# Error in Radar-Derived Soil Moisture due to Roughness Parameterization: An Analysis Based on Synthetical Surface Profiles

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## Abstract

**:**

## 1. Introduction

- A series of surface height points (roughness profile) is defined along a 1-dimensional surface transect, mostly sampled by means of meshboard, pin profilometer or laser techniques [6]. Generally, profiles used in practice have a length between 1 m and 4 m [6–9], and the horizontal spacing between height points usually lies between 1 mm [5] and 2 cm [7].
- From this profile a linear trend is removed to compensate for the possibility that the measurement device was not aligned perfectly parallel to a horizontal reference surface [8].

## 2. Soil moisture retrieval technique

## 3. Sensitivity of soil moisture retrieval to RMS height and correlation length

## 4. Sensitivity of soil moisture retrieval to roughness parameterization techniques

#### 4.1. Generation of synthetical 1-dimensional surface profiles

_{t}the height at coordinate t, a

_{t}white noise and φ a weight factor which can be found from the Yule-Walker equations as [35]:

#### 4.2. Profile length

#### Experimental setup

^{−1}into soil moisture content, using the roughness parameters from the sampled profiles.

#### Errors on roughness parameterization

#### Errors on soil moisture retrieval

#### 4.3. Number of profile measurements

#### Experimental setup

^{−1}into soil moisture content, using the series of averaged roughness parameters from sampled profiles. Only 4-m profiles are considered in this part of the experiment, as these are frequently used in practice.

#### Errors on roughness parameterization

#### Errors on soil moisture retrieval

#### 4.4. Spacing between height points

#### Experimental setup

#### Errors on roughness parameterization

#### Errors on soil moisture retrieval

#### 4.5. Instrument accuracy

#### Experimental setup

#### Errors on roughness parameterization

#### Errors on soil moisture retrieval

#### 4.6. Trend removal

#### Experimental setup

#### Errors on roughness parameterization

#### Errors on soil moisture retrieval

## 5. Conclusions

## Acknowledgments

## References and Notes

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**Figure 1.**Backscatter coefficients calculated for different values of RMS height and correlation length and a moisture content of 25 vol% for (a) an ASAR VV configuration and (b) a PALSAR HH configuration.

**Figure 2.**Sensitivity of the soil moisture retrieval to RMS height (vol%/cm) for an ASAR VV configuration and a moisture content of (a) 5 vol%, (b) 15 vol%, (c) 25 vol%, and (d) 35 vol%.

**Figure 3.**Sensitivity of the soil moisture retrieval to correlation length (vol%/cm) for an ASAR VV configuration and a moisture content of (a) 5vol%, (b) 15vol%, (c) 25vol%, and (d) 35vol%.

**Figure 4.**Sensitivity of the soil moisture retrieval to RMS height (vol%/cm) for a PALSAR HH configuration and a moisture content of (a) 5 vol%, (b) 15 vol%, (c) 25 vol%, and (d) 35 vol%.

**Figure 5.**Sensitivity of the soil moisture retrieval to correlation length (vol%/cm) for a PALSAR HH configuration and a moisture content of (a) 5vol%, (b) 15vol%, (c) 25 vol%, and (d) 35 vol%.

**Figure 6.**Mean and standard deviations of RMS height and correlation length for different profile lengths, sampled from large profiles with (a) (s,l) = (1 cm,5 cm) and (b) (s,l) = (1 cm,40 cm).

**Figure 7.**Mean and standard deviations of inverted soil moisture contents for different profile lengths. Inverted soil moisture contents are derived using roughness parameters from sampled profiles, originating from large profiles with (s,l) equal to (a), (b) (2 cm,5 cm), (c), (d) (1 cm,5 cm), (e), (f) (2 cm,40 cm) and (g), (h) (1 cm,40 cm), and initial moisture contents of (a), (c), (e), (g) 5 vol% and (b), (d), (f), (h) 25 vol%.

**Figure 8.**Number of profiles required to obtain a standard deviation of RMS height or correlation length less than 10% of the mean for different profile lengths. Sampled profiles originate from large profiles with (s,l) equal to (a) (1 cm,5 cm) and (b) (1 cm,40 cm).

**Figure 9.**Mean and standard deviations of inverted soil moisture contents for different numbers of profiles used. Inverted soil moisture contents are derived using roughness parameter series from sampled 4-m profiles, originating from large profiles with (s,l) equal to (a), (b) (1 cm,5 cm) and (c), (d) (1cm,40 cm), and for (a), (c) ASAR VV and (b), (d) PALSAR HH. Considered initial moisture contents are 5 vol% (crosses), 15 vol% (circles), 25 vol% (stars) and 35 vol% (squares).

