# 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

## References

- Walker, W.R. Lessons for the Last Half Century of Irrigation Engineering Research—Where to Now? In IV Congreso Nacional y III Congreso Iberoamericano de Riego y Drenaje; La Molina National Agrarian University: Lima, Peru, 2011. [Google Scholar]
- Anderson, M.C.; Allen, R.G.; Morse, A.; Kustas, W.P. Use of Landsat thermal imagery in monitoring evapotranspiration and managing water resources. Remote Sens. Environ.
**2012**, 122, 50–65. [Google Scholar] [CrossRef] - Allen, R.; Irmak, A.; Trezza, R.; Hendrickx, J.M.H.; Bastiaanssen, W.; Kjaersgaard, J. Satellite-based ET estimation in agriculture using SEBAL and METRIC. Hydrol. Process.
**2011**, 25, 4011–4027. [Google Scholar] [CrossRef] - Kogan, F.N.F. Operational space technology for global vegetation assessment. Bull. Am. Meteorol. Soc.
**2001**, 82, 1949–1964. [Google Scholar] [CrossRef] - National Geospatial Advisory Committee (NGAC). The Value Proposition for Ten Landsat Applications; NGAC: Washington, DC, USA, 2012; pp. 1–6. [Google Scholar]
- Hassan-Esfahani, L.; Torres-Rua, A.F.; Ticlavilca, A.M.; Jensen, A.; McKee, M. Topsoil moisture estimation for precision agriculture using unmmaned aerial vehicle multispectral imagery. In Proceedings of the 2014 IEEE Geoscience and Remote Sensing Symposium, Quebec, QC, Canada, 13–18 July 2014; pp. 3263–3266.
- Petropoulos, G. Remote Sensing of Energy Fluxes and Soil Moisture Content; CRC Press: Boca Raton, FL, USA, 2013. [Google Scholar]
- Hassan-Esfahani, L.; Torres-Rua, A.; McKee, M. Assessment of optimal irrigation water allocation for pressurized irrigation system using water balance approach, learning machines, and remotely sensed data. Agric. Water Manag.
**2015**, 153, 42–50. [Google Scholar] [CrossRef] - Islam, S.; Engman, T.; Kivelson, M.; Islam, S.; Engman, T. Why bother for 0.0001% of Earth's water: Challenges for soil moisture research. Eos Trans. Am. Geophys. Union
**1996**, 77, 420. [Google Scholar] [CrossRef] - Small, E.; Kurc, S. The Influence of Soil Moisture on the Surface Energy Balance in Semiarid Environments; Water Resources Research Institute: Las Cruces, NM, USA, 2001. [Google Scholar]
- Liu, Y.Y.; Parinussa, R.M.; Dorigo, W.A.; de Jeu, R.A.M.; Wagner, W.; van Dijk, A.I.J.M.; McCabe, M.F.; Evans, J.P. Developing an improved soil moisture dataset by blending passive and active microwave satellite-based retrievals. Hydrol. Earth Syst. Sci.
**2011**, 15, 425–436. [Google Scholar] [CrossRef][Green Version] - Yin, J.; Zhan, X.; Zheng, Y.; Liu, J.; Fang, L.; Hain, C.R. Enhancing Model Skill by Assimilating SMOPS Blended Soil Moisture Product into Noah Land Surface Model. J. Hydrometeorol.
**2015**, 16, 917–931. [Google Scholar] [CrossRef] - Gassman, P.; Williams, J.; Benson, V.; Izaurralde, R.C.; Hauck, L.M.; Jones, C.A.; Atwood, J.D.; Kiniry, J.R.; Flowers, J.D. Historical Development and Applications of the EPIC and APEX Models; Center for Agricultural and Rural Development, Iowa State University: Ames, IA, USA, 2005. [Google Scholar]
- Andris, C.; Cowen, D.; Wittenbach, J. Support Vector Machine for Spatial Variation. Trans. GIS
**2013**, 17, 41–61. [Google Scholar] [CrossRef] - Bachour, R.; Walker, W.R.; Ticlavilca, A.M.; McKee, M.; Maslova, I. Estimation of Spatially Distributed Evapotranspiration Using Remote Sensing and a Relevance Vector Machine. J. Irrig. Drain. Eng.
