# Microwave Imaging Radiometers by Aperture Synthesis Performance Simulator (Part 2): Instrument Modeling, Calibration, and Image Reconstruction Algorithms

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## Abstract

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## 1. Introduction

#### 1.1. Instrument Software Module

#### 1.2. Calibration Software Module

- internal calibration, including 1-0 unbalance correction for 1 bit/2 level correlator offsets, the Fringe Washing Function (FWF) at the origin, and eventually the shape correction, estimation/measurement of the system temperatures, offset errors correction, and receiver quadrature error correction using uncorrelated and two-level correlated noise injection.
- external calibration including the flat target response characterization, rredundant space calibration, phase/amplitude closures, and use of an external beacon.

#### 1.3. Image Reconstruction Software Module

#### 1.4. Performance Analysis and Visualization Modules

- Snapshot Mode: in this mode the processing of a single brightness temperature snapshot, corresponding to the instrument integration period, can be performed to make the comparison against the reference BT maps. It is possible to define a sequence of calibration steps to be performed prior to the snapshot processing (e.g., 10 snapshots of internal calibration: centralized noise injection or user-defined; three snapshots of external flat target response).

- Monte-Carlo Mode: in this mode the instrument performance can be evaluated by analyzing a sequence of consecutive snapshots for a given constant brightness temperature map. For each snapshot, a set of system/instrument random errors can be included in the simulation and the component parameters that are fixed and are allowed to vary are defined. Additionally, a simplified error mode is available, in which only the receivers’ amplitude, phase and correlator offset errors are defined for each receiver.

- Time Evolution Mode: the simulation in this mode allows the processing of a single snapshot or several snapshots to analyze time-dependent processes, including time-dependent input BT maps, instrument aging, thermal drifts (antennas’ and receivers’ physical temperatures along with the orbit and date), gain stability, or other instrument component model time-dependent parameters (e.g., antenna position and orientation).

## 2. Materials and Methods

#### 2.1. Modeling of Instrument System and Subsystems

_{B}is the brightness temperature, T

_{REC}is the physical temperature of the receivers, F

_{n1,2}are the normalized antenna voltage patterns at a given polarization, ${\tilde{\text{r}}}_{mn}^{\text{ii}}$, ${\tilde{\text{r}}}_{mn}^{\text{qi}}$ are the fringe-washing functions (real and imaginary parts), and Ω

_{m,n}are the antenna solid angles [1].

_{0}the complex fringe-wash function at the origin, $\text{CN}=\text{}{\text{T}}_{0}\xb7\left(\mathrm{I}-\text{}{\text{S}}_{\text{NI}}\xb7{\mathrm{S}}_{\text{NI}}^{*}\right)\xb7{\mathsf{\delta}}_{\text{pq}}$, and T

_{0}= 290 K.

#### 2.1.1. Antenna Array Component

_{mn}, v

_{mn}, w

_{mn}) being measured at each particular time (antenna positions may be moving) are determined. Figure 3 shows sample SAIRPS antenna arrays: (a) static arrays; (b) rotating arrays; and (c) (u, v) points (baselines) measured by the rotating arrays.

_{n}, Δy

_{n}, Δz

_{n}) with respect to the antenna nominal positions (x

_{n}(t), y

_{n}(t), z

_{n}(t)) also account for (x

_{n}(t), y

_{n}(t), z

_{n}(t)) + (Δx

_{n}, Δy

_{n}, Δz

_{n}) in the “direct problem”, meaning the computation of the instrument’s observables, while the estimated (measured on the ground or known from models) antenna phase centers (x

_{n}(t), y

_{n}(t), z

_{n}(t)) are used in the “inverse problem” or image reconstruction from the instrument’s observables.

_{n m,n}term. However, these equations implicitly assume that the antennas only pick up radiation from the half hemisphere around the antenna boresight. This is not the case in practice, and both the front and back lobes must be accounted for. The brightness temperature contribution coming from the platform(s) supporting the antennas has been neglected. Other contributions to the brightness temperature coming from the back of the antenna are computed by running the Radiative Transfer Module (companion paper, Part I), but looking in the opposite direction. The equations, including the back lobes, turn out to be exactly the same as Equation (2), but changing “w” to “−w” in the baseline definition, $\Delta r=-{\mathsf{\lambda}}_{0}\left(\text{u}\mathsf{\xi}+\text{v}\mathsf{\eta}-\mathrm{w}\sqrt{1-{\mathsf{\xi}}^{2}-{\mathsf{\eta}}^{2}}\right)$, and, obviously, using the BT distribution present in the back side of the array. The real and imaginary parts of the visibility samples computed from the back lobes (90° < θ ≤ 180°) must be added to the ones computed from the fore lobe (0° ≤ θ ≤ 90°).

