# Multiscale Assimilation of Sentinel and Landsat Data for Soil Moisture and Leaf Area Index Predictions Using an Ensemble-Kalman-Filter-Based Assimilation Approach in a Heterogeneous Ecosystem

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Case Study

_{a}) is 14.6 °C, with a mean July T

_{a}of 23.7 °C.

#### 2.1.1. Field Data

_{2}/H

_{2}O infrared gas analyzer positioned adjacent to each other at the top of the tower. These two instruments measured velocity, temperature, and gas concentrations for the estimation of sensible heat flux, evapotranspiration (ET), and CO

_{2}exchanges (F

_{c}) with the standard eddy covariance method (e.g., [59]). Half-hourly statistics were computed.

^{2}gDM

^{−1}) and woody vegetation (0.005 m

^{2}gDM

^{−1}) species were measured directly by weighing the dry biomass.

#### 2.1.2. Remote Sensing Data

#### 2.2. The Proposed Assimilation Approach

#### 2.2.1. Optical Remote Sensing Data for LAI Estimate

_{N}operator using an empirical approach (e.g., [18,60,61]):

_{1}, β

_{2}, and β

_{3}are coefficients for vegetation species. Note that analogous solutions should be derivable from different ${\mathsf{\Gamma}}_{L}\left(\mathrm{NDVI}\right)$ relationships.

_{v,t}fraction of tree cover in the field, which was estimated as NDVI/NDVI

_{max}following Detto et al. [6], with NDVI

_{max}as the maximum value of NDVI in the investigated field.

#### 2.2.2. Radar Images for Soil Moisture Retrieval

^{2}+ 11.44 NDVI − 0.5982

_{θ}operator [65]:

#### 2.2.3. The Ecohydrological Model

#### 2.2.4. The Land Surface Model

_{rz}is the root zone depth; I

_{bs}is the infiltration rate on bare soil; I

_{t}and I

_{gr}are the throughfall rates infiltrating into the soil covered by trees and grass, respectively; q

_{D}is the rate of drainage out of the bottom of the root zone; E

_{bs}is the rate of bare soil evaporation; E

_{t}and E

_{g}are the rates of transpiration of trees and grass, respectively; f

_{v,g}is the fraction of grass cover; and f

_{bs}is the fraction of bare soil [14].

_{D}rate was estimated using the unit head gradient assumption (Table 1; [11,14]).

_{t}and E

_{g}were estimated distinctly using the Penman–Monteith equation (e.g., [70], p. 224) for each vegetation component. Canopy resistances, accounting for environmental stresses, were estimated using a typical Jarvis [71] approach (Table 1). The actual rate of bare soil evaporation was determined as $\alpha \left(\theta \right)PE$, where α(θ) is a rate-limiting function, estimated by the polynomial function of Parlange et al. [72], and PE is the potential evaporation estimated by the Penman equation (e.g., [70], Equations 10.15, 10.16, and 10.19). Hence, the total evapotranspiration was estimated as:

_{2}flux [57]:

_{c,t}and F

_{c,g}are the carbon exchange of trees and grass, respectively, and R

_{bs}is the soil respiration. Carbon exchange rates for each PFT (i.e., F

_{c,t}, F

_{c,g}) were computed as the difference between photosynthesis and growth respiration (Table 2). Soil respiration was estimated as a function of the temperature (Table 2, [57,73,74,75]). The model parameters are presented in Table 3.

**Table 1.**Equations of drainage (q

_{D}), canopy resistance (r

_{c}) with stress functions of soil moisture (θ), air temperature (T

_{a}) and vapor pressure deficit (VPD), sensible heat flux (H), net radiation (R

_{n}), soil heat flux (G), and surface temperature (T

_{s}) in the LSM. Parameters are defined in Table 3.

