dt

,
Number of daughter nuclides D increases=P0-P
The activity A at any given time is thus related to the initial
activity A0

RADIOACTIVE EQUILLIBRIUM
At equilibrium no. of daughter elements decays same no is created
here λ is decay constant N is daughter elements

This equilibrium is reached at time 𝑡𝑒𝑞 is given by

 Peak energies of Potassium Uranium and Thorium are 1.46,

1.76, 2.62 MeV respectively.

Method of
Rubidium–strontium
Similar to eq.

Eqn. used for dating is:

Where

Potassium–argon
Argon–argon

Value of J is found from control samples.

Uranium–lead

Lead–lead isochrones

WELL LOGGING
Overburden pressure

Gamma ray Index 𝑰𝑮𝑹 /shale volume

Volume of shale
FROM SP LOG
𝑽𝒔𝒉 = 𝟏 − 𝑷𝑺𝑷/𝑺𝑺𝑷
Electron number density is

Relation b/w effective number density and bulk density

Apparent density

Porosity density relation

Or,

Specific Photo-Electric Absorption Index

Volumetric Photo-Electric Absorption Index

Volumetric photo-electric absorption index U

Shale volume

Transit time and velocity(ft/s) relation FROM SONIC LOG

Porosity from sonic log

Arp’s formula
Archie’s First Law

Bulk resistivity of a rock Ro fully saturated with an aqueous fluid of resistivity

Rw is directly proportional to the resistivity of the fluid

F is called the Formation Factor and describes the effect of the presence of the
rock matrix.

Energies of neutrons used

Fast neutrons >0.5 MeV
Intermediate 102 to 105 eV
Epithermal 0.1 to 100 eV
Thermal neutrons < 0.1 eV
Combining Archie’s Laws

Porosity (sonic)

Formation-water resistivity

MOVABLE HYDROCARBON
Water saturation of the flushed zone (Sxo)

Gamma rays with energy >3 MeV (pair production).

Gamma rays with energy 0.5 to 3 MeV These gamma rays undergo
Compton scattering.
Gamma rays with energy <0.5 MeV This process is called photo-
electric adsorption, and is important in the Litho-Density tool.
 INVERSION
Least squares solution to the inverse problem Gm = d

UNDERDETERMINED PROBLEMS

When the equation Gm = d does not provide enough information to determine

uniquely all the model parameters, the problem is said to be underdetermined
when there are more unknowns than data.
We shall refer to underdetermined problems that have nonzero prediction error
as mixed-determined problems, to distinguish them from purely
underdetermined problems error.
The Purely Underdetermined Problem
Assume that there are fewer equations than unknown model parameters, that is,
N < M, and that there are no inconsistancies in these equations. It is therefore
possible to find more than one solution for which the prediction error E is zero.
(In fact, we shall show that underdetermined linear inverse problems have an
infinite number of such solutions.)

OVERDETERMINED PROBLEMS
When there is too much information contained in the equation Gm = d for it to
possess an exact solution, we speak of it as being overdetermined. This is the
case in which we can employ least squares to select a “best” approximate
solution.
Mixed-Determined Problems

This estimate of the model parameters is called the damped least squares
solution

The generalized inverse of

a) the overdetermined least squares problem
b) the minimum length underdetermined solution 𝐺 −𝑔 =
The Data Resolution Matrix
This matrix describes how well the predictions match the data

The Model Resolution Matrix

R=
The Unit Covariance Matrix
If the data are assumed to be uncorrelated and all have equal variance σ2
the unit covariance matrix is given by
Because the unit covariance matrix, like the data and model resolution matrices,
is independent of the actual values and variances of the data, it is a useful tool in
experimental design.
Householder Transformations
The minimum length, least squares, and constrained least squares solutions can
be found through simple transformations of the equation Gm = d.
Transformations of this type that do not change the length of the vector
components are called unitary transformations. They may be interpreted as
rotations an reflections of the coordinate axes. Unitary transformations satisfy

 Discrete ill-posed problems:

The naive solution A -1 b is useless.
Condition number
The condition number associated with the linear equation Ax = b gives a bound
on how inaccurate the solution x will be after approximation. Note that this is
before the effects of round-off error are taken into account; conditioning is a
property of the matrix. In particular, one should think of the condition number
as being (very roughly) the rate at which the solution, x, will change with
respect to a change in b. Thus, if the condition number is large, even a small
error in b may cause a large error in x. On the other hand, if the condition
number is small then the error in x will not be much bigger than the error in b
Find the row sum norm of the following matrix [A]

Properties of Matrix Norm

Properties of the condition number

FREDHOLM EQUATION

In these problems, f(x) typically represents the input (excitation) to a medium

(system), κ stands for the properties of the medium, the kernel and g constitutes
the output (response).
A problem is ill-posed if any one of the following criteria is not satisfied: a
solution exists; the solution is unique; and the solution depends continuously on
the data (stability). As a consequence the effect of noise (which is always
present in physical measurements) becomes very important: even a small error
in the observed value can cause very large variations in the solution.
Such techniques that incorporate smoothness constraints to transform an ill-
posed problem into a well-posed problem are called regularization methods.
Tikhonov regularization
For
When the problem is not well posed (either because of nonexistence or nonuniqueness
of) then the standard approach (known as ordinary least squares) leads to an
overdetermined (Overfitted), or more often an underdetermined (underfitted) system
of equations. Most real-world phenomena operate as lowpass filters in the forward
direction where maps to. Therefore in solving the inverse problem, the inverse
mapping operates as a highpass filter that has the undesirable tendency of amplifying
noise (eigenvalues / singular values are largest in the reverse mapping where they
were smallest in the forward mapping).
Ordinary least squares seeks to minimize the sum of squared residuals, which can be
compactly written as

