# Estimation of Surface Soil Moisture in Irrigated Lands by Assimilation of Landsat Vegetation Indices, Surface Energy Balance Products, and Relevance Vector Machines

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Soil Moisture Measurement and Estimation

#### 2.2. The Mapping Evapotranspiration at High Resolution with Internalized Calibration Model (METRIC) Model

_{n}is net radiation at the surface; G is the soil heat flux; H is the sensible heat flux to the air; and LE is the latent heat flux (or actual evapotranspiration: the energy used to evaporate water). Every component of the SEB equation is expressed in Watts/m

^{2}. ET models such as METRIC, SEBAL, and others, make use of the SEB equation to estimate actual evapotranspiration (LE) by solving for the other three components of Equation (1) [3]. The METRIC model solves for each of the parameters of the SEB equation with the following procedure:

_{n}Equation (2) is computed for each pixel using albedo (α) and transmittances computed from short wavebands (Landsat first to fifth and seventh reflectance bands, or ρ

_{1}:ρ

_{5}and ρ

_{7}), using broadband emissivity (ε

_{0}) computed from the thermal band (Landsat sixth band).

_{s↓}, R

_{L↑}, and R

_{L↓}are the incoming shortwave, emitted outgoing longwave, and the incoming longwave radiation, respectively (W/m

^{2}) [48,50].

_{s}), along with the R

_{n}estimates [51]:

_{air}), air specific heat (c

_{p}), estimated aerodynamic resistance to heat transport (r

_{ah}), and surface-to-air temperature differences (dT) predicted from thermal infrared radiances.

#### 2.3. The Relevance Vector Machine

**t**on the inputs

**x**, thus achieving accurate predictions of

**t**for previously unseen values of

**x**[52]. The general form of the RVM function is presented in Equations (5) and (6):

**w**is the model “weights” and

**Φ(x)**is a basis function where

**Φ**=[φ

_{1}… φ

_{M}] is the N * M “design” matrix whose columns comprise the complete set of M “basic vectors”,

**y**or y

_{n}is the RVM predictand, and ε

_{n}is the difference or residual [53]. The main feature of the RVM is the target function

**y**Equation (5), which attempts to minimize the difference

**ε**with respect to

**t**Equation (6).

**t**, p(t

_{n}|x

_{n}) is Gaussian distributed, N(t

_{n}|y

_{n},σ

^{2}), as is the difference

**ε**, N(0,σ

^{2}). Using these two assumptions, the likelihood of the set

**{x, t}**can be written as:

**w**and σ, usually by maximum likelihood methods [56], a common approach is to impose on them constraints based on an assumed “prior” Gaussian probability distribution Equation (8):

**α**being a vector of Q hyperparameters, individually related to each weight value. From Bayes’ rule, the “posterior” probability over all unknown parameters over the set is:

**w|t**, α,σ

^{2}) ∼ N(

**m, Σ**), where the mean

**m**and the covariance

**Σ**are given by:

**A**= diag(

**α**). Solving for

**m**and

**Σ**involves finding the hyperparameters (

**α**and σ

^{2}) that maximize the second term in Equation (9):

**α**) and p(σ

^{2})), it is possible to maximize the evidence or likelihood function (the first term in Equation (12)):

_{opt}and σ

_{opt}

^{2}(which affects the estimation of

**m**and

**Σ**). This optimization algorithm uses an efficient sequential addition and deletion of candidate basis functions described by [53]. Thus, the basis functions from the training set that are associated with non-zero weights (not deleted during optimization) are called “relevance vectors”.