**2014**, 140, 04014029. [Google Scholar] [CrossRef] - Batt, H.A.; Stevens, D.K. How to Utilize Relevance Vectors to Collect Required Data for Modeling Water Quality Constituents and Fine Sediment in Natural Systems: Case Study on Mud Lake, Idaho. J. Environ. Eng.
**2014**, 140, 06014003. [Google Scholar] [CrossRef] - Kaheil, Y.H.; Gill, M.K.; McKee, M.; Bastidas, L.A.; Rosero, E. Downscaling and Assimilation of Surface Soil Moisture Using Ground Truth Measurements. IEEE Trans. Geosci. Remote Sens.
**2008**, 46, 1375–1384. [Google Scholar] [CrossRef] - Gill, M.K.; Kemblowski, M.W.; McKee, M.; Gill, M.K.; Kemblowski, M.W.; McKee, M. Soil Moisture Data Assimilation Using Support Vector Machines and Ensemble Kalman Filter. J. Am. Water Resour. Assoc.
**2007**, 43, 1004–1015. [Google Scholar] [CrossRef] - Khader, A.I.; McKee, M. Use of a relevance vector machine for groundwater quality monitoring network design under uncertainty. Environ. Model. Softw.
**2014**, 57, 115–126. [Google Scholar] [CrossRef] - Zaman, B.; McKee, M. Spatio-temporal Prediction of Root Zone Soil Moisture Using Multivariate Relevance Vector Machines. Open J. Mod. Hydrol.
**2014**, 4, 80–90. [Google Scholar] [CrossRef] - Zaman, B.; McKee, M.; Neale, C.M.U. Fusion of remotely sensed data for soil moisture estimation using relevance vector and support vector machines. Int. J. Remote Sens.
**2012**, 33, 6516–6552. [Google Scholar] [CrossRef] - Margulis, S.A.; McLaughlin, D.; Entekhabi, D.; Dunne, S. Land data assimilation and estimation of soil moisture using measurements from the Southern Great Plains 1997 Field Experiment. Water Resour. Res.
**2002**, 38, 35:1–35:18. [Google Scholar] [CrossRef] - Sànchez-Marrè, M.; Béjar, J.; Comas, J.; Brocca, L.; Melone, F.; Moramarco, T. Soil Moisture Monitoring at Different Scales for Rainfall-Runoff Modelling. In Proceedings of International Congress on Environmental Modelling and Software (iEMSs 2008), Barcelona, Spain, 7–10 July 2008; Volume 1, pp. 407–414.
- Njoku, E.G.; Wilson, W.J.; Yueh, S.H.; Dinardo, S.J.; Li, F.K.; Jackson, T.J.; Lakshmi, V.; Bolten, J. Observations of soil moisture using a passive and active low-frequency microwave airborne sensor during SGP99. IEEE Trans. Geosci. Remote Sens.
**2002**, 40, 2659–2673. [Google Scholar] [CrossRef] - Sabater, J.M.; Jarlan, L.; Calvet, J.-C.; Bouyssel, F.; De Rosnay, P. From Near-Surface to Root-Zone Soil Moisture Using Different Assimilation Techniques. J. Hydrometeorol.
**2007**, 8, 194–206. [Google Scholar] [CrossRef] - Moran, M.S.; Peters-Lidard, C.D.; Watts, J.M.; McElroy, S. Estimating soil moisture at the watershed scale with satellite-based radar and land surface models. Can. J. Remote Sens.
**2004**, 30, 805–826. [Google Scholar] [CrossRef] - Entekhabi, D.; Njoku, E.G.; O’Neill, P.E.; Kellogg, K.H.; Crow, W.T.; Edelstein, W.N.; Entin, J.K.; Goodman, S.D.; Jackson, T.J.; Johnson, J.; et al. The soil moisture active passive (SMAP) mission. Proc. IEEE
**2010**, 98, 704–716. [Google Scholar] [CrossRef] - Piles, M.; Entekhabi, D.; Camps, A. A Change Detection Algorithm for Retrieving High-Resolution Soil Moisture From SMAP Radar and Radiometer Observations. IEEE Trans. Geosci. Remote Sens.
