# On the Optimal Design of Doppler Scatterometers

## Abstract

**:**

## 1. Introduction

## 2. Optimizing Doppler Scatterometer Parameters Given SNR

#### 2.1. Error Models

#### 2.1.1. Measurement Noise and Wind Speed Error

#### 2.1.2. Radial Velocity Errors

#### 2.2. Optimizing ${K}_{p}$

#### 2.3. Optimizing Radial Velocity Variance

## 3. Optimizing Doppler Scatterometer Parameters over Winds

#### Optimizing Wind Estimates

## 4. Other Design Considerations

#### 4.1. Frequency and Polarization

#### 4.2. Incidence Angle

#### 4.3. Range Ambiguities and Contiguous Scans

#### 4.4. System Spatial Resolution

## 5. Design Examples

## 6. Conclusions

- The system bandwidth should be chosen so that the signal-to-noise ratio is approximately 1. This somewhat counterintuitive result can be understood as balancing the number of looks and the SNR in the ${K}_{p}$ equation, and will typically lead to higher bandwidths than in historical designs (e.g., QuikSCAT).
- Varying the inter-pulse period as a function of scan angle so that the Doppler bandwidth is appropriately sampled (but not over-sampled) can have significant benefits in the radial velocity performance. One should use the opportunity presented by longer pulse correlation times to separate the pulses as much as possible, while lengthening them to improve the SNR per pulse.
- As high a frequency should be chosen as possible, all other things being equal.
- The fast change in brightness with incidence angle strongly suggests that near-nadir incidence angles be used, as in SKIM. However, increasing the antenna length can mitigate this significantly. Near-nadir incidence angles have additional disadvantages in terms of temporal revisit and mapping errors, due to the reduced swath. The incidence angle is probably the parameter that needs most optimization to balance random measurement errors and interpolation mapping errors.
- Varying the PRF has significant advantages for the continuity of the along-track coverage and minimizing range ambiguities.
- It is possible, with systems that are at the present state of the art, to achieve the performance goals outlined by Chelton et al. [2]. A high-power system will exceed these requirements, but may have greater engineering challenges.

- By varying the spacing between the pulses while keeping the total burst energy constant, one can expect significant improvements (up to an order of magnitude in the along-track direction) for the radial velocity error, as shown in Figure 5.
- By varying the PRF, one can increase substantially (up to two orders of magnitude in the along-track direction) the unambiguous footprint, as shown in Figure 9. This allows ground coverage without gaps with a moderate-sized antenna at antenna realizable spin-rates. This is beneficial in the mission mechanical design.

