# Analysis of a Least-Squares Soil Moisture Retrieval Algorithm from L-band Passive Observations

^{*}

## Abstract

**:**

## 1. Introduction

- The use of no a priori information in the $CF$ vs. the use of a priori information about all the auxiliary parameters excluding ${s}_{m}$ on the cost function.
- The effect of the presence of a vegetation canopy.
- The effect of the soil moisture content (dry/moist/wet).
- The retrieval formulation using the vertical and horizontal polarizations separately or using the first Stokes parameter.

## 2. Methodology

#### 2.1. Scenario Definition

**Table 1.**Selected original values of soil moisture (${s}_{m}$), soil roughness ($HR$), soil temperature (${T}_{s}$), vegetation albedo (ω) and vegetation opacity (τ) for the six master scenarios. ${\sigma}_{{p}_{i}}^{0}$ is the nominal uncertainty of parameter ${p}_{i}$.

${s}_{m}$ [m${}^{3}$/m${}^{3}$] | $HR$ | $Ts$ [K] | ω | τ [Np] | ||

(${\sigma}_{{s}_{m}}^{0}=0.04$) | (${\sigma}_{HR}^{0}=0.05$) | (${\sigma}_{Ts}^{0}=2$) | (${\sigma}_{\omega}^{0}=0.1$) | (${\sigma}_{\tau}^{0}=0.1$) | ||

Bare | dry soil | 0.02 | 0.2 | 300 | 0 | 0 |

moist soil | 0.2 | 0.2 | 300 | 0 | 0 | |

wet soil | 0.4 | 0.2 | 300 | 0 | 0 | |

Vegetation-covered | dry soil | 0.02 | 0.2 | 300 | 0 | 0.24 |

moist soil | 0.2 | 0.2 | 300 | 0 | 0.24 | |

wet soil | 0.4 | 0.2 | 300 | 0 | 0.24 |

#### 2.2. Forward Model

#### 2.3. Retrieval Algorithm

^{th}observation, and M is the number of parameters ${p}_{i}$ to be retrieved. ${\sigma}_{{p}_{i0}}$ represents the uncertainty on the a priori parameter ${p}_{i0}$, and its value is used to parameterize the constraint on the parameter ${p}_{i}$ in the retrievals: ${p}_{i}$ can be set to be free (${\sigma}_{{p}_{i0}}=100$, no a priori information is used), it can be constrained to be more or less close to the reference value ${p}_{i0}$, or it can be constant (${\sigma}_{{p}_{i0}}<{10}^{-3}$, assuming high accuracy on the a priori information). Note that ${p}_{i0}$ are specified a priori, whereas ${p}_{i}$ values are adjusted during the minimization process.

**Table 2.**Selected standard deviations of soil moisture (${s}_{m}$), soil roughness ($HR$), soil temperature (${T}_{s}$), vegetation albedo (ω) and vegetation opacity (τ) for the two selected cost function configurations $C{F}_{1}$ and $C{F}_{2}$.

${\sigma}_{{s}_{m}}$ [m${}^{3}$/m${}^{3}$] | ${\sigma}_{HR}$ | ${\sigma}_{Ts}$ [K] | ${\sigma}_{\omega}$ | ${\sigma}_{\tau}$ [Np] | |

$C{F}_{1}$ | 100 | 100 | 100 | 100 | 100 |

$C{F}_{2}$ | 100 | 0.05 | 2 | 0.1 | 0.1 |

## 3. Sensitivity Analysis

**Figure 2.**Cost functions formulated using ${T}_{I}$ over a bare dry soil scenario. Contours of $HR$ vs ${s}_{m}$ (a) and ${T}_{s}$ vs ${s}_{m}$ (b), with no constraints on the cost function ($C{F}_{1}$). Contours of $HR$ vs ${s}_{m}$ (c) and ${T}_{s}$ vs ${s}_{m}$ (d), adding constraints on all parameters, except for ${s}_{m}$ ($C{F}_{2}$).

**Figure 3.**Cost functions formulated using ${T}_{I}$ over a vegetation-covered dry soil scenario. Contours of $HR$ vs. ${s}_{m}$ (a) and ${T}_{s}$ vs ${s}_{m}$ (b), with no constraints on the cost function ($C{F}_{1}$). Contours of $HR$ vs ${s}_{m}$ (c) and ${T}_{s}$ vs ${s}_{m}$ (d), adding constraints on all parameters, except for ${s}_{m}$ ($C{F}_{2}$).

