### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Model Structure and Applicability
- 3. Defining the Model
- 4. Realizing the Model
- 5. Summary
- References
- Supporting Information

[1] Many new Earth remote-sensing instruments are embracing both the advantages and added complexity that result from interferometric or fully polarimetric operation. To increase instrument understanding and functionality, a model of the signals these instruments measure is presented. A stochastic model is used as it recognizes the nondeterministic nature of any real-world measurements, while also providing a tractable mathematical framework. A wide-sense stationary, ergodic, Gaussian-distributed model structure is proposed. Temporal and spectral correlation measures provide a statistical description of the physical properties of coherence and polarization-state. From this relationship, the model is mathematically defined. A method of realizing the model (necessary for applications such as synthetic calibration-signal generation) is given, and computer simulation results are presented. The signals are constructed using the output of a multi-input, multi-output linear filter system, driven with white noise.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Model Structure and Applicability
- 3. Defining the Model
- 4. Realizing the Model
- 5. Summary
- References
- Supporting Information

[2] Radio-interferometers and polarimeters are being used more widely in remote sensing for probing the Earth's lands and oceans [e.g., *Swift et al.*, 1991; *Ruf*, 1988; *Yueh et al.*, 1997]. While these techniques have a rich history in the space sciences, Earth-viewing versions are relatively rare. For example, results from only two synthesis imaging radio-interferometer for airborne remote sensing appear in the literature. The Electronically-Scanned Thinned Array Radiometer (ESTAR) [*Le Vine et al.*, 2001; *Ruf and Principe*, 2003] uses interferometric imaging across-track and real aperture imaging along-track to image land and ocean surfaces from an aircraft. Recently, a Y-shaped 2-D synthesis imager was proposed for spaceflight [*Kerr*, 1998]. Earth Sensing using radio-polarimetry has a similar history, with several airborne instruments appearing in the last decade [*Yueh et al.*, 1995; *Piepmeier and Gasiewski*, 2001; *Bobak et al.*, 2001; *Lahtinen et al.*, 2001] and the launch of a spaceborne Earth-viewing polarimeter in 2003 [*Gaiser*, 1999].

[3] There are distinct differences in the operation and calibration techniques of ground-based space-viewing telescopes versus orbiting Earth-viewing instruments. Operationally, Earth-viewing imagers have a fraction of the integration time available compared to radio-telescopes: seconds versus hours. Additionally, radio-telescope elements have extremely narrow beams (∼0.5°) compared to low-Earth orbit (LEO) synthesis imagers, whose wide beams (∼60–90°) are required to image the entire Earth at least once per day. Perhaps more important are the calibration differences. Interferometric radio-telescopes are typically calibrated using extrasolar point sources, whereby the system can be characterized to within a common gain coefficient [*Thompson et al.*, 1991]. Furthermore, if the flux of the point source is known, the interferometer can be absolutely calibrated. There are no obvious point sources when looking downward at the Earth, at least ones that emit energy within the protected spectrum allocated for passive observing. Furthermore, because of their large beam widths, Earth-viewing interferometers cannot selectively view a single extrasolar point source. (Besides, the time and operations required for regularly rotating a spacecraft for calibration would cause data loss and increase mission risk.) Therefore, a different calibration technique must be devised for orbiting Earth-imaging systems. This paper lays the groundwork for such a technique by rigorously examining the signals that these radiometers measure.

[4] While polarimetry and interferometry can be investigated independently, they are based upon a common concept—measuring the interdependence of two signals. For polarimetry these signals are the amplitudes of some orthogonal pair. In Earth remote sensing, vertical and horizontal polarizations are chosen because they correspond to the Earth's natural polarization basis as viewed from LEO. Two-beam interferometry involves measuring the coherence of two signals separated in space and/or time. In order to fully understand these instruments, it is desirable to have an accurate model of the types of signal pairs they measure. This type of model can be employed in mathematical analysis, computer simulations and for the generation of synthetic calibration signals (one possible on-orbit calibration tool).

[5] The developed model's structure and its applicability to useful physical problems is discussed in Section 2. Using this framework, Section 3 shows how a comprehensive model can be determined from the physical properties of the modeled wave(s). Section 4 gives a computational method for realizing the model. A polarimetric example is carried through Sections 2 and 3. This polarimetric example is implemented in Section 4 in order to demonstrate the ideas presented. A Summary is presented in the final section.

### 4. Realizing the Model

- Top of page
- Abstract
- 1. Introduction
- 2. Model Structure and Applicability
- 3. Defining the Model
- 4. Realizing the Model
- 5. Summary
- References
- Supporting Information

[24] The previous sections have developed a stochastic model for a pair of signals. This model is defined by the temporal and spectral correlation functions of *A*(*t*), *B*(*t*), *C*(*t*) and *D*(*t*). These are found from physical signal properties. This is a clear mathematical model that has the potential to be useful in theoretical analysis. However, for applications such as computer simulations and synthetic signal generation it is necessary to create realizations of these random processes. It is sufficient to create realizations of the real variables *A*(*t*), *B*(*t*), *C*(*t*) and *D*(*t*), as the other variables (*P*(*t*), *Q*(*t*), *X*(*t*) and *Y*(*t*)) can be created by deterministic functions of these four realizations. To do this an assumption about the probability distribution of the processes will need to be made. The obvious choice is to assume a Gaussian distribution. This can be justified by appealing to the Central Limit Theorem and has the advantage that it can be defined by the second-order statistics the model gives. This results in the required wide-sense stationary, ergodic process.

