A problem of interest in the analysis of geophysical time series involves a simple model

Solution

A problem of interest in the analysis of geophysical time series involves a simple model for observed data containing a signal and a re?ected version of the signal with unknown ampli?cation factor a and unknown time delay δ. For example, the depth of an earthquake is proportional to the time delay δ for the P wave and its re?ected form P on a seismic record. Assume the signal is white and Gaussian with variance σ2, and consider the generating model x(t) = s(t) + a*s(t−δ).

(a)    Prove the process x(t) is stationary. If |a| <1, show that ∞ s(t)=(−a)jx(t -jδ)  , j=0 is  a  mean  square  convergent  representation  for  the  signal  s(t), for

t = 1, ±1, ±2,.. ..

As E[s(t)]=0 --> E[x(t)]=0.

Process white gaussian is of 0 mean and sigma2 variance.

(b)   If the time delay δ is assumed to be known, suggest an approximate computational method for estimating the parameters a and σ2 using maximum likelihood and the Gauss–Newton   method.

Var(X)=(1+a^2)Var(s)  --> Var(X)=(1+a^2)(sigma2) --> for a given Z one can find X and Y with Z=(1+X)Y, where X=a^2 and Y=variance of the signal.

(c)    If the time delay δ is an unknown integer, specify how we could estimate the parameters including δ. Generate a n = 500 point series with a = .9,  σ2= 1  and  δ = 5.  Estimate  the  integer time delay δ by searching over δ = 3, 4,..., 7.

Var(X)=(1+0.9^2)*1=1.81 -->s=sqrt(1.81)=1.34

X=rand(500,1,0,1.34)

delta=4