Keywords: Modeling, Squint Angle, Optimization, Accuracy.
Abstract By using radargrammetric (range/Doppler) equations for an SAR image, radar's trajectory polynomial modelings require statistical evaluation in order to determine the model optimality. Thus, a least-squares estimation technique, a parametric significance hypothesis testing, and the necessary optimization criteria are introduced, in detail enough for a basic algorithmic understanding. A second-order polynomial expansion is then employed to describe the time-varying squint-angle parameter. Experiments with an airborne SAR image reveal that the time dependence of an imaging radar squint parameter can have an impact on planimetric position accuracy, as would be expected. Capability to adequately model airborne/spaceborne SAR orientation parameters still merits research and development.
1. Introduction
A proper geometric image processing frequently relies on a sound functional model. For a side-looking SAR (synthetic aperture radar) image, consisting of sequential, scanned range-lines, a sound functional model again depends on some time-varying parameter descriptions. In most instances, time polynomial expansions can be used to describe the radar antenna's positions/velocities along a flight path (Lee et al., 2000; Tannous and Pikeroen, 1994; Toutin and Gray, 2000). This is logically done because the unknown position/velocity parameters explicitly exist in the radargrammetric (range and Doppler) equations. Other geometric parameters, such as the squint angle, the pixel spacing and the near-range slant delay, are less obvious as far as their time-dependent modelings are concerned.
This paper addresses a time polynomial modeling for the squint parameter and reports the planimetric accurary gain that can be expected. The least-squares estimator is statistical in nature, therefore sufficient control point coordinate measurements should be made available in an airborne SAR-image. Following a brief introduction of the necessary analytic equations, estimation results will be presented and analyzed.
2. Analytic Relationships
In order to understand what our parameter estimation algorithm does, it is of interest to give the next relevant relationships.
2.1 Functional and Stochastic Models
For the geometric SAR-image processing, the range and Doppler (radargrammetric) conditions are of fundamental importance (Curlander et al., 1987; Dowman, 1992; Gelautz et al., 1998; Leberl, 1979):
where R
ji (m) is the slant range between the radar antenna position
(X
oj,Y
oj,Z
oj) at station j and the object-space target position
(X
x,Y
y,
z) at point i ; see Figure 1 for notation illustrations (Wu and Lin, 2000).
The squint angle t (deg) is such as interpreted in Figure 2, where the instantaneous along-track antenna unit velocity
(u
x,u
y,u
z ) is also shown. In Eq. (3),
M
b(m/pixel) stands for the pixel spacing.
r
i,j (pixels) is the i-th cross-track, pertaining to station j, image coordinate measurement.
R
n (m) represents a constant slant range delay.
Figure 1. Range vector r and position vectors s and p in a given topocentric rectangular system of coordinates denoted by unit vectors x, y, z
Figure 2. Range vector r* in a radar antenna system of rectangular coordinates defined by unit vectors u, v, and w; squint angle t and off-nadir look angle
W after Leberl (1976)
The along-track image coordinate measurement is denoted by
t
j (pixels, or seconds), serving as an argument. The trajectory parameters
X
oj(t), Y
oj(t) and Z
oj(t) are further expanded in polynomials:
The polynomial coefficients/parameters
(a
k,b
k,c
k ) with k = 0, 1, …, will be estimated in a least-squares sense, and be individually checked for their significance by using statistical hypothesis testings. Recapitulating, we now have unknown parameters
(T,M
b,a
k,...,b
k,...,c
k) and image point measurements
(...,r
i,t
j,...) in our functional model. The measurement error variance-covariance matrix then serves as the accompanying stochastic model.
2.2 Time-Varying Squint Parameter
As a logical consequence, the squint angle t is expanded in a second-order polynomial, leading to an extension of the underlying functional model:
Here, the fixed-order polynomial modeling is rather tentative. One may notice from Eq. (4) that higher-order time polynomial terms can give rise to numerical stability problems. The acceptance of any initial convergent parameter solution is statistically based on the global model chi-square distribution test (Leick, 1995), at an
a % significance level.