_{0}, ϕ

_{0}) by two parameters,

_{0}, ϕ

_{0}) is the direction of the maximum of the antenna pattern (not necessarily the array boresight +z), A

_{a}and A

_{f}are the magnitudes of the amplitude (with respect to 1) or phase (radians) ripples, n

_{a}and n

_{f}are the frequencies of the amplitude and phase ripples, and Φ

_{a}and Φ

_{f}are the phases of the amplitude and phase ripples to randomize them and avoid all ripples being in phase at the antenna maximum; $\epsilon $ is an arbitrary small number to avoid singularities.

_{Ωm}= ${\overline{\mathsf{\eta}}}_{\mathsf{\Omega}}$ average plus deviations Δη

_{Ωm}) must also be taken into account to properly compute the overall frequency response and noise figure of each receiving channel. The antenna ohmic losses are modeled as a perfectly matched attenuator of parameters${\left|{S}_{12}\right|}^{2}={\left|{S}_{21}\right|}^{2}=1/{\eta}_{\mathsf{\Omega}}$, and ${\left|{S}_{11}\right|}^{2}={\left|{S}_{22}\right|}^{2}=0$, right after an ideal lossless antenna. The antenna physical temperature (T

_{ph}(t)) is also used to compute the actual noise introduced by the antenna ohmic losses, since the noise power at the antenna output is given by a different antenna temperature: ${T}_{A}^{\prime}={T}_{A}{\eta}_{\mathsf{\Omega}}+{T}_{ph}\left(1-{\eta}_{\mathsf{\Omega}}\right)$, and the added noise at the antenna input is given by $\Delta {T}_{R}={T}_{ph}\left(1/{\eta}_{\mathsf{\Omega}}-1\right)$.

_{21}) between two parallel or collinear half-wavelength dipoles as a function of the distance in terms of the wavelength. As expected, the mutual coupling between collinear dipoles (in red) is much weaker and decreases faster than between the parallel dipoles (in blue). Figure 5 shows the simulated antenna coupling (S

_{21}) in amplitude and phase between two co-planar half-wavelength dipoles at (0,0) and (0,λ) as a function of the angle between their axes. As expected, coupling is at maximum when they are parallel and at minimum (zero) when they are perpendicular. This model allows us to simulate in a realistic way complex three-dimensional (3D) arrays with arbitrary configurations and antenna orientations such as the one shown in Figure 5c.

_{mn}of the visibility matrix $\stackrel{=}{V}$ corresponds to the cross-correlation between the signals collected by antennas m and n, and the $\stackrel{=}{C}$ matrix is defined as:

_{in}and Z

_{L}are the input and load impedances of the antennas (assumed all to be the same for the sake of simplicity), and $\stackrel{=}{Z}$ is the Z-parameter matrix and it is given by:

_{1m}is the mutual impedance between elements 1 and m, and d is the basic antenna spacing (in wavelengths).

#### 2.1.2. Receiver Component

#### Modeling of Receiving Chains Frequency Response

#### Noise Modeling

_{s}is the corresponding source reflection coefficient, and Γ

_{0}the output reflection coefficient, computed as:

_{ph}is the physical temperature.

_{1}and c

_{2}in two ports, and it is detailed in [6]. These noise waves add to the outgoing signal waves b

_{1}and b

_{2}(Figure 7), as follows:

_{1}and c

_{2}and their correlation is given by Equations (23a)–(23d) from [6] (except for the Boltzmann’s constant term, omitted below).

