RANDOM NOISE IN SEISMIC
DATA:
TYPES, ORIGINS,
ESTIMATION, AND
REMOVAL
Principle Investigator: Dr. Tareq Y. Al-Naffouri
Co-Investigators:
Ahmed Abdul Quadeer
Babar Hasan Khan
Ahsan Ali
ACKNOWLEDGEMENTS

Saudi Aramco

Schlumberger

SRAK

KFUPM
OUTLINE
Introduction
 A breif overview of Noise and Stochastic Process
 Linear Estimation Techniques for Noise Removal





Least Squares
Minimum-Mean Squares
Expectation Maximization
Kalman Filter
Random Matrix Theory
 Conclusion

INTRODUCTION
Seismic exploration has undergone a digital
revolution – advancement of computers and
digital signal processing
 Seismic signals from underground are weak and
mostly distorted – noise!
 The aim of this presentation – provide an
overview of some very constructive concepts of
statistical signal processing to seismic
exploration

WHAT IS NOISE?
Noise simply means unwanted signal
 Common Types of Noise:

Binary and binomial noise
 Gaussian noise
 Impulsive noise

WHAT IS A STOCHASTIC PROCESS?
Broadly – processes which change with time
 Stochastic – no specific patterns

TOOLS USED IN STOCHASTIC PROCESS?

Statistical averages - Ensemble

E ( X tn ) 
n
x
 t p( xt )dxt


Autocorrelation function

Autocovariance function
LINEAR ESTIMATION TECHNIQUES
FOR NOISE REMOVAL
LINEAR MODEL

Consider the linear model

Mathematically,

In Matrix form,
or
LEAST SQUARES & MINIMUM MEAN
SQUARES ESTIMATION
LEAST SQUARES & MINIMUM MEAN
SQUARES ESTIMATION

Advantages:
Linear in the observation y.
 MMSE estimates blindly given the joint 2nd order
statistics of h and y.


Problem: X is generally not known!

Solution: Joint Estimation!
JOINT CHANNEL AND DATA RECOVERY
EXPECTATION MAXIMIZATION ALGORITHM
One way to recover both X and h is to do so
jointly.
 Assume we have an initial estimate of h then X
can be estimated using least squares from



The estimate
can in turn be used to obtain
refined estimate of h
The procedure goes on iterating between x and h
EXPECTATION MAXIMIZATION ALGORITHM

Problems:
Where do we obtain the initial estimate of h from?
 How could we guarantee that the iterative procedure
will consistently yield better estimates?

UTILIZING STRUCTURE TO ENHANCE
PERFORMANCE

Channel constraints:
Sparsity
 Time variation


Data Constraints
Finite alphabet constraint
 Transmit precoding
 Pilots

KALMAN FILTER
A filtering technique which uses a set of
mathematical equations that provide efficient
and recursive computational means to estimate
the state of a process.
 The recursions minimize the mean squared error.
 Consider a state space model

FORWARD BACKWARD KALMAN FILTER

Estimates the sequence h0, h1, …, hn optimally
given the observation y0, y1, …, yn.
FORWARD BACKWARD KALMAN FILTER

Forward Run:
FORWARD BACKWARD KALMAN FILTER


Backward Run: Starting from λT+1|T = 0 and i =
T, T-1, …, 0
The desired estimate is
COMPARISON OVER OSTBC MIMOOFDM SYSTEM
USE OF RANDOM MATRIX
THEORY FOR SEISMIC SIGNAL
PROCESSING
INTRODUCTION TO RANDOM MATRIX THEORY
Wishart Matrix
PDF of the eigenvalues
EXAMPLE: ESTIMATION OF POWER AND
THE NUMBER OF SOURCES
COVARIANCE MATRIX AND ITS ESTIMATE
EIGEN VALUES OF CX
FREE PROBABILITY THEORY
R-Transform
S-Transform
??
APPROXIMATION OF CX
CONCLUSIONS
The Ideas presented here are commonly used in
Digital Communication
 But when applied to seismic signal processing
can produce valuable results, with of course some
modifications
 For Example: Kalman Filter, Random Matrix
Theory