**x′**, the predictive distribution for the corresponding target t and the prediction confidence σ

^{2}(

**x′**) can be computed as:

**t**(

**y**) for an unseen input data

**x′**is:

**y**is determined by the variance of this distribution σ

^{2}(

**x′**) given by:

## 3. Material and Methods

#### 3.1. Area of Study

#### 3.2. Weather Information

#### 3.3. Remote Sensing Data

#### 3.4. Surface Soil Moisture Data Collection

^{3}/m

^{3}(±3% θ) [63,64]. A laboratory validation of the GS3 sensor [6] in Central Utah agricultural lands (near the area of study) determined that custom sensor calibration was not required. To account for the spatial footprint of the individual Landsat pixel (30 m by 30 m), several soil moisture measurements were made for each sampling site and posteriorly averaged within the pixel footprint. Table 2 presents details of the crops at each sampling site. The crop types tend to fully cover the surface after the initial development and until the harvest stage. The irrigation method used in the area of study is level basin, in which water covers the entire field during an irrigation event, with no drainage. This type of irrigation is possible because of the laser leveling practice continuously used in the Lower Sevier River Basin.

^{3}/m

^{3}(dry soil to water layer) for the four dates. Similar soil moisture ranges for similar crop types in Central Utah are reported by [6] at up to 0.55 m

^{3}/m

^{3}. Higher soil moisture values represent the basin water in the field due to an irrigation event [62,63]. No distinctive pattern of soil moisture ranges can be identified among crop types or sampling sites (Table 2), due to dissimilar farm management, including irrigation scheduling (on-demand water distribution).

#### 3.5. Potential Surface Soil Moisture Predictors

**a)****Atmospheric Variables:**These potential predictors are weather parameters obtained from a typical agricultural weather station and are presented in Table 3. Since precipitation records for the area of study have been historically scarce, rainfall was omitted as a potential predictor.**b)****Landsat Indices:**The Landsat individual surface reflectance, band ratios, vegetation indices, and cap-tasseled images that were calculated for each of the four dates are summarized in Table 4.**c)****Evapotranspiration and Energy Balance Components**

#### 3.6. Relevance Vector Machine Calibration

**a)****The Relevance Vector Machine (RVM)**

**b)****Stratified Cross-Validation**

^{th}-fold. For this study, the number of folds, K, equals four (e.g., May 13 fold#1, May 29 fold #2, etc.); therefore, the CV technique is “stratified” by the collection dates, per se, which avoids bias or randomness effects on the model-performance statistics and ensures repeatability of results.

**c)****Soil Moisture Predictors Identification**

^{3}/m

^{3}) and the Coefficient of Efficiency or Nash–Sutcliffe Coefficient (η, no units) [70]. These two parameters have been used extensively in hydrological and earth science applications [71]. In addition, because of the number of potential predictors incorporated into the RVM calibration with stratified CV and FPS, there can be cases where two or more different predictor subsets have similar predictive power (similar statistical results). Occam Razor philosophy indicates that, ceteris paribus, simpler models/parameters must have priority over more complex ones. Thus, simpler predictor subsets took priority. This implied that the predictors listed in Table 3 took precedence over those named in Table 4, which in turn took precedence over the ones identified in Table 5.

**d)****Surface Soil Moisture Model Selection**

## 4. Results and Discussion

#### 4.1. Potential Predictors

#### 4.2. Model Development

^{3}/m

^{3}, η ≈ 1), which have a poor statistical performance on the CV results. In the same figure, for each forward predictor selection iteration, the best RVM models based on stratified CV results (blue dots) do not suffer from these overfitting conditions.

^{3}/m

^{3}). Outside agricultural fields, it seems that the RVM is able to infer surface soil moisture in fallow or non-cultivated areas, although no ground information is available to confirm this. The developed RVM model does not recognize open water surfaces, given that no information about these locations (in the southern portion of the area of study) was provided during the calibration process. Finally, the RVM seems to have limited performance in locations with mixed pixel information (fields with canals or houses), which creates an identifiable perimeter for most fields. This is because Landsat 30-m/pixel resolution does not allow for a direct discrimination of non-agricultural pixels. A spatial filter based on the National Land Cover Database NLCD [73] and used by the METRIC ET algorithm can help to discriminate these mixed pixels in Landsat images and the soil moisture maps of this study.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**): Landsat Path 38 Row 33 scene footprint (dashed lines) and area of study (black). (

**b**): gap-filled Landsat 7 ETM+ false view (29 May 2012) of the area of study and soil moisture sampling sites (stars) in Central Utah.