_{0}= 290 K, Z

_{0}is the normalization impedance, T

_{min}is the minimum noise temperature that will be attained when the device is charged with Γ

_{opt}at the input, Γ

_{opt}is the input reflection coefficient which provides the best noise performance, and R

_{n}is the equivalent noise "resistance". These four parameters (Γ

_{opt}is complex) are the ones that describe the noise performance of an active device, and they can be mapped into the four noise wave parameters ${\left|{c}_{1}\right|}^{2}$, ${\left|{c}_{2}\right|}^{2}$, and ${c}_{1}{c}_{2}^{*}$ (which is also complex). In order to refer $\overline{{\left|{c}_{2}\right|}^{2}}$ and $\overline{{c}_{1}{c}_{2}^{*}}$ at the input, it suffices to divide Equation (23b) by ${\left|{S}_{21}\right|}^{2}$, and Equation (23c) by ${S}_{21}^{*}$ (see Equation (10) in [6]).

_{11}and S

_{21}parameters (amplitude and phase, X- and Y-polarizations, I- and Q-channels); and (c) the end-to-end frequency responses (amplitude and phase) for X- and Y-polarizations, I- and Q-channels.

#### Modeling the Fringe Washing Function

_{rec}·(N

_{rec}-1)/2 pairs: 1-2, 1-3, 1-4, … 1-N

_{rec}; 2-3, 2-4, … 2-N

_{rec}; 3-4, 3-5…3-N

_{rec}…) from a number of complex samples of the individual receiver frequency responses:

_{n m,n}is the frequency response of receiver m or n, respectively (I- or Q-branches), normalized to unity, and ${B}_{m,n}$ are the noise bandwidths. The symbol F

^{−1}is the inverse Fourier transform and u() is the step unit function. Note that the quadrature channel frequency transfer function used in ${\tilde{r}}_{mn}^{qi}$ includes a 90° phase shift introduced in the local oscillator. Figure 10 shows sample fringe-washing functions for the X- and Y-polarizations, II and QI pairs, computed using Equation (25).

_{s}, 0, T

_{s}), mimicking the fitting in Equation (27). On the other hand, since the shape of the FWF is an even function, and the relative bandwidth is very small, it can be simply ignored and set to 1. Note that the phase and amplitude at the origin are the values that need to be calibrated.

#### 2.1.3. Correlator Model

_{p}and Y

_{p}are the positions where the steps of the transfer function occur (Figure 11). In some particular and simplified cases, Equation (29) can be obtained analytically, for instance in the case of quantifying with two levels (one bit, $\text{Q}=\mathrm{P}=1$ and ${\Delta}_{0}^{x}={\Delta}_{0}^{y}=2$). In this case, Equation (28) becomes the well-known solution stated [8]:

_{m}and I

_{n}channels (or Q

_{m}and Q

_{n}) used to compute the real part of the visibility sample, and ${\mathrm{T}}_{\text{Skew},\text{imag}}$ is the skew error between the Q

_{m}and I

_{n}channels (or I

_{m}and Q

_{n}) used to compute the imaginary part of the visibility sample.

- the aperture jitter, which is related to the random sampling time variations in the ADC caused by thermal noise in the sample and hold circuit. The aperture jitter is commonly modeled as an independent Gaussian variable with zero-mean and a standard deviation of ${\mathsf{\sigma}}_{\text{Jitter}}={\mathsf{\sigma}}_{\text{app}}$.
- The clock jitter, which is a parameter of the clock generator that drives the ADC with the clock signal. The clock jitter is modeled as a Wiener process, i.e., a continuous-time, non-stationary random process with independent Gaussian increments with a standard deviation equal to ${\mathsf{\sigma}}_{\text{Jitter}}=\sqrt{{\mathsf{\sigma}}_{\text{clk}}\xb7\left|\mathrm{n}-\mathrm{m}\right|\xb7{\mathrm{T}}_{\mathrm{s}}}$.

_{i}. This spectrum can be approximated by:

_{ph}is the physical temperature of the LO, G is the gain of the LO driver (assuming that the LO needs to be amplified for distribution to different channels, if not G = 1), and Isolation is the LO-IF isolation of the mixer.

#### 2.1.4. Methodology to Include Instrument Errors

#### Realistic Instrument Modeling

_{xy}(${R}_{xy}=\mathrm{q}\left[{\mathsf{\rho}}_{xy}\right]$, Equation (29), with x = I

_{m}or Q

_{m}, and y = I

_{n}). The addition of thermal noise applies both in observation and noise injection modes.

#### Uncorrelated Noise Model

_{m}= g

_{n}, and ${\sigma}_{{g}_{m}\left(x\right)}^{2}={\sigma}_{{g}_{n}\left(y\right)}^{2}$.