Equations | Source |
---|---|

Drainage ${q}_{D}={k}_{s}{\left(\frac{\theta}{{\theta}_{s}}\right)}^{2b+3}$ | [11] |

Canopy resistance ${r}_{c}=\frac{{r}_{s,min}}{LAI{\left[{f}_{1}\left(\theta \right){f}_{2}\left({T}_{a}\right){f}_{3}\left(VPD\right)\right]}^{-1}}$ | [69] |

${f}_{1}\left(\theta \right)=\{\begin{array}{c}0\\ \frac{\theta -{\theta}_{wp}}{{\theta}_{lim}-{\theta}_{wp}}\\ 1\end{array}\begin{array}{c}if\theta \le {\theta}_{wp}\\ if{\theta}_{wp}\theta {\theta}_{lim}\\ if\theta \ge {\theta}_{lim}\end{array}$ ${f}_{2}\left({T}_{a}\right)=\{\begin{array}{l}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{1em}{T}_{a}\le {T}_{a,\mathrm{min}}\hspace{1em}\mathrm{and}\hspace{1em}{T}_{a}>{T}_{a,\mathrm{max}}\hfill \\ 1-\frac{{T}_{a,opt}-{T}_{a}}{{T}_{a,opt}-{T}_{a,\mathrm{min}}}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{1em}{T}_{a,\mathrm{min}}{T}_{a}{T}_{a,opt}\hfill \\ 1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{1em}{T}_{a,opt}\le {T}_{a}\le {T}_{a,\mathrm{max}}\hfill \end{array}$ f _{3} = 1 − ω log(VPD) | [74] |

Sensible heat flux $H={\rho}_{a}{c}_{p}{C}_{H}u\left({T}_{s}-{T}_{a}\right)$, where C _{H} the heat transfer coefficient | [14] |

Net radiation ${R}_{n}={R}_{swin}\left(1-\alpha \right)+\epsilon \left({R}_{lwin}-\sigma {T}_{s}{}^{4}\right)$, with shortwave incoming ration, R _{swin}; longwave incoming ration, R_{lwin}, estimated based on Equation 6 in Brutsaert (1982); α—albedo; ε—emissivity; σ—the Stefan–Boltzmann constant | [14] |

Soil heat flux G = R _{n} − H − LE | [14] |

Surface temperature $\frac{d{T}_{s}}{dt}={C}_{T}G-\frac{2\pi}{\tau}\left({T}_{s}-{T}_{a}\right)$, where T _{2} is the mean T_{s} value over one day, τ, and C_{T} is the soil thermal coefficient$\frac{d{T}_{2}}{dt}=\frac{1}{\tau}\left({T}_{s}-{T}_{2}\right)$ | [14] |

**Table 2.**Equations of the vegetation dynamic model components. Parameters are defined in Table 3.

Ecophysiological Term | Equations | Source |
---|---|---|

Photosynthesis | $Ph={\epsilon}_{P}\left(PAR\right){f}_{PAR}PAR\frac{1.37{r}_{a}+1.6{r}_{c,min}}{1.37{r}_{a}+1.6{r}_{c}}$ ${\epsilon}_{P}\left(PAR\right)={a}_{0}+{a}_{1}PAR+{a}_{2}PA{R}^{2}$ ${f}_{PAR}=1-{e}^{-{k}_{e}LAI}$ | [76] |

Allocation | For the tree cover: ${a}_{a}=\frac{{\xi}_{a}}{1+\Omega \left[2-\lambda -{f}_{1}\left(\theta \right)\right]}$ ${a}_{s}=\frac{{\xi}_{s}+\Omega \left(1-\lambda \right)}{1+\Omega \left[2-\lambda -{f}_{1}\left(\theta \right)\right]}$ ${a}_{r}=\frac{{\xi}_{r}+\Omega \left(1-{f}_{1}\left(\theta \right)\right)}{1+\Omega \left[2-\lambda -{f}_{1}\left(\theta \right)\right]}$ ${\xi}_{a}+{\xi}_{s}+{\xi}_{r}=1;$ $\lambda ={e}^{-{k}_{e}LAI}$ | [14] |

For grass cover: ${a}_{a}=\frac{{\xi}_{a}+\Omega \lambda}{1+\Omega \left[1+\lambda -{f}_{1}\left(\theta \right)\right]}$ ${a}_{r}=\frac{{\xi}_{r}+\Omega \left(1-{f}_{1}\left(\theta \right)\right)}{1+\Omega \left[1+\lambda -{f}_{1}\left(\theta \right)\right]}$ ${\xi}_{a}+{\xi}_{r}=1$ | [14] | |

