(0,0)
F ig u r e 3.1: 2-D InSAR vector diagram (all vectors lie in the same plane).
P a r a m e te r S y m b o l V a r ia n c e a"
Location antenna 1 Location antenna 2 Baseline vector Point target location Centre frequency 1 Centre frequency 2 Propagation speed 1 Propagation speed 2 Range from 1 to point Range from 2 to point Round trip delay 1 Round trip delay 2 Absolute phase difference Unwrapped phase at point Phase offset (constant)
Ai = {Aly,Al^) A2 = {A2y, A2z) b= {by,bz) f l / 2 Cl C2 ri T2 t\ t2 u + f C ^A x ~ = i^A-2y,(^A2z) àp = (crp„,crpj <7/. a/. O r , ori r-2 O ^tx O t , Co o
90 Chapter 3. Analysis of a Simplified 2-D InSAR Model
3 .2 .1 Genergd M o d e llin g P r o c e d u r e
A general approach to modelling a physical system is depicted in Figure 3.2. Essen tially, the procedure consists of describing the physical system by a set of parameters (some of which we want to know), and a set of equations which define the relationships between the various parameters. In order to estimate the model parameters, a set of measurements is made. The model parameters are then calculated from these mea surements using a parameter estimation technique, which, for complicated systems, is usually in the form of a numerical optimization algorithm. Because the estimates are calculated from noisy data, they too will be in error. It is therefore necessary to char acterize the accuracy of the estimates in some way, perhaps by means of a confidence interval.
We shall apply this general approach to the InSAR reconstruction problem.
Noisy measurements
Estimation algorithm
Best estimates of model parameters (satisfying constraints)
Error estimates in the form of a covariance matrix
Model Constraint
parameters equations (^ priori information)
F ig u re 3.2: Illustration of modelling and parameter estimation procedure.
3 .2 .2 I n S A R M o d e llin g P r o c e d u r e
In this simplified InSAR analysis, the problem of reconstructing the 3-D surface from the unwrapped phase is decomposed^ into two distinct stages, depicted in Figure 3.3:
1. Estimation of the model parameters: antenna locations, radar parameters, phase unwrapping constants etc.
2. Estimation of the spatial location of a point on the interferogram. For every point on the interferogram, the spatial location is estimated using the estimated model parameters together with the particular radar measurements associated with the point.
The inputs to the InSAR model optimization procedure are:
is noted that it is in principle possible to view the problem as one big optimization problem, rather than this simplified decomposition into two distinct stages. A more general approach is described in Chapter 9.
3.2. Problem Definition and Description of 2-D Geometry and Varia,bles 91
Measurements: calibration points & initial estimates of model parameters Covariance matrix R Estimation of model parameters using numerical optimization
Best estimates of system model parameters Covariance matrix R Model parameters Constraint equations {a priori information) Interferogram point measurements (tpt2.9u) Covariance matrix R Estimation of point location Best estimate of reconstructed point Covariance matrix R Model parameters Constraint equations {a priori information)
F ig u r e 3 .3 : D ecom position of the InSAR procedure into two steps: 1) E stim ation of m odel parameters. 2) Estim ation of surface point location.
92 Chapter 3. ^Inalysis of a Simplified 2-D InSAR Model
• Calibration point measurements.
• Errors associated with the calibration point measurements (in the form of a covariance matrix holding all variances and non-zero covariances).
• Model information in the form of defined parameters and constraint equations relating parameters.
The outputs of the optimization procedure are: • The optimized model parameter estimates.
• The errors in the estimates of the model parameters (in the form of a covariance matrix holding all variances and covariances).
The topographic surface is then reconstructed on a point by point basis, by calcu lating the spatial location of every point sample on the interferogram. The problem of reconstructing a single point may be viewed as an optimization problem with inputs:
• The sample information for a particular point on the interferogram (delay and phase measurements).
• The estimated errors in the point sample measurements. • The optimized model parameter estimates.
• The errors in the estimates of the model parameters (in the form of a covariance matrix holding all variances and covariances).
• Model information in the form of defined parameters and constraint equations relating parameters.
and outputs:
• The estimate of the spatial location of the point.
• An estimate of the error associated with the spatial location estimate. In the next section applicable estimation techniques are reviewed.
3.3
R ev iew o f A p p licable E stim ation Techniques
In this section we review two techniques from classical estimation theory^ : “chi-square estimation” and “weighted least-squares” .
^Although in this work we apply techniques from classical estimation theory, the author is a pro ponent of the Bayesian approach, which entails a fundamentally different philosophy. We accept the point estimators derived from classical theory only in the sense that they are equivalent to the peak of a Bayesian posterior distribution under the hypothesis of a uniform “prior”. For details on the difference between classical and Bayesian inference see ( 0 ’Hagan 1994).