**2009**, 47, 4125–4131. [Google Scholar] [CrossRef] - Jackson, R.D.; Idso, S.B.; Reginato, R.J.; Pinter, P.J. Canopy temperature as a crop water stress indicator. Water Resour. Res.
**1981**, 17, 1133–1138. [Google Scholar] [CrossRef] - Wetzel, P.; Atlas, D.; Woodward, R.; Werzel, P.; Atlas, D.; Woodward, R. Determining soil moisture from geosynchronous satellite infrared data. J. Climatol. Appl. Meteorol.
**1984**, 23, 375–391. [Google Scholar] [CrossRef] - Ahmad, M.-D.; Bastiaanssen, W.G.M. Retrieving Soil Moisture Storage in the Unsaturated Zone Using Satellite Imagery and Bi-Annual Phreatic Surface Fluctuations. Irrig. Drain. Syst.
**2003**, 17, 141–161. [Google Scholar] [CrossRef] - Hendrickx, J.M.H.; Hong, S.; Friesen, J.; Compaore, H.; van de Giesen, N.C.; Rodgers, C.; Vlek, P.L.G. Mapping Energy Balance Fluxes and Root Zone Soil Moisture in the White Volta Basin using Optical Imagery. Proc. SPIE
**2006**, 6239, 62390Q:1–62390Q:12. [Google Scholar] - Scott, C.A.; Bastiaanssen, W.G.M.; Ahmad, M.-D. Mapping Root Zone Soil Moisture Using Remotely Sensed Optical Imagery. J. Irrig. Drain. Eng.
**2003**, 129, 326–335. [Google Scholar] [CrossRef] - Levitt, D.G.; Simpson, J.R.; Huete, A.R. Estimates of surface soil water content using linear combinations of spectral wavebands. Theor. Appl. Climatol.
**1990**, 42, 245–252. [Google Scholar] [CrossRef] - Musick, H.B.; Pelletier, R.E. Response of some Thematic Mapper band ratios to variation in soil water content. Photogramm. Eng. Remote Sens.
**1986**, 52, 1661–1668. [Google Scholar] - Miller, J.D.; Yool, S.R. Mapping forest post-fire canopy consumption in several overstory types using multi-temporal Landsat TM and ETM data. Remote Sens. Environ.
**2002**, 82, 481–496. [Google Scholar] [CrossRef] - Idso, S.B.; Jackson, R.D.; Reginato, R.J.; Kimball, B.A.; Nakayama, F.S. The Dependence of Bare Soil Albedo on Soil Water Content. J. Appl. Meteorol.
**1975**, 14, 109–113. [Google Scholar] [CrossRef] - Carlson, T.N.; Dodd, J.K.; Benjamin, S.G.; Cooper, J.N. Satellite Estimation of the Surface Energy Balance, Moisture Availability and Thermal Inertia. J. Appl. Meteorol.
**1981**, 20, 67–87. [Google Scholar] [CrossRef] - Li, J.; Islam, S. Estimation of root zone soil moisture and surface fluxes partitioning using near surface soil moisture measurements. J. Hydrol.
**2002**, 259, 1–14. [Google Scholar] [CrossRef] - Saxton, K.E.; Johnson, H.P.; Shaw, R.H. Modeling evapotranspiration and soil moisture. Trans. ASAE
**1974**, 17, 673–677. [Google Scholar] [CrossRef] - Saxton, K.; Willey, P. The SPAW Model for Agricultural Field and Pond Hydrologic Simulation. In Watershed Models, 1st ed.; CRC Press: Boca Raton, FL, USA, 2005; pp. 400–435. [Google Scholar]
- Holtan, H.N.; Lopez, N.C. USDAHL-70 Model of Watershed Hydrology; Agricultural Research Service, U.S. Department of Agriculture: Washington, DC, USA, 1971. [Google Scholar]
- Holtan, H.N. USDAHL-74 Revised Model of Watershed Hydrology a United States Contribution to the International Hydrological Decade; Agricultural Research Service, U.S. Department of Agriculture: Washington, DC, USA, 1975. [Google Scholar]
- Peck, E.L. Catchment Modeling with the United States National Weather Service River Forecast System; NOAA: Washington, DC, USA, 1976. [Google Scholar]
- Shuttleworth, W.J. Terrestrial Hydrometeorology, 1st ed.; John Wiley & Sons, Ltd.: Chichester, UK, 2012. [Google Scholar]
- Das, N.N.; Mohanty, B.P. Root Zone Soil Moisture Assessment Using Remote Sensing and Vadose Zone Modeling. Vadose Zo. J.