_{q}for three and 15 levels, with sampling at exactly the Nyquist rate, is presented. The three-level curve converges more quickly to zero than the 15-level curve, because the shape of the FWF is sharper due to the higher non-linear distortion at the three-level than at the 15-level quantization. This effect is equivalent to having more uncorrelated samples. Despite the mean of the correlation being lower for the three-level than the 15-level quantization, the variance of the correlation is also lower in this case.

_{q}= N = B·τ, where N is the total number of uncorrelated samples, B is the noise bandwidth, and τ is the integration time. The evaluation of Equation (35) may be impractical from the implementation point of view, due to excessive time spent in the evaluation of the summation. In [8], the concept of effective integration time is used:

#### Correlated Noise Model

_{A}and T

_{R}are the antenna temperature and receiver’s noise temperature, and $\Delta \text{f}={\mathrm{f}}_{0}-{\mathrm{f}}_{\text{LO}}$, where ${\mathrm{f}}_{0}$ is the central frequency and ${\mathrm{f}}_{\text{LO}}$ is the local oscillator frequency, B is the receivers’ noise bandwidth, and the triangle function is defined as $\mathsf{\Lambda}\left(x\right)=1-\left|x\right|;\text{}\left|x\right|\le 1$.

#### Simplified Instrument Modeling

_{A})), phase errors (${\mathsf{\u03f5}}_{\mathsf{\varphi}}$: mean value + zero-mean Gaussian random variable determined by its standard deviation N(0, σ

_{ϕ}) or a uniform random variable from −π to +π: U(−π, +π)), and offset errors (${\u03f5}_{offset,\text{}r}$ and ${\u03f5}_{offset,\text{}i}$: mean value + zero-mean Gaussian random variable determined by its standard deviation for the real and imaginary parts N(0, ${\sigma}_{{V}_{offset,\text{}r}}$) and N(0, ${\sigma}_{{V}_{offset,\text{}i}}$)). Antenna co- and cross-polar pattern errors include a model for a sinusoidal oscillation of amplitude and phase equal to zero-mean Gaussian random variables determined by their standard deviations.

#### 2.2. Modeling of Instrument Calibration

#### 2.2.1. Internal Calibration

#### Calibration by Noise Injection (NI)

#### PMS Calibration

#### Visibilities Denormalization

#### Correlator Transfer Inversion

_{mn}) are defined as the actual cross-correlation (either analog ρ

_{xy}or digital R

_{xy}) divided by the product of the standard deviation of the two signals being cross-correlated (σ

_{x}·σ

_{y}). Then, the inverse of the correlator transfer function (Equation (28) and Figure 12a) is performed, averaging the number of ADC sigma levels, the reference correlator function is computed in the instrument simulation.

#### Quadrature Error Correction

#### FWF at the Origin

#### FWF Shape Computation

#### 2.2.2. External Calibration

#### Calibration by Redundant Space Calibration (RSC)

_{m,n}and ${g}_{m,n}$. Without loss of generality the phases and amplitudes of all elements can be referred to as an arbitrary arm antenna “i”, whose phase is set to zero (f

_{i}= 0), and its amplitude to one (g

_{i}= 1, G

_{i}= 0).

#### Calibration by Phase/Amplitude Closures

#### Calibration Using External Sources

#### The Flat Target Transformation

#### 2.3. Image Reconstruction Algorithms

- Compensation of Polarization Rotation

- A Priori BT Compensation

_{REC}” in Equation (2)), sky as seen by the antenna back lobes, and the sea, land, sun and moon contributions [21], from which the differential visibilities are computed:

- Windowing of the Visibility Samples

#### 2.3.1. Non-Uniform Fast Fourier Transform

_{REC}term, the obliquity factor, and the antenna patterns terms).

#### 2.3.2. G-Matrix Method

- -
- Due to the finite spatial frequency coverage of the visibility samples, the solution of Equation (54) is not unique. Typically, a minimum norm solution is sought.
- -
- In addition, since there is usually aliasing because of the sampling of the spatial frequency domain (i.e., minimum antenna spacing larger than half a wavelength for a linear, rectangular U-, L, or T-array, or larger than the wavelength over $\sqrt{3}$ for an equilateral triangle, hexagonal or Y-shaped array), the solution is only defined in a region smaller than the unit circle ${\xi}^{2}+{\eta}^{2}\le 1$.
- -
- All redundant baselines or only the non-redundant ones can be used. However, it is useful to perform a weighted average of the redundant baselines first (rows of the G-matrix) to reduce the noise.