**Figure 10.**Mean and standard deviations of inverted soil moisture contents for different horizontal spacings used in the parameterization of roughness from 4-m profiles with (s,l) = (1 cm,5 cm). Considered spacings are (a) 2mm, (b) 5mm, (c) 10mm and (d) 15mm.

**Figure 11.**Autocorrelation functions derived for the same roughness profile with (s,l) = (1 cm,5 cm), sampled with a spacing of respectively 1mm and 15 mm.

**Figure 12.**Boxplots of inverted soil moisture contents, calculated using roughness parameters from resampled profiles with 15-mm spacing, and exponential (E) and Gaussian (G) autocorrelation functions, for initial moisture contents of 5, 15, 25 and 35 vol%, and all defined sensor configurations, for profiles with (s,l) equal to (a) (2 cm,5 cm), (b) (1 cm,5 cm), (c) (2cm,40 cm) and (d) (1cm,40 cm).

**Figure 13.**(a) Original simulated 4-m roughness profile with (s,l) = (1 cm,5 cm) and a horizontal spacing of 1 mm, added to (b) a linear trend with slope of 0.025 m/m, and (c) a cosine trend with a wavelength of 5m and an amplitude of 5 cm

**Figure 14.**Mean inverted soil moisture contents for different radar configurations, using roughness parameters obtained after (a) linear detrending over the 4-m profile, (b) piecewise 1-m detrending, (c) second-order polynomial detrending and (d) third-order polynomial de-trending of 10 synthetical roughness profiles of (s,l) = (1 cm,5 cm) with added linear trend.

**Figure 15.**Mean inverted soil moisture contents for different radar configurations, using roughness parameters obtained after (a) linear detrending over the 4-m profile, (b) piecewise 1-m detrending, (c) second-order polynomial detrending and (d) third-order polynomial de-trending of 10 synthetical roughness profiles of (s,l) = (1 cm,5 cm) with added cosine trend.

Model | Parameter | Value |
---|---|---|

Integral Equation Model | ||

ENVISAT ASAR configuration | Frequency | 5.3 GHz (C-band) |

Polarization | HH or VV | |

Incidence angle | 23° | |

ALOS PALSAR configuration | Frequency | 1.27 GHz (L-band) |

Polarization | HH or VV | |

Incidence angle | 34.3° | |

4-Component Dielectric Mixing Model | Bulk density | 1.2 g/cm^{3} |

Specific density | 2.65 g/cm^{3} | |

Sand content | 15% | |

Clay content | 11.4% | |

Temperature | 15°C |

**Table 2.**Average values of s and l obtained for different horizontal spacings. Standard deviations are added between brackets.

Sample spacing | |||||
---|---|---|---|---|---|

‘Truth’ (1 mm) | 2mm | 5mm | 10mm | 15mm | |

s (cm) | 1 | 0.99 (0.00) | 0.98 (0.00) | 0.97 (0.01) | 0.96 (0.02) |

l (cm) | 5 | 5.02 (0.04) | 5.15 (0.11) | 5.34 (0.22) | 5.47 (0.40) |

s (cm) | 1 | 1.00 (0.00) | 1.00 (0.00) | 1.00 (0.00) | 0.99 (0.00) |

l (cm) | 40 | 40.02 (0.04) | 40.07 (0.21) | 40.20 (0.41) | 40.23 (0.37) |

s (cm) | 2 | 1.99 (0.01) | 1.97 (0.01) | 1.93 (0.02) | 1.91 (0.01) |

l (cm) | 5 | 5.07 (0.05) | 5.17 (0.08) | 5.43 (0.19) | 5.76 (0.52) |

s (cm) | 2 | 2.00 (0.00) | 2.00 (0.00) | 2.00 (0.01) | 1.99 (0.01) |

l (cm) | 40 | 40.03 (0.05) | 40.16 (0.28) | 40.30 (0.45) | 40.26 (0.58) |

Correct (s,l) | RMSE on s or l due to | |||||
---|---|---|---|---|---|---|

1mm noise | 2mm noise | 5mm noise | ||||

s (cm) | l (cm) | s (cm) | l (cm) | s (cm) | l (cm) | |

(1 cm,5 cm) | 0.0019 | 0.0316 | 0.0074 | 0.0775 | 0.0443 | 0.3782 |

(1 cm,40 cm) | 0.0017 | 0.1673 | 0.0068 | 0.4764 | 0.0421 | 2.9972 |

(2 cm,5 cm) | 0.0015 | 0.0447 | 0.0043 | 0.0316 | 0.0215 | 0.1265 |

(2 cm,40 cm) | 0.0009 | 0.0707 | 0.0035 | 0.1871 | 0.0219 | 0.9301 |

**Table 4.**RMSE values of the retrieved soil moisture contents due to roughness parameterization errors, introduced by instrument noise.