**2006**, 5, 296. [Google Scholar] [CrossRef] - Allen, R.G.; Tasumi, M.; Morse, A.; Trezza, R.; Wright, J.L.; Bastiaanssen, W.; Kramber, W.; Lorite, I.; Robison, C.W. Satellite-Based Energy Balance for Mapping Evapotranspiration with Internalized Calibration (METRIC)—Model. J. Irrig. Drain. Eng.
**2007**, 133, 395–406. [Google Scholar] [CrossRef] - Liang, S. Quantitative Remote Sensing of Land Surfaces; Wiley-Interscience: Hoboken, NJ, USA, 2004. [Google Scholar]
- Allen, R.G.; Pereira, L.S.; Raes, D.; Smith, M. Crop Evapotranspiration: Guidelines for Computing Crop Requirements; FAO Irrigation and Drainage Paper 56; Food and Agriculture Organization of the United Nations (FAO): Rome, Italy, 1998. [Google Scholar]
- Allen, R.G.; Tasumi, M.; Morse, A.; Trezza, R.; Wright, J.L.; Bastiaanssen, W.; Kramber, W.; Lorite, I.; Robison, C.W. Satellite-Based Energy Balance for Mapping Evapotranspiration with Internalized Calibration (METRIC)—Applications. J. Irrig. Drain. Eng.
**2007**, 133, 395–406. [Google Scholar] [CrossRef] - Tasumi, M. Progress in Operational Estimation of Regional Evapotranspiration Using Satellite Imagery; University of Idaho: Moscow, ID, USA, 2003. [Google Scholar]
- Tipping, M.E. The Relevance Vector Machine. Adv. Neural Inf. Process. Syst.
**2000**, 12, 652–658. [Google Scholar] - Tipping, M.E.; Faul, A. Fast marginal likelihood maximisation for sparse Bayesian models. In Ninth International Workshop on Artificial Intelligence and Statistics; AISTATSS: Key West, FL, USA, 2003. [Google Scholar]
- Al-Arab, M.; Torres-Rua, A.; Ticlavilca, A.; Jensen, A.; McKee, M. Use of high-resolution multispectral imagery from an unmanned aerial vehicle in precision agriculture. In 2013 IEEE International Geoscience and Remote Sensing Symposium—IGARSS, Melbourne, VIC, Australia, 21–26 July 2013; pp. 2852–2855.
- Elarab, M.; Ticlavilca, A.M.; Torres-Rua, A.F.; Maslova, I.; McKee, M. Estimating chlorophyll with thermal and broadband multispectral high resolution imagery from an unmanned aerial system using relevance vector machines for precision agriculture. Int. J. Appl. Earth Obs. Geoinf.
**2015**, 43, 32–42. [Google Scholar] [CrossRef] - Tipping, M.E. Sparse Bayesian Learning and the Relevance Vector Machine. J. Mach. Learn. Res.
**2000**, 1, 211–244. [Google Scholar] - Fletcher, T. Relevance Vector Machines Explained; University College London: London, UK, 2010. [Google Scholar]
- Tipping, M.E.; Faul, A. Analysis of sparse Bayesian learning. Adv. Neural Inf. Process. Syst.
**2002**, 14, 383–389. [Google Scholar] - DRI. Community Environmental Monitoring Program|CEMP—DOE. 2014. Available online: cemp.dri.edu (accessed on 1 January 2012).