regularization.
Singular Value Decomposition (SVD)
The SVD theorem states

Where the columns of U are the left singular vectors, S (the same dimensions as
A) has singular values and is diagonal; and VT has rows that are the right singular
Vectors.
Calculating the SVD consists of finding the eigenvalues and eigenvectors of AAT and ATA.
The eigenvectors of ATA make up the columns of V, the eigenvectors of AAT make up the
columns of U. Also, the singular values in S are square roots of eigenvalues from AAT or
ATA. The singular values are the diagonal entries of the S matrix and are arranged in
descending order. The singular values are always real numbers ,if the matrix A is a real
matrix, then U and V are also real.
Backus-Gilbert method
The Backus–Gilbert method, also known as the optimally localized
average (OLA) method is named for its discoverers. Backus–Gilbert instead seeks to
impose stability constraints, so that the solution would vary as little as possible if the
input data were resampled multiple times. But regularisation methods seek for
smoothness of solution.
SIMULATED ANNEALING
Metropolis introduced a simple algorithm for simulating the evolution of a solid in a heat
bath to thermal equilibrium. Their algorithm is based on Monte Carlo techniques and
generates a sequence of states of the solid in the following way. Given a current state i of
the solid with energy Ei then a subsequent state j is generated by applying a perturbation
mechanism which transforms the current state into a next state by a small distortion, for
instance by displacement of a particle. The energy of the next state is Ej. If the energy
difference, Ej — Ei, is less than or equal to zero, the state j is accepted as the current
state. If the energy difference is greater than zero, then the state j is accepted with a
probability given by

where T denotes the temperature of the heat bath and kB is a physical constant called the
Boltzmann constant. . The acceptance rule described above is known as the Metropolis
criterion and the algorithm that goes with it is known as the Metropolis algorithm. It is
known that, if the lowering of the temperature is done sufficiently slowly, the solid can
reach thermal equihbrium at each temperature. In the Metropohs algorithm this is
achieved by generating a large number of transitions at a given value of the temperature.
Thermal equilibrium is characterized by the Boltzmann distribution, which gives the
probabiHty of the solid of being in a state / with energy Ei at temperature T, and which is
given by

Genetic Algorithms
Once the problem is encoded in a chromosomal manner and a fitness measure for
discriminating good solutions from bad ones has been chosen, we can start to evolve
solutions to the search problem using the following steps:
1 Initialization. The initial population of candidate solutions is usually generated
randomly across the search space.
2 Evaluation. Once the population is initialized or an offspring population is created, the
fitness values of the candidate solutions are evaluated.
3 Selection. Selection allocates more copies of those solutions with higher fitness values
and thus imposes the survival-of-the-fittest mechanism on the candidate solutions. The
main idea of selection is to prefer better solutions to worse ones, and many selection
procedures have been proposed to accomplish this idea, including roulette-wheel
selection, stochastic universal selection, ranking selection and tournament selection, some
of which are described in the next section.
4 Recombination. Recombination combines parts of two or more parental solutions to
create new, possibly better solutions (i.e. offspring). There are many ways of
accomplishing this and competent performance depends on a properly designed
recombination mechanism. The offspring under recombination will not be identical to any
particular parent and will instead combine parental traits in a novel manner .
5 Mutation. While recombination operates on two or more parental chromosomes,
mutation locally but randomly modifies a solution. Again, there are many variations of
mutation, but it usually involves one or more changes being made to an individual’s trait
or traits. In other words, mutation performs a random walk in the vicinity of a candidate
solution.
6 Replacement. The offspring population created by selection, recombination, and
mutation replaces the original parental population. Many replacement techniques such as
elitist replacement, generation-wise replacement and steady-state replacement methods
are used in GAs.
7 Repeat steps 2–6 until a terminating condition is met.
Artificial neural networks
It is a computational system inspired by the Structure ,Processing Method, Learning Ability of a
biological brain
Characteristics of Artificial Neural Networks
A large number of very simple processing neuron-like processing elements. A large number of
weighted connections between the elements Distributed representation of knowledge over the
connections Knowledge is acquired by network through a learning process
Why Artificial Neural Networks?
- Massive Parallelism - Distributed representation - Learning ability - Generalization ablity - Fault
tolerance
 An arrangement of one input layer of activations feeding forward to one output layer of
neurons is known as a simple Perceptron.
REMOTE SENSING
EM Spectrum of wavelengths
Geo-stationary Satellites: Geostationary satellites are the satellites which revolve round the earth
above the equator at the height of about 36,000 to 41,000 km., in the direction of earth’s rotation.
They make one revolution in 24 hours, synchronous with the earth’s rotation .

Sun-synchronous Satellites: Sun-synchronous satellites are the satellites which revolved round
the earth in north-south direction (pole to pole) at the height of about 300 to 1000 km. (Fig.2)
They pass over places on earth having the same latitude twice in each orbit at the same local sun-
time, hence are called sun-synchronous satellites. Through these satellites, the entire globe is
covered on regular basis and gives repetitive coverage on periodic basis