#### G-Matrix Inversion Using Moore-Penrose Pseudo-Inverse

#### G-Matrix with an Extended CLEAN Iteration

#### G Matrix Algorithm—Other Iterative Algorithms

#### 2.4. Performance Analysis

_{i}. The following metrics are computed inside a predefined area within the alias-free field of view.

- Radiometric bias, which is computed as the spatial average of the error image:$$\Delta {T}_{bias}=\frac{1}{N}{\displaystyle \sum}_{l=1}^{N}\{{\langle {\widehat{T}}_{B}\left({\xi}_{l},{\eta}_{l},{t}_{i}\right)\rangle}_{t}-{T}_{B}\left({\xi}_{l},{\eta}_{l}\right)\}$$
- Radiometric accuracy, which is computed as the spatial standard deviation in the spatial coordinates of the error image:$$\Delta {T}_{accuracy}=\sqrt{\frac{1}{N-1}{\displaystyle \sum}_{l=1}^{N}{\{{\langle {\widehat{T}}_{B}\left({\xi}_{l},{\eta}_{l},{t}_{i}\right)\rangle}_{t}-{T}_{B}\left({\xi}_{l},{\eta}_{l}\right)\}}^{2}}$$
- Radiometric sensitivity, which is computed as the standard deviation in the time domain of the error image (over time, random errors are zero mean):$$\Delta {T}_{sensitivity}=\sqrt{\frac{1}{M-1}{\displaystyle \sum}_{l=1}^{M}{\{{\widehat{T}}_{B}\left(\xi ,\eta ,{t}_{l}\right)-{\langle {\widehat{T}}_{B}\left(\xi ,\eta ,{t}_{i}\right)\rangle}_{t}\}}^{2}}$$

## 3. Simulation Results and Discussion

- A MIRAS-like instrument (single payload of ESA SMOS mission) at 1.413 GHz in LEO orbit consisting of a Y-shaped array with three arms spaced 120°, 23 antennas per arm spaced 0.5770 wavelengths, and a circular aperture of 0.35 wavelengths each. Calibration is performed using two-level centralized noise injection, the four-point calibration for the power measurement systems, and the flat target response.
- A C-band (6.9 GHz) Low Redundancy Linear Array (LRLA) in a LEO orbit with 18 antennas, a minimum antenna spacing of 0.635 wavelengths, and an antenna aperture of 0.35 × eight wavelengths. Calibration is performed using two-level centralized noise injection, the four-point calibration for the power measurement systems, and the flat target response.
- A GAS-like instrument, a GEO sounder at 53.6 GHz (longitude = 0°) consisting of a Y-shaped array with three arms spaced 120°, 12 antennas per arm spaced 2.885 wavelengths, and a circular aperture of 0.35 wavelengths each.

_{rms}), than for the real and imaginary parts of the BTs at the cross-polarization (XY), which also exhibit a different behavior over land and the ocean. This is because the cross-polar antenna patterns are not included in the G-matrix formulation, meaning they are included in the formulation of the direct problem (computation of the visibility samples), but not in the inverse problem (image reconstruction). The inclusion of all the cross-polar patterns requires the use of Equation (54) instead of Equation (55). The benefits of using Equation (54) or Equation (55) are a trade-off between computation time (the matrix to be inverted in Equation (54) is much larger than that in Equation (55)), error amplification (due to a poorer condition number of the matrix in Equation (54)), and error compensation (due to the inclusion of the cross-polar antenna patterns). The actual benefit is depending on the particular values of the cross-polar patterns and has to be evaluated on a case-by-case basis.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 3.**Sample SAIRPS antenna arrays: (

**a**) static arrays; (

**b**) rotating arrays; and (

**c**) (u, v) points (baselines) measured by the rotating arrays.

**Figure 4.**Simulated antenna coupling (S

_{21}) between two parallel or collinear half-wavelength dipoles as a function of the distance normalized to the wavelength.

**Figure 5.**(

**a**) Simulated amplitude and (

**b**) phase of mutual coupling coefficient (S

_{21}) between two co-planar half-wavelength dipoles at (0,0) and (0, λ) as a function of the angle of rotation between them. (

**c**) Sample 3D random array of half-wavelength dipoles, with random orientations, and amplitude [dB] of the mutual coupling matrix.