**Figure 4.**Cost functions formulated using ${T}_{hh}-{T}_{vv}$ with no constraints. Contours of $HR$ vs. ${s}_{m}$ (a) and ${T}_{s}$ vs ${s}_{m}$ (b) over a bare dry soil scenario. Contours of $HR$ vs. ${s}_{m}$ (c) and ${T}_{s}$ vs ${s}_{m}$ (d) over a bare moist soil scenario. Contours of $HR$ vs. ${s}_{m}$ (e) and ${T}_{s}$ vs ${s}_{m}$ (f) over a bare wet soil scenario.

**Figure 5.**Cost functions formulated using ${T}_{hh}-{T}_{vv}$ with no constraints. Contours of τ vs. ${s}_{m}$ (a) over a vegetation-covered dry scenario. Contours of τ vs. ${s}_{m}$ (b) over a vegetation-covered moist scenario. Contours of τ vs. ${s}_{m}$ (c) over a vegetation-covered wet scenario.

## 4. Analysis with Simulated SMOS Data

#### 4.1. Simulation Strategy

- –
- The geophysical models and the ancillary data used in the L2 Processor Simulator are the same as in SEPS, so that the effect of the model used is not affecting the results.
- –
- The performance of the $CF$ configuration is not dependent on ${\sigma}_{{F}_{n}}$, since the absolute accuracy of the radiometric measurements is available on the SEPS output and is used in L2 Processor Simulator.
- –
- To reduce the computational time, the search limits of the retrieved variables in the $CF$ have been reduced within reasonable bounds, namely $0\le {s}_{m}\le 0.5$ m${}^{3}$/m${}^{3}$, $250\le {T}_{s}\le 350$ K, $0\le HR\le 5$, $0\le \tau \le 3$ Np, and $0\le \omega \le 0.3$ [23].
- –
- The reference values of the parameters on the $CF$ (${p}_{i0}$) are randomly determined from a normal distribution with the nominal standard deviations on Table 1, added to the original values.
- –
- Homogeneous pixels have been assumed in the simulations to evidence the contribution of each parameter in the results and facilitate the analysis. However, further studies will be required to assess the limitations imposed by heterogeneity of vegetation cover and soil characteristics within a satellite footprint.

#### 4.2. Simulation Results

Scenario | Retrieved ${s}_{m}$ error | $C{F}_{1}$ ($\raisebox{1ex}{$HR=0.2$}\!\left/ \!\raisebox{-1ex}{$HR=1$}\right.$) | $C{F}_{2}$ ($\raisebox{1ex}{$HR=0.2$}\!\left/ \!\raisebox{-1ex}{$HR=1$}\right.$) | ||

Earth | Stokes | Earth | Stokes | ||

Bare Dry Soil | Mean | $\raisebox{1ex}{$0.149$}\!\left/ \!\raisebox{-1ex}{$0.185$}\right.$ | $\raisebox{1ex}{$0.106$}\!\left/ \!\raisebox{-1ex}{$0.140$}\right.$ | $\raisebox{1ex}{$0.026$}\!\left/ \!\raisebox{-1ex}{$0.038$}\right.$ | $\raisebox{1ex}{$0.010$}\!\left/ \!\raisebox{-1ex}{$0.021$}\right.$ |

Std. dev. | $\raisebox{1ex}{$0.157$}\!\left/ \!\raisebox{-1ex}{$0.179$}\right.$ | $\raisebox{1ex}{$0.164$}\!\left/ \!\raisebox{-1ex}{$0.160$}\right.$ | $\raisebox{1ex}{$0.092$}\!\left/ \!\raisebox{-1ex}{$0.102$}\right.$ | $\raisebox{1ex}{$0.024$}\!\left/ \!\raisebox{-1ex}{$0.039$}\right.$ | |

RMS | $\raisebox{1ex}{$0.216$}\!\left/ \!\raisebox{-1ex}{$0.257$}\right.$ | $\raisebox{1ex}{$0.196$}\!\left/ \!\raisebox{-1ex}{$0.211$}\right.$ | $\raisebox{1ex}{$0.096$}\!\left/ \!\raisebox{-1ex}{$0.108$}\right.$ | $\raisebox{1ex}{$0.027$}\!\left/ \!\raisebox{-1ex}{$0.044$}\right.$ | |

Bare Moist Soil | Mean | $\raisebox{1ex}{$0.069$}\!\left/ \!\raisebox{-1ex}{$0.059$}\right.$ | $\raisebox{1ex}{$0.018$}\!\left/ \!\raisebox{-1ex}{$0.056$}\right.$ | $\raisebox{1ex}{$-0.014$}\!\left/ \!\raisebox{-1ex}{$-0.050$}\right.$ | $\raisebox{1ex}{$-0.006$}\!\left/ \!\raisebox{-1ex}{$0.006$}\right.$ |