[25] Once the probability density functions are known, it is possible to create a sampled realization directly. If N points of data were required, the correlation matrix for these points could be calculated and a multivariate, Gaussian random number generator applied (such as the mvnrnd command found in the MATLAB software package.) However, difficulties arise when the dimensionality of this p.d.f. is considered. There are four codependent outputs at every sample so a signal of length *N* would have a p.d.f. of dimension 4*N*. For example, a signal of 125 MHz bandwidth requires a minimum sampling rate of 250 × 10^{6} samples per second—thus a two-second signal would require 500 × 10^{6} samples and result in a probability density function of dimension two billion. This is computationally impractical so a simpler method must be found.

[26] A common technique in one dimension is to use a linear filter to shape noise into a desired spectral shape. The problem here is more complicated as the function is from one dimension (time) into four dimensions (*A*(*t*), *B*(*t*), *C*(*t*), *D*(*t*)). A generalized filter structure with *N* inputs (*I*_{1}(*t*), *I*_{2}(*t*), …, *I*_{N}(*t*)) and *M* outputs (*O*_{1}(*t*), *O*_{2}(*t*), …, *O*_{M}(*t*)) is given below (this structure is presented in *Jenkins and Watts* [1968]).

The filter responses must be chosen so that the desired correlation functions are realized. These functions are given by the expressions below [*Jenkins and Watts*, 1968].

This set of equations has a simpler form in the Fourier domain, as the convolutions become multiplications.

The final step takes advantage of conjugate symmetry (which is guaranteed by the fact that *h*_{mn}(*t*) is real for all *mn*.)

[27] A general set of solutions for the system of equations given by equation (12) is nontrivial as the system is nonlinear. Simplifications can be made by making certain assumptions—the first of which is that the input random processes are independent, white, Gaussian, unit variance and zero mean (so = δ(*i* − *j*).) This ensures the outputs will be Gaussian and zero mean as required; and that equation (12) reduces to the form given below.

In this case only four output processes are required (*M* = 4.) This means we have ten independent equations—one for each of the correlation functions given in equation (7) (there are 16 equations but there is redundancy, as shown by the bracketed term in equation (11)). A solution is presented for the case of four input processes (i.e., *N* = 4) although the method can be applied to larger dimensions provided that *M* = *N* and the specified spectra satisfy the properties of a spectral matrix. The equations for this problem (as defined by equation (13)) are given below (the ν dependence has been dropped for clarity.) The outputs are denoted by *A*, *B*, *C*, *D* as before, while the inputs are numbered 1, 2, 3, 4.

Although these equations are nonlinear, they can still be solved relatively easily. The solution method developed is most succinctly expressed in matrix notation.

Using this notation, equation (14) can be rewritten as shown below.

For matrices the ^{†} operation represents a conjugate transpose. In order to find a suitable set of filters it is sufficient to solve equation (17) at the frequency points of interest. It is shown in *Strang* [1976] that because is positive semidefinite (always the case for spectral matrices such as this [*Jenkins and Watts*, 1968]), equation (17) can always be solved. A method for doing this is outlined below.

[29] By comparing equation (18) with equation (17) it can be seen that the filter responses can be calculated by as follows:

Equation (19) shows how equation (14) can be solved by finding the eigenvalues and eigenvectors of the spectral matrix at each frequency point. It is easy to show that this method will produce filter spectra that are conjugate-symmetric, which is necessary to ensure that the filters have a real response. Reordering the eigenvalues and eigenvectors will still result in a valid solution at a single frequency point but care should be taken not to do this when constructing functions over many frequency points. Doing so would result in a sharp discontinuity in the filter spectra produced which would increase the length of the filter impulse response.

[30] In this section it has been shown how independent, white, Gaussian noise functions (which are easily generated) can be fed into a system of linear filters to produce realizations of the model. The filter sets are generated using equation (19) which depends on the model parameters. This process is computationally tractable and produces results like those shown in the example below.

[31] Example: The example functions coming from the signal in section 2.2.2 were realized using the methodology given in this section. This process was carried out using MATLAB and plots of the resulting spectra can be seen in Figure 1. It can be seen that the frequency axes extend outside the region [−*BW*, *BW*] — this corresponds to oversampling. The frequency axes are normalized so that the range of unaliased frequencies falls between −0.5 and 0.5.

[32] The estimates of the realized spectra were found using periodogram averaging. Four hundred spectra were averaged in each case, and a rectangular window was used in the time domain to remove the noisy terms associated with a large ∣τ∣. Each plot has 399 frequency points and the filters were truncated to 199 taps. Figure 1 shows that the resulting spectra agree closely with those specified. The small differences can be accounted for by the necessary truncation of the generation filters and by the fact that a finite number of spectra were used to create the periodogram average. The above model realization can be readily translated into space-qualifiable microwave circuitry for use in the characterization Earth-imaging correlation radiometers (a subject for a separate paper).