(s,l) of original profile | Soil moisture content (vol%) | RMSE (vol%) of retrieved soil moisture due to | |||
---|---|---|---|---|---|

1 mm noise | 2 mm noise | 5 mm noise | |||

(1 cm,5 cm) | 5 | 0.04 | 0.11 | 0.54 | |

(1 cm,5 cm) | 15 | 0.11 | 0.30 | 1.39 | |

(1 cm,5 cm) | 25 | 0.21 | 0.57 | 2.61 | |

(1 cm,5 cm) | 35 | 0.34 | 0.92 | 4.15 | |

(1 cm,40 cm) | 35 | 0.39 | 2.08 | 8.31 | |

(2 cm,5 cm) | 35 | 0.24 | 0.26 | 0.51 | |

(2 cm,40 cm) | 35 | 0.23 | 0.46 | 2.76 |

**Table 5.**Average values of s and l, calculated after detrending of 4-m profiles. Standard deviations are added between brackets.

Correct roughness values | ||||||||
---|---|---|---|---|---|---|---|---|

s (cm) | l (cm) | s (cm) | l (cm) | s (cm) | l (cm) | s (cm) | l (cm) | |

1 | 5 | 1 | 40 | 2 | 5 | 2 | 40 | |

Detrending type | Linear trended surface | |||||||

Linear 4-m | 0.99 (0.01) | 4.93 (0.13) | 0.91 (0.07) | 30.09 (7.48) | 1.97 (0.03) | 4.76 (0.25) | 1.80 (0.18) | 31.57 (6.00) |

Piecewise 1-m | 0.92 (0.03) | 3.80 (0.45) | 0.51 (0.04) | 8.31 (1.34) | 1.85 (0.06) | 3.77 (0.40) | 0.98 (0.12) | 7.45 (1.12) |

Second-order | 0.98 (0.02) | 4.67 (0.29) | 0.81 (0.10) | 24.22 (7.40) | 1.94 (0.04) | 4.51 (0.26) | 1.57 (0.19) | 22.00 (7.58) |

Third-order | 0.96 (0.02) | 4.36 (0.30) | 0.64 (0.05) | 15.02 (3.96) | 1.93 (0.03) | 4.37 (0.26) | 1.41 (0.16) | 17.84 (6.18) |

Cosine trended surface | ||||||||

Linear 4-m | 2.64 (0.13) | 50.07 (2.25) | 2.78 (0.42) | 54.30 (4.68) | 3.07 (0.26) | 28.13 (8.31) | 3.01 (0.74) | 44.81 (7.06) |

Piecewise 1-m | 0.95 (0.04) | 4.13 (0.49) | 0.58 (0.06) | 9.40 (1.47) | 1.85 (0.05) | 3.79 (0.36) | 1.00 (0.10) | 7.79 (1.47) |

Second-order | 1.32 (0.13) | 13.48 (6.67) | 1.10 (0.35) | 25.47 (8.16) | 2.17 (0.13) | 7.19 (2.03) | 1.88 (0.27) | 28.38 (8.27) |

Third-order | 1.04 (0.04) | 5.33 (0.53) | 0.79 (0.12) | 18.16 (4.72) | 1.95 (0.05) | 4.53 (0.30) | 1.47 (0.12) | 18.81 (5.13) |

© 2009 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Lievens, H.; Vernieuwe, H.; Álvarez-Mozos, J.; De Baets, B.; Verhoest, N.E.C.
Error in Radar-Derived Soil Moisture due to Roughness Parameterization: An Analysis Based on Synthetical Surface Profiles. *Sensors* **2009**, *9*, 1067-1093.
https://doi.org/10.3390/s90201067

**AMA Style**

Lievens H, Vernieuwe H, Álvarez-Mozos J, De Baets B, Verhoest NEC.
Error in Radar-Derived Soil Moisture due to Roughness Parameterization: An Analysis Based on Synthetical Surface Profiles. *Sensors*. 2009; 9(2):1067-1093.
https://doi.org/10.3390/s90201067

**Chicago/Turabian Style**

Lievens, Hans, Hilde Vernieuwe, Jesús Álvarez-Mozos, Bernard De Baets, and Niko E.C. Verhoest.
2009. "Error in Radar-Derived Soil Moisture due to Roughness Parameterization: An Analysis Based on Synthetical Surface Profiles" *Sensors* 9, no. 2: 1067-1093.
https://doi.org/10.3390/s90201067