**Figure 2.**Measured surface soil moisture (bottom row) and corresponding Landsat indices and metric energy balance products (upper rows) for the four Landsat overpass dates considered in the study.

**Figure 3.**ETrF maps using METRIC model and Landsat 7 ETM+ images for the area of study. Note the absence of gaps in the maps due to the use of the natural neighboring interpolation technique on the original DN bands.

**Figure 4.**Measured surface soil moisture comparison with METRIC fractional evapotranspiration (top), Landsat surface temperature (mid), and NDVI (bottom) values for the four dates considered in this study.

**Figure 5.**Example of RMSE and η values for fourty-six FPS iterations using RVM with a bubble kernel function. Subplot title indicated variable selected for the i-th FPS iteration. Red dots are statistical result values for considered kernel width values using all (Data). Blue dots are all (Data) statistical results of selected stratified CV inputs for each iteration.

**Figure 6.**Performance of the soil moisture RVM model, including the 95% confidence interval (CI) using the bubble kernel using all available data: (top) measured vs. RVM soil moisture values, (lower left) residual distribution, (lower center), residual histogram and (lower right) 1:1 measured vs. RVM surface soil moisture plot.

**Figure 7.**Spatial Estimation of Surface Soil Moisture for the four Landsat 7 ETM+ dates using the RVM model with the bubble kernelkernel. Note: Soil moisture values higher than 0.5 m

^{3}/m

^{3}indicate fields under irrigation.

DOY | Date | Average Agronomic/Irrigation Conditions |
---|---|---|

136 | 13 May | Alfalfa first cut (harvest) period, severe decrease in on-demand water orders, limited to corn and small grains. |

152 | 29 May | Alfalfa cutting period was ending, large farm areas with no crop cover, minimal on-demand water orders. |

168 | 14 June | On-demand water orders increasing, second alfalfa-growing cycle in progress. |