Std. dev. | $\raisebox{1ex}{$0.122$}\!\left/ \!\raisebox{-1ex}{$0.160$}\right.$ | $\raisebox{1ex}{$0.134$}\!\left/ \!\raisebox{-1ex}{$0.143$}\right.$ | $\raisebox{1ex}{$0.085$}\!\left/ \!\raisebox{-1ex}{$0.105$}\right.$ | $\raisebox{1ex}{$0.039$}\!\left/ \!\raisebox{-1ex}{$0.054$}\right.$ | |

RMS | $\raisebox{1ex}{$0.140$}\!\left/ \!\raisebox{-1ex}{$0.171$}\right.$ | $\raisebox{1ex}{$0.135$}\!\left/ \!\raisebox{-1ex}{$0.154$}\right.$ | $\raisebox{1ex}{$0.085$}\!\left/ \!\raisebox{-1ex}{$0.116$}\right.$ | $\raisebox{1ex}{$0.039$}\!\left/ \!\raisebox{-1ex}{$0.054$}\right.$ | |

Bare Wet Soil | Mean | $\raisebox{1ex}{$-0.056$}\!\left/ \!\raisebox{-1ex}{$-0.100$}\right.$ | $\raisebox{1ex}{$-0.081$}\!\left/ \!\raisebox{-1ex}{$-0.090$}\right.$ | $\raisebox{1ex}{$-0.052$}\!\left/ \!\raisebox{-1ex}{$-0.113$}\right.$ | $\raisebox{1ex}{$-0.038$}\!\left/ \!\raisebox{-1ex}{$-0.031$}\right.$ |

Std. dev. | $\raisebox{1ex}{$0.084$}\!\left/ \!\raisebox{-1ex}{$0.142$}\right.$ | $\raisebox{1ex}{$0.096$}\!\left/ \!\raisebox{-1ex}{$0.130$}\right.$ | $\raisebox{1ex}{$0.050$}\!\left/ \!\raisebox{-1ex}{$0.088$}\right.$ | $\raisebox{1ex}{$0.032$}\!\left/ \!\raisebox{-1ex}{$0.037$}\right.$ | |

RMS | $\raisebox{1ex}{$0.101$}\!\left/ \!\raisebox{-1ex}{$0.173$}\right.$ | $\raisebox{1ex}{$0.125$}\!\left/ \!\raisebox{-1ex}{$0.158$}\right.$ | $\raisebox{1ex}{$0.072$}\!\left/ \!\raisebox{-1ex}{$0.143$}\right.$ | $\raisebox{1ex}{$0.050$}\!\left/ \!\raisebox{-1ex}{$0.048$}\right.$ |

Scenario | Retrieved ${s}_{m}$ error | $C{F}_{1}$ | $C{F}_{2}$ | ||

Earth | Stokes | Earth | Stokes | ||

Dry Soil + Canopy | Mean | 0.169 | 0.170 | 0.060 | 0.049 |

Std. dev. | 0.162 | 0.169 | 0.116 | 0.053 | |

RMS | 0.235 | 0.240 | 0.131 | 0.072 | |

Moist Soil + Canopy | Mean | 0.076 | 0.095 | 0.003 | 0.048 |

Std. dev. | 0.143 | 0.121 | 0.120 | 0.076 | |

RMS | 0.162 | 0.153 | 0.120 | 0.090 | |

Wet Soil + Canopy | Mean | -0.062 | -0.040 | -0.061 | -0.021 |

Std. dev. | 0.119 | 0.102 | 0.093 | 0.050 | |

RMS | 0.134 | 0.109 | 0.111 | 0.054 |

Scenario | Retrieved τ error | $C{F}_{1}$ | $C{F}_{2}$ | ||

Earth | Stokes | Earth | Stokes | ||

Dry Soil + Canopy | Mean | 0.439 | 0.369 | 0.110 | 0.036 |

Std. dev. | 0.888 | 0.606 | 0.307 | 0.085 | |

RMS | 0.991 | 0.709 | 0.326 | 0.092 | |

Moist Soil + Canopy | Mean | 0.224 | 0.100 | 0.049 | 0.025 |

Std. dev. | 0.732 | 0.342 | 0.267 | 0.078 | |

RMS | 0.765 | 0.356 | 0.272 | 0.082 | |

Wet Soil + Canopy | Mean | 0.187 | 0.019 | 0.053 | -0.029 |

Std. dev. | 0.714 | 0.208 | 0.274 | 0.056 | |

RMS | 0.738 | 0.209 | 0.279 | 0.063 |

**Figure 6.**Retrieved soil moisture RMSE of simulated SMOS observations versus pixel position in the swath; Simulations over the dry (red, dashed lines), moist(green, solid lines), and wet (blue, dashed-dotted lines) scenarios of Table 1. First row: bare soil scenarios, second row: vegetation-covered scenarios. Left column: with no constraints on the cost function ($C{F}_{1}$), right column: adding constraints on all parameters, except ${s}_{m}$ ($C{F}_{2}$). In each plot: first Stokes parameter (left side) and Earth reference frame (right side). Vertical lines denote the Narrow Swath.