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

Symbol | Description |
---|---|

${\sigma}_{0}$ | Normalized radar cross section. |

$\delta x$ | Error in parameter x |

${K}_{p}$ | Coefficient of variation $=\delta {\sigma}_{0}/{\sigma}_{0}$ |

v | Speed |

$\varphi $ | Azimuth angle. |

${v}_{r}$ | Radial velocity. |

$\mathsf{\Phi}$ | Pulse-pair phase. |

${v}_{r}$ | Radial velocity |

$\lambda $ | EM wavelength |

k | EM wavenumber $=2\pi /\lambda $ |

${\tau}_{p}$ | Inter-pulse period. |

${\sigma}_{x}$ | Standard deviation of parameter x. |

${N}_{r}$ | Number of range samples. |

${N}_{b}$ | Number of independent pulse pair samples |

$\gamma $ | Total correlation coefficient. |

${\gamma}_{N}$ | Thermal noise correlation. |

${\gamma}_{T}$ | Temporal correlation. |

${\gamma}_{D}$ | Doppler correlation. |

${T}_{B}$ | Burst time. |

${T}_{c}$ | Pulse-pair correlation time. |

${T}_{W}$ | Water correlation time. |

${v}_{p}$ | Platform speed |

SNR | Signal-to-noise ratio. |

$\nu $ | Noise-to-signal ratio $={\mathrm{SNR}}^{-1}$ |

${\sigma}_{\varphi a}$ | Gaussian azimuth antenna pattern standard deviation. |

L | Antenna length |

$\eta $ | Azimuth beamwidth factor. |

$\tau $ | ${\tau}_{p}/{T}_{c}$ |

B | System bandwidth |

X | Desired range resolution. |

$\theta $ | Incidence angle |

c | Speed of light. |

${B}_{1}$ | Minimum bandwidth $=c/2Xsin\theta $ |

${\mathrm{SNR}}_{1}$ | ${N}_{r}\mathrm{SNR}$ |

${J}_{v}$ | Noise-only minimization function (Equation (20)) |

${N}_{ro}$ | Optimal number of range looks |

${K}_{pmin}$ | Optimum ${K}_{p}$ |

${J}_{\tau \nu}$ | Temporal and noise minimization function (Equation (25)) |

${\tau}_{o}$ | Optimal $\tau $ |

${\nu}_{o}$ | Optimal $\nu $ |

${\sigma}_{vr\phantom{\rule{0.166667em}{0ex}}o}$ | Optimal radial velocity |

${v}_{ML}$ | Most likely wind speed. |

s | Wind speed normalized by most likely wind speed. |

$f\left(s\right)$ | Wind speed probability density function. |

${\sigma}_{0ML}$ | ${\sigma}_{0}$ for most likely wind speed |

${\sigma}_{0NE}$ | Noise-equivalent ${\sigma}_{0}$ |

${\nu}_{ML}$ | $\nu $ for most likely wind speed |

${\sigma}_{0r}$ | ${\sigma}_{0}$ relative to most likely wind $={\nu}_{ML}/\nu $ |

$<>$ | Average over wind speeds |

${\Delta}_{az}$ | Antenna azimuth resolution |

${X}_{az}$ | Final azimuth resolution after averaging |

${N}_{S}$ | Number of contiguous range-direction footprints |

D | Size of range footprint |

${T}_{R}$ | Rotation period |

R | Inner radius of the scanned annulus |

${D}_{0}$ | Width of the annulus (i.e., the along-track swath) |

$d\left(\varphi \right)$ | Along-track direction in order to ensure along-track continuity |

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**Figure 1.**Burst-mode timing diagram conventions. ${N}_{p}$ chirped pulses with inter-pulse duration separation ${\tau}_{p}$ (gray) are transmitted during a transmit cycle of duration ${T}_{p}$, and the return pulses (blue) are range compressed so that the duration of the return pulses is dominated by the illuminated swath. It is assumed that the interval between transmit pulses is minimized, and the pulse duration, ${t}_{p}$, obeys ${T}_{B}\approx {N}_{p}{t}_{p}$. (To get the exact results below, multiply the SNR by ${t}_{p}/{\tau}_{p}$.) To form the measurement of interest, ${N}_{r}$ range samples are averaged for each received pulse (blue). Additional looks are obtained by averaging over the return pulses, but, since pulses are correlated for times smaller than ${T}_{c}$, only ${N}_{b}={T}_{B}/{T}_{c}$ independent samples are obtained. In general, for fast pulsing, there will be overlaps (range ambiguities) for parts of the received pulse, limiting the useful swath.

**Figure 2.**Ka-band ${\sigma}_{0}$ (not in dB) averaged over all azimuth angles divided by ${\sigma}_{0ML}$ ∼ 7.8 × 10${}^{3}$ (∼−22 dB), the cross section of the most likely wind speed value, ${v}_{ML}=5.9$ m/s, as a function of the wind speed divided by ${v}_{ML}$. (Red line) Data from [13]; (Blue line) Quadratic fit. (Purple line) Rayleigh distribution of normalized wind speed.