184 | 30 June | Second alfalfa growing cycle at maturity stage. |

Site | Crop | Site | Crop |
---|---|---|---|

1 | alfalfa | 11 | corn |

2 | alfalfa | 12 | alfalfa |

3 | grain | 13 | alfalfa |

4 | grain | 14 | alfalfa |

5 | corn | 15 | alfalfa |

6 | corn | 16 | alfalfa |

7 | corn | 17 | alfalfa |

8 | grain | 18 | alfalfa |

9 | grain | 19 | alfalfa |

10 | grain | 20 | alfalfa |

Potential Predictors | Abrev. | Units |
---|---|---|

Max. Daily Air Temp. | Tmax | °C |

Min. Daily Air Temp. | Tmin | °C |

Ave. Daily Dew Temp. | DewP | °C |

Ave. Daily Wind Speed | Wind | m/s |

Ave. Solar Radiation | Rs | W/m^{2} |

Extra. Solar Radiation | Rso | W/m^{2} |

Daily Reference ET | ETr | mm/d |

Potential Predictors | Abrev. | Units |
---|---|---|

Reflectance Bands 1 to 5 and 7 | ρ | No units |

Surface Temperature | T_{surf} | |

Band 1/Band 2 ratio | ρ1/ρ2 | °C |

Band 1/Band 3 ratio | ρ1/ρ3 | No units |

Band 1/Band 4 ratio | ρ1/ρ4 | |

Band 1/Band 5 ratio | ρ1/ρ5 | |

Band 1/Band 7 ratio | ρ1/ρ7 | |

Band 2/Band 3 ratio | ρ2/ρ3 | |

Band 2/Band 4 ratio | ρ2/ρ4 | |

Band 2/Band 5 ratio | ρ2/ρ5 | |

Band 2/Band 7 ratio | ρ2/ρ7 | |

Band 3/Band 4 ratio | ρ3/ρ4 | |

Band 3/Band 5 ratio | ρ3/ρ5 | |

Band 3/Band 7 ratio | ρ3/ρ7 | |

Band 4/Band 5 ratio | ρ4/ρ5 | |

Band 4/Band 7 ratio | ρ4/ρ7 | |

Band 5/Band 7 ratio | ρ5/ρ7 | |

Red NDVI | NDVI | |

Green NDVI | GNDVI | |

Blue NDVI | BNDVI | |

Normalized Burn Ratio | NBR | |

Brightness | Bri | |

Greenness | Gre | |

Wetness | Wet | |

Haze | Haz | |

Normalized Diff. Water Index | NDWI | |

Leaf Area Index | LAI | |

Surface Albedo | α | m^{2}/m^{2} |

Emissivity | ε | No units |

Potential Predictors | Abrev. | Units |
---|---|---|

Net Radiation | Rn | W/m^{2} |

Ground Heat Flux | G | W/m^{2} |

Latent Heat Flux | ET_{24} | mm/d |

Sensible Heat Flux | H | W/m^{2} |

Water Evaporation | E_{water} | W/m^{2} |

Fractional Reference ET | ETrF | no units |

Date | Air T_{max} | Air T_{min} | AirT_{dew} | Wind Speed | Solar Rad | Ext Rad | ET_{ref} |
---|---|---|---|---|---|---|---|

°C | °C | °C | m/s | W/m^{2} | W/m^{2} | mm/d | |

13-May | 24.6 | 4.5 | −5.9 | 0.60 | 314.4 | 456.9 | 6.39 |

29-May | 27.8 | 7.5 | −5.5 | 0.67 | 323.3 | 475.3 | 6.43 |

14-Jun | 30.9 | 14.9 | −3.1 | 0.76 | 331.5 | 482.9 | 6.82 |

30-Jun | 35.9 | 18.0 | 0.1 | 1.16 | 339.8 | 483.9 | 7.02 |

Kernel Type | Inputs (in Order of Selection) | Stratified CV | All (Data) | ||
---|---|---|---|---|---|

RMSE m^{3}/m^{3} | η | RMSE m^{3}/m^{3} | Η | ||

Bubble | ρ4, ETrF, LAI,ρ3/ρ4 | 0.10 | 0.42 | 0.06 | 0.66 |

TPS | Rn, Rs, Tmin, Tmax, Rso | 0.14 | 0.16 | 0.12 | 0.30 |

Cauchy | Rn, Rso, Tmin | 0.12 | 0.24 | 0.10 | 0.40 |

Laplace | Rn, Rso, Tmin, LAI | 0.12 | 0.24 | 0.08 | 0.50 |

R | Tmax, Rn, Tmin, ETr | 0.14 | 0.14 | 0.12 | 0.26 |

Gauss | Rn, Rso, Tmin | 0.12 | 0.26 | 0.12 | 0.32 |

Cubic | Rn, Tmin, Tmax, Rs | 0.14 | 0.14 | 0.12 | 0.28 |

© 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**

Torres-Rua, A.F.; Ticlavilca, A.M.; Bachour, R.; McKee, M.
Estimation of Surface Soil Moisture in Irrigated Lands by Assimilation of Landsat Vegetation Indices, Surface Energy Balance Products, and Relevance Vector Machines. *Water* **2016**, *8*, 167.
https://doi.org/10.3390/w8040167

**AMA Style**

Torres-Rua AF, Ticlavilca AM, Bachour R, McKee M.
Estimation of Surface Soil Moisture in Irrigated Lands by Assimilation of Landsat Vegetation Indices, Surface Energy Balance Products, and Relevance Vector Machines. *Water*. 2016; 8(4):167.
https://doi.org/10.3390/w8040167

**Chicago/Turabian Style**

Torres-Rua, Alfonso F., Andres M. Ticlavilca, Roula Bachour, and Mac McKee.
2016. "Estimation of Surface Soil Moisture in Irrigated Lands by Assimilation of Landsat Vegetation Indices, Surface Energy Balance Products, and Relevance Vector Machines" *Water* 8, no. 4: 167.
https://doi.org/10.3390/w8040167