**Figure 6.**Receiver block diagram (single channel). POLSW1,2 is the polarization switch control signal that selects the input.

**Figure 7.**The schematic representation of a two-port circuit element using scattering parameters and noise waves (definition from [6]).

**Figure 8.**Schematic representation of visibility offset induced by finite antenna coupling and non-zero cross-correlation between outgoing noise waves (${c}_{1}^{n}$).

**Figure 9.**Sample receivers model (X and Y-polarizations, I and Q-channels) generated by the SAIRPS framework is according to the inputs defined by the user: (

**a**) Noise factor (dB) as a function of frequency; (

**b**) S-parameters (amplitude in linear units, phase in radians). S-parameters of in-phase (I) and quadrature (Q) branches of X and Y-pol. receivers as a function of frequency (amplitude (lin) and, phase (rad)); and (

**c**) Frequency response (amplitude in (dB), phase in degrees). Frequency response of in-phase (I) and quadrature (Q) branches of X and Y-pol. receivers as a function of frequency (amplitude (dB) and phase (deg)) (amplitude (lin) and, phase (rad)).

**Figure 10.**Sample amplitude (

**left**) and phase (

**right**) fringe-washing functions for the X- and Y-polarizations, II and QI pairs. In the case of an instrument simulated in fully polarimetric mode, the XY pairs are also computed.

**Figure 11.**Sample generic analog-to-digital transfer function: g

_{i}(x) = ${\sum}_{m=0}^{M-1}{\Delta}_{m}^{i}$ u(x − X

_{m}), X

_{m}are the quantization levels, and ${\Delta}_{m}^{i}$ = X

_{m+1}− X

_{m}the quantization steps. The function plotted has compression gain and different steps in the analog (horizontal axis) and digitized (vertical axis) domains.

**Figure 12.**(

**a**) Relationship between the non-linear and the ideal correlation for different digitization schemes. Other schemes are computed for each simulation, including errors in the quantization thresholds etc.; (

**b**) Root mean square error and ADC span window relationship for different equally spaced quantification levels.

**Figure 13.**Impact of the clock inaccuracies on the correlation value. T

_{skew}is the sampling rate offset between the two signals being correlated (x and y), and T

_{Jitter}is its fluctuations due to clock inaccuracies.

**Figure 14.**Typical L(f) function (adapted from [11]).

**Figure 15.**Effect of the quantization on the cross-correlation and spectrum; (

**a**) cross-correlation (FWFs) for different quantization levels; and (

**b**) their corresponding spectra.

**Figure 16.**Impact on the quantized correlation of different sampling frequencies, FWF deformation, for two quantization levels, a bandwidth 2B (assuming a band-pass signal).

**Figure 17.**Evolution of the standard deviation of the cross-correlation as a function of the number of samples N

_{q}for three, seven, and 15 levels, with sampling at exactly the Nyquist rate.

**Figure 18.**Calibration of the visibility samples and internal noise injection processing in centralized noise injection mode.

**Figure 20.**Real part (

**left**), imaginary part (

**center**), and absolute value (

**right**) of the visibility samples measured by a Y-shaped array (baselines × 10) corresponding to a quasi-point source.

**Figure 21.**Real part (

**left**), imaginary part (

**center**), and absolute value (

**right**) of the visibility samples measured by a Y-shaped array (baselines × 10) corresponding to a flat target, a $\sqrt{cos\left(\theta \right)}$ antenna voltage pattern (so as to compensate for the obliquity factor), and FWF equal to one.

**Figure 22.**Sample results of the NUFFT algorithm for six array configurations: First row shows the antenna positions (red dots) and the (u, v) points, second row shows a reconstructed synthetic image, and third row shows the impulse response of the array or equivalent array factor. (

**a**) LEFT: Y-array with 10 elements per arm and uniform spacing between antenna elements d = λ/√3. CENTER: Y-array with 10 elements per arm and geometrical spacing between antenna elements d = λ/√3, increasing at a ratio of 1.02. RIGHT: Rectangular array with 10 elements per leg and uniform spacing between antenna elements d = λ/√3; (

**b**) LEFT: Linear array with 30 elements uniformly spaced d = λ/√3, CENTER: Circular array with 31 elements randomly spaced. RIGHT: Random 2D array with σ

_{x}= σ

_{y}= 10/3.