**Figure 3.**Ratio of $\left(\right)$ to the minimum achievable value of ${J}_{\nu}$ as a function of $\nu $ and for lower wind speed thresholds of 1 m/s (aqua), 2 m/s (magenta), and 3 m/s (gray). In red is the optimal curve given in Equation (20). The location of the optimal $\nu $ is indicated by a circle. For Equation (20), the optimum $\nu =1$, but the optimum value becomes smaller when the wind speed distribution is accounted for, as in Section 3.

**Figure 4.**(

**Upper row**) Two dimensional of the optimal function as a function of $\nu $ and $\tau $ for Equation (25) (

**left**) and for the wind averaged case (

**right**), Equation (34), choosing a threshold wind speed of 2 m/s (c.f., Section 3). (

**lower row**) Cross sections at optimal $\nu $ (

**left**) and optimal $\tau $ (

**right**) for the optimal case (red), and for various lower wind-speed thresholds: of 1 m/s (aqua), 2 m/s (magenta), and 3 m/s (gray).

**Figure 5.**Degradation in performance in ${J}_{\tau \nu}$ for fixed PRF relative to the value where the PRF is allowed to vary as a function of scan angle for varying antenna length L. On the left panel, the PRF is optimized for broadside viewing (e.g., as in SKIM), while, on the right, it is optimized to have lowest error in the mid-swath. As can be seem, performance can degrade by close to an order of magnitude for the worst scan angles if the PRF is not varied over the scan. A temporal correlation time of 1 ms was used for these calculations.

**Figure 6.**(

**left**) Ratio of ${K}_{p}$ as a function of wind speed to ${K}_{p}$ for the maximum-likelihood wind velocity for the optimal value of $\nu $ and a lower value representing lower bandwidth: the performance is robust to picking the bandwidth; (

**right**) ratio of the radial velocity as a function of wind speed to the radial velocity at the maximum-likelihood wind for various values of $\tau $.

**Figure 7.**Relative radial velocity standard deviation referenced to the performance at the SKIM incidence angles, all other parameters being held constant. The penalty in the projection factor is outweighed by the greater brightness at near-nadir incidence.

**Figure 9.**(

**Left**) Width of the range-PRF unambiguous swath as a function of the normalized cross-track distance from the nadir path. A temporal correlation time of 1 ms is assumed, and fixed antenna area. (

**Right**) Cartoon illustrating the symbols used and showing how ambiguity and overlap requirements vary as a function of flight direction and scan angle. The blue annulus shows the region that has no range ambiguities using the broadside PRF. In green is the area that has no range ambiguities when looking along the flight path. The area enclosed by the red lines shows the area required to be covered to ensure along-track swath continuity.

**Figure 10.**Surface projected radial velocity error as a function of normalized cross-track distance from nadir for a peak output power of 100 W and varying antenna length, L. This performance assume only one pencil beam.

**Figure 11.**Expected surface velocity standard deviation as a function of normalized cross-track distance from nadir for the along-track (

**left**) and cross-track (

**right**) surface velocity components for a peak output power of 100 W and varying antenna length, L.

**Figure 12.**Surface-projected radial velocity error as a function of normalized cross-track distance from nadir for antenna lengths, L, of 4 m (red) and 5 m (blue) and for peak output power of 100 W (no marker), 400 W (filled circles), 1.5 kW (empty rectangles).

**Figure 13.**Expected surface velocity standard deviation as a function of normalized cross-track distance from nadir for the along-track (

**left**) and cross-track (

**right**) surface velocity components for antenna lengths, L, of 4 m (red) and 5 m (blue) and for peak output power of 100 W (no marker), 400 W (filled circles), 1.5 kW (empty rectangles).