- LPSO. Landsat Project Science Office—Landsat 7 Science Data Users Handbook; NASA: Greenbelt, MD, USA, 2008. [Google Scholar]
- Allen, R.; Trezza, A.R.; Tasumi, M.; Kjaersgaard, J. Metric Application Manual for Landsat Satellite Imagery; University of Idaho: Kimberly, ID, USA, 2012. [Google Scholar]
- Decagon Devices Inc. GS3 Operator’s Manual; Decagon Devices Inc.: Pullman, WA, USA, 2014. [Google Scholar]
- Decagon Devices Inc. GS3 User Manual; Decagon Devices Inc.: Pullman, WA, USA, 2012. [Google Scholar]
- Cobos, D.R.; Chambers, C.; Clarke, T.R.; Chambers, C. Calibrating ECH2O Soil Moisture Sensors; Decagon Devices Inc.: Pullman, WA, USA, 2010. [Google Scholar]
- Vector Anomaly Ltd. Vector Anomaly: Sparsebayes Software. 2009. Available online: http://www.vectoranomaly.com/downloads/downloads.htm (accessed on 20 November 2011).
- Arlot, S.; Celisse, A. A survey of cross-validation procedures for model selection. Stat. Surv.
**2010**, 4, 40–79. [Google Scholar] [CrossRef] - Ruß, G. From Spatial Data Mining in Precision Agriculture to Environmental Data Mining. In Computational Intelligence in Intelligent Data Analysis; Springer Berlin Heidelberg: Berlin, Germany, 2013; pp. 263–273. [Google Scholar]
- Guyon, I.; Elisseeff, A. An introduction to variable and feature selection. J. Mach. Learn. Res.
**2003**, 3, 1157–1182. [Google Scholar] - Crist, E.P.; Cicone, R.C. A Physically-Based Transformation of Thematic Mapper Data—The TM Tasseled Cap. IEEE Trans. Geosci. Remote Sens.
**1984**, 3, 256–263. [Google Scholar] [CrossRef] - Nash, J.; Sutcliffe, J. River flow forecasting through conceptual models part I—A discussion of principles. J. Hydrol.
**1970**, 10, 282–290. [Google Scholar] [CrossRef] - Moriasi, D.; Arnold, J.; van Liew, M. Model evaluation guidelines for systematic quantification of accuracy in watershed simulations. Trans. ASABE
**2007**, 50, 885–900. [Google Scholar] [CrossRef] - Tucker, C.J. Red and photographic infrared linear combinations for monitoring vegetation. Remote Sens. Environ.
**1979**, 8, 127–150. [Google Scholar] [CrossRef] - Homer, C.; Huang, C.; Yang, L.; Wylie, B.; Coan, M. Development of a 2001 national land-cover database for the United States. Photogramm. Eng. Remote Sens.
**2004**, 70, 829–840. [Google Scholar] [CrossRef] - Torres-Rua, A.F.; Ticlavilca, A.M.; Walker, W.R.; McKee, M. Machine Learning Approaches for Error Correction of Hydraulic Simulation Models for Canal Flow Schemes. J. Irrig. Drain. Eng.
**2012**, 138, 999–1010. [Google Scholar] [CrossRef] - Huang, C.; Townshend, J.R.G.; Liang, S.; Kalluri, S.N.V.; DeFries, R.S. Impact of sensor’s point spread function on land cover characterization: Assessment and deconvolution. Remote Sens. Environ.
**2002**, 80, 203–212. [Google Scholar] [CrossRef] - Wenny, B.; Helder, D.; Hong, J.; Leigh, L.; Thome, K.; Reuter, D. Pre- and Post-Launch Spatial Quality of the Landsat 8 Thermal Infrared Sensor. Remote Sens.
**2015**, 7, 1962–1980. [Google Scholar] [CrossRef] - Markham, B.L.; Storey, J.C.; Williams, D.L.; Irons, J.R. Landsat sensor performance: history and current status. IEEE Trans. Geosci. Remote Sens.
**2004**, 42, 2691–2694. [Google Scholar] [CrossRef] - Tasumi, M.; Trezza, R.; Allen, R.G.; Wright, J.L. Operational aspects of satellite-based energy balance models for irrigated crops in the semi-arid U.S. Irrig. Drain. Syst.
**2005**, 19, 355–376. [Google Scholar] [CrossRef] - Morton, C.G.; Huntington, J.L.; Pohll, G.M.; Allen, R.G.; McGwire, K.C.; Bassett, S.D. Assessing Calibration Uncertainty and Automation for Estimating Evapotranspiration from Agricultural Areas Using METRIC. J. Am. Water Resour. Assoc.
**2013**, 49, 549–562. [Google Scholar] [CrossRef]

**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