**Figure 23.**Equivalent array factor or impulse response for the GAS-like instrument using a Blackmann window.

**Figure 24.**Simulation results for the GAS-like instrument in dual-polarization mode. (

**a**) Input sky BT maps in the instrument reference frame; (

**b**) Image reconstruction using the conjugate gradient (CG) method; (

**c**) Image reconstruction for an ideal instrument using a NUFFT; (

**d**,

**e**) Reconstruction errors in Figure 23b,c; (

**f**) Pixel mean error of the actual reconstruction; (

**g**) Pixel standard deviation; (

**h**) Total pixel rms error.

**Figure 25.**(

**a**) Input BT consisting of a 1000 K–0 K step, (

**b**) reconstructed BT images for an ideal instrument using the NUFFT, computed radiometric sensitivity assuming (

**c**) uncorrelated noise and (

**d**) correlated noise in the visibility samples.

**Figure 26.**Correlator transfer function impact for a SMOS-like instrument: (

**a**) input BT; (

**b**) simulation results using an ideal noise-free instrument with one-bit/two-level correlators; and (

**c**) with three-bit correlators.

**Figure 27.**Three-bit correlator transfer function computed by integration of the analytical formula clearly shows a dependence on the value of sigma (standard deviation of the input signal).

**Figure 28.**(

**a**) Original BT maps; (

**b**) Actual reconstruction using the G-matrix and the conjugate gradient; (

**c**) Error of actual image reconstruction; (

**d**) Pixel rms error.

**Figure 29.**(

**a**) Original BT map; (

**b**) Reconstructed BT image using the TSVD, with threshold value = 1; (

**c**) Reconstructed BT image using the conjugate gradient; (

**d**) Reconstructed BT image for an ideal instrument using the NUFTT; (

**e**) Reconstructed BT image using the extended CLEAN algorithm.

**Table 1.**Typical window functions with radial symmetry $\rho =\sqrt{{\mathrm{u}}^{2}+{\mathrm{v}}^{2}+{\mathrm{w}}^{2}}$.

Window Function | Formula |
---|---|

Rectangular | $W\left(\rho \right)=1$ |

Triangular | $W\left(\rho \right)=1-\frac{\rho}{{\rho}_{max}}$ |

Hamming | $W\left(\rho \right)=0.54+0.461\mathrm{cos}\left(\pi \frac{\rho}{{\rho}_{max}}\right)$ |

Hanning | $W\left(\rho \right)=0.5+0.5\mathrm{cos}\left(\pi \frac{\rho}{{\rho}_{max}}\right)$ |

Blackman | $W\left(\rho \right)=0.42+0.5\mathrm{cos}\left(\pi \frac{\rho}{{\rho}_{max}}\right)0.5+0.08\text{cos}\left(2\pi \frac{\rho}{{\rho}_{max}}\right)$ |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Camps, A.; Park, H.; Kang, Y.; Bandeiras, J.; Barbosa, J.; Vieira, P.; Friaças, A.; D’Addio, S. Microwave Imaging Radiometers by Aperture Synthesis Performance Simulator (Part 2): Instrument Modeling, Calibration, and Image Reconstruction Algorithms. *J. Imaging* **2016**, *2*, 18.
https://doi.org/10.3390/jimaging2020018

**AMA Style**

Camps A, Park H, Kang Y, Bandeiras J, Barbosa J, Vieira P, Friaças A, D’Addio S. Microwave Imaging Radiometers by Aperture Synthesis Performance Simulator (Part 2): Instrument Modeling, Calibration, and Image Reconstruction Algorithms. *Journal of Imaging*. 2016; 2(2):18.
https://doi.org/10.3390/jimaging2020018

**Chicago/Turabian Style**

Camps, Adriano, Hyuk Park, Yujin Kang, Jorge Bandeiras, Jose Barbosa, Paula Vieira, Ana Friaças, and Salvatore D’Addio. 2016. "Microwave Imaging Radiometers by Aperture Synthesis Performance Simulator (Part 2): Instrument Modeling, Calibration, and Image Reconstruction Algorithms" *Journal of Imaging* 2, no. 2: 18.
https://doi.org/10.3390/jimaging2020018