L | 2 m | 3 m | 4 m | 5 m |
---|---|---|---|---|

Transmit Power | $\begin{array}{c}120\phantom{\rule{0.166667em}{0ex}}\mathrm{W}\\ 400\phantom{\rule{0.166667em}{0ex}}\mathrm{W}\\ 1.5\phantom{\rule{0.166667em}{0ex}}\mathrm{kW}\end{array}$ | $\begin{array}{c}120\phantom{\rule{0.166667em}{0ex}}\mathrm{W}\\ 400\phantom{\rule{0.166667em}{0ex}}\mathrm{W}\\ 1.5\phantom{\rule{0.166667em}{0ex}}\mathrm{kW}\end{array}$ | $\begin{array}{c}120\phantom{\rule{0.166667em}{0ex}}\mathrm{W}\\ 400\phantom{\rule{0.166667em}{0ex}}\mathrm{W}\\ 1.5\phantom{\rule{0.166667em}{0ex}}\mathrm{kW}\end{array}$ | |

Altitude | 700 km | 700 km | 700 km | 700 km |

Total Swath | 1706 km | 1706 km | 1706 km | 1706 km |

$\lambda $ | $0.008$ m | $0.008$ m | $0.008$ m | $0.008$ m |

Polarization | VV | VV | VV | VV |

$\theta $ | ${56}^{\xb0}$ | ${56}^{\xb0}$ | ${56}^{\xb0}$ | ${56}^{\xb0}$ |

Antenna Width | 0.88 m | 0.58 m | 0.44 m | 0.35 m |

2-way Gain | 104.5 dB | 104.5 dB | 104.5 dB | 104.5 dB |

Azimuth Beamwidth | $0.{28}^{\xb0}$ | $0.{18}^{\xb0}$ | $0.{14}^{\xb0}$ | $0.{11}^{\xb0}$ |

Azimuth Resolution | 5.5 km | 3.6 km | 2.7 km | $2.2$ km |

Elevation Beamwidth | $0.{63}^{\xb0}$ | $0.{94}^{\xb0}$ | $1.{26}^{\xb0}$ | $1.{57}^{\xb0}$ |

Range Footprint | 22.3 km | 33.5 km | 44.7 km | 55.9 km |

1-look Bandwidth | 36 kHz | 36 kHz | 36 kHz | 36 kHz |

1-look Elevation Resolution | 5.0 km | 5.0 km | 5.0 km | 5.0 km |

Antenna Efficiency | 70% | 70% | 70% | 70% |

Transmit Loss | –1.4 dB | –1.4 dB | –1.4 dB | –1.4 dB |

Burst Length | 1.5 ms | 1.5 ms | 1.5 ms | 1.5 m |

System Temperature | $841{\phantom{\rule{0.166667em}{0ex}}}^{\xb0}$ K | $841{\phantom{\rule{0.166667em}{0ex}}}^{\xb0}$ K | $841{\phantom{\rule{0.166667em}{0ex}}}^{\xb0}$ K | $841{\phantom{\rule{0.166667em}{0ex}}}^{\xb0}$ K |

1-look Noise Equivalent ${\sigma}_{0}$ | $\begin{array}{c}-52.2\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}\\ -54.8\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}\\ -57.6\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}\end{array}$ | $\begin{array}{c}-50.4\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}\\ -53.0\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}\\ -55.9\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}\end{array}$ | $\begin{array}{c}-49.1\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}\\ -51.8\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}\\ -54.6\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}\end{array}$ | $\begin{array}{c}-48.2\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}\\ -50.8\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}\\ -53.7\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}\end{array}$ |

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**MDPI and ACS Style**

Rodriguez, E.
On the Optimal Design of Doppler Scatterometers. *Remote Sens.* **2018**, *10*, 1765.
https://doi.org/10.3390/rs10111765

**AMA Style**

Rodriguez E.
On the Optimal Design of Doppler Scatterometers. *Remote Sensing*. 2018; 10(11):1765.
https://doi.org/10.3390/rs10111765

**Chicago/Turabian Style**

Rodriguez, Ernesto.
2018. "On the Optimal Design of Doppler Scatterometers" *Remote Sensing* 10, no. 11: 1765.
https://doi.org/10.3390/rs10111765