Identification of reflection coefficients from noisy data by means of extended minimum variance estimators: A critical examination

Recently, a new class of time-domain state space models has been developed (Ref. 1) to describe layered media systems. When layers are uniform, the resulting state equations are referred to as uniform causal functional equations (UCFE). An example of a UCFE is: x (t + ??) = Ax (t) + b [m(t) + w(t)] (1) where, for a K-layer system, x (t) is a 2K x 1 state vector comprised of K upgoing states and K downgoing states, m(t) is the source signature, w(t) is a random process which reflects uncertainty about our knowledge of m(t), and A and b are matrices (of appropriate dimensions) which are functions of reflection coefficients r0, r1,..., rK which characterize the system. Additionally, ?? is the one-way travel time for each layer. A surface measurement (i.e., seismogram) y(t), where y(t) = h´ x(t) + n(t) (2) is also assumed available. This measurement is corrupted by measurement noise, n(t) and is in terms of vector h which is also a function of some of the reflection coefficients.