**Figure 7.**Retrieved vegetation opacity RMSE of simulated SMOS observations versus pixel position in the swath; Simulations over the vegetation-covered dry (red, dashed lines), moist(green, solid lines), and wet (blue, dashed-dotted lines) scenarios of Table 1, (a) with no constraints on the cost function ($C{F}_{1}$), and (b) adding constraints on all parameters, except ${s}_{m}$ ($C{F}_{2}$). In each plot: first Stokes parameter (left side) and Earth reference frame (right side). Vertical lines denote the Narrow Swath.

## 5. Conclusions and Discussion

- –
- The use of adequate ancillary information on the cost function significantly improves the accuracy of ${s}_{m}$ retrievals, and is needed to satisfy the SMOS science requirement of 0.04 m${}^{3}$/m${}^{3}$. Using $C{F}_{2}$ constraints (Table 2), ${s}_{m}$ RMSE retrievals of ≈ 0.07 to 0.09 m${}^{3}$/m${}^{3}$ are obtained using $({T}_{hh},{T}_{vv})$, and of ≈ 0.03 to 0.05 m${}^{3}$/m${}^{3}$ using ${T}_{I}$ over bare soil scenarios. As expected, there is a strong decrease of the brightness temperatures sensitivity to ${s}_{m}$ in the presence of vegetation, and ${s}_{m}$ RMSE retrievals of ≈ 0.11 to 0.13 m${}^{3}$/m${}^{3}$ are obtained using $({T}_{hh},{T}_{vv})$, and of ≈ 0.05 to 0.09 m${}^{3}$/m${}^{3}$ using ${T}_{I}$ (with $\tau =0.24,\omega =0$).
- –
- The use of adequate constraints on the cost function ($C{F}_{2}$) highly improves the accuracy of τ estimations and is therefore critical to derive VWC maps from SMOS at the required accuracy of 0.2 kg/m${}^{2}$; Preliminary calculations indicate that VWC maps with an accuracy of ≈ 1.9 to 2.2 kg/m${}^{2}$ could be estimated from τ retrievals using $({T}_{hh},{T}_{vv})$, and of ≈ 0.4 to 0.6 kg/m${}^{2}$ using ${T}_{I}$.
- –
- More accurate soil moisture estimates have been obtained over wet soils than over dry soils (bare and with low vegetation), except for the case of retrievals using ${T}_{I}$ and $C{F}_{2}$. Regarding τ retrievals, more accurate estimates have been obtained over wet soils than over dry soils in all the configurations.
- –
- Better ${s}_{m}$ retrievals have been obtained when using ${T}_{I}$ than when using ${T}_{hh}-{T}_{vv}$. Also, the formulation in terms of ${T}_{I}$ leads to better τ retrievals in all the configurations. These results suggest that, although ${T}_{hh}-{T}_{vv}$ is the formulation generally adopted in most studies, the use of ${T}_{I}$ should not be disregarded. In addition, ${T}_{I}$ is more robust in the presence of geometric rotations and Faraday rotation (at any spatial scale) than $({T}_{hh},{T}_{vv})$. These effects have been perfectly corrected on the simulations, but are critical from an operational point of view.
- –
- Due to SMOS observation geometry, better accuracies could be obtained if only the Narrow Swath (640-km, the central part of the FOV) is used. The use of adequate constraints ($C{F}_{2}$) and the retrieval formulation in terms of ${T}_{I}$ provide the most accurate ${s}_{m}$ and τ retrievals over all scenarios in the case of considering either the nominal or the Narrow Swath.

## Acknowledgements

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**MDPI and ACS Style**

Piles, M.; Vall-llossera, M.; Camps, A.; Talone, M.; Monerris, A. Analysis of a Least-Squares Soil Moisture Retrieval Algorithm from L-band Passive Observations. *Remote Sens.* **2010**, *2*, 352-374.
https://doi.org/10.3390/rs2010352

**AMA Style**

Piles M, Vall-llossera M, Camps A, Talone M, Monerris A. Analysis of a Least-Squares Soil Moisture Retrieval Algorithm from L-band Passive Observations. *Remote Sensing*. 2010; 2(1):352-374.
https://doi.org/10.3390/rs2010352

**Chicago/Turabian Style**

Piles, María, Mercè Vall-llossera, Adriano Camps, Marco Talone, and Alessandra Monerris. 2010. "Analysis of a Least-Squares Soil Moisture Retrieval Algorithm from L-band Passive Observations" *Remote Sensing* 2, no. 1: 352-374.
https://doi.org/10.3390/rs2010352