Respiration | Maintenance and growth respirations of biomass components: ${R}_{l,\mu}={m}_{a}{f}_{4}\left(T\right){B}_{l}$$;{R}_{l,\gamma}={g}_{a}{a}_{a}{P}_{g}$ ${R}_{s,\mu}={m}_{s}{f}_{4}\left(T\right){B}_{s}$$;{R}_{s,\gamma}={g}_{s}{a}_{s}{P}_{g}$ ${R}_{r,\mu}={m}_{r}{f}_{4}\left(T\right){B}_{r}$$;{R}_{r,\gamma}={g}_{r}{a}_{r}{P}_{g}$ | [77] |

${f}_{4}\left(T\right)={Q}_{10}^{\frac{{T}_{m}}{10}}$ with T_{m} = mean daily temperature | [76] | |

$\mathrm{Soil}\mathrm{respiration}\phantom{\rule{0ex}{0ex}}{R}_{bs}={R}_{10}{Q}_{N}^{\frac{{T}_{m}}{10}}$ where R _{10} is the reference respiration rate at 10 °C and Q_{N} is the soil respiration sensitivity to temperature | [57] | |

Senescence | ${S}_{l}={d}_{a}{B}_{l}$ ${S}_{s}={d}_{s}{B}_{s}\phantom{\rule{0ex}{0ex}}{S}_{r}={d}_{r}{B}_{r}$ | [14,77] |

Litterfall | ${L}_{a}={k}_{a}{B}_{d}$ | [14,77] |

Parameter | Description | Value | |
---|---|---|---|

Grass | Tree | ||

LSM–VDM parameters | |||

r_{s,min} [s m^{−1}] | Minimum stomatal resistance | 100 | 300 |

T_{min} [°K] | Minimum temperature | 272.15 | 272.15 |

T_{opt} [°K] | Optimal temperature | 295.15 | 285.15 |

T_{max} [°K] | Maximum temperature | 313.15 | 318.15 |

θ_{wp} [-] | Wilting point | 0.08 | 0.04 |

θ_{lim,} [-] | Limiting soil moisture for vegetation | 0.20 | 0.17 |

ω [KPa^{−1}] | Slope of the f_{3} relation | 0.6 | 0.6 |

Only VDM parameters | |||

c_{l} [m^{2} gDM^{−1}] | Specific leaf areas of the green biomass in growing season | 0.01 | 0.005 |

c_{d} [m^{2} gDM^{−1}] | Specific leaf areas of the dead biomass | 0.01 | 0.003 |

k_{e} [-] | PAR extinction coefficient | 0.5 | 0.5 |

ξ_{a} [-] | Parameter controlling allocation to leaves | 0.6 | 0.55 |

ξ_{s} [-] | Parameter controlling allocation to stem | - | 0.1 |

ξ_{r} [-] | Parameter controlling allocation to roots | 0.4 | 0.35 |

Ω [-] | Allocation parameter | 0.8 | 0.8 |

m_{a} [d^{−1}] | Maintenance respiration coefficients for aboveground biomass | 0.032 | 0.001 |

g_{a} [-] | Growth respiration coefficients for aboveground biomass | 0.28 | 0.69 |

m_{r} [d^{−1}] | Maintenance respiration coefficients for root biomass | 0.007 | 0.002 |

g_{r} [-] | Growth respiration coefficients for root biomass | 0.1 | 0.1 |

Q_{10} [-] | Temperature coefficient in the respiration process | 2.45 | 2.42 |

d_{a} [d^{−1}] | Death rate of aboveground biomass | 0.05 | 0.0045 |

d_{r} [d^{−1}] | Death rate of root biomass | 0.003 | 0.005 |

k_{a} [d^{−1}] | Rate of standing biomass pushed down | 0.05 | 0.35 |

Only LSM parameters | |||

z_{om,v} [m] | Vegetation momentum roughness length | 0.05 | 0.5 |

z_{ov,v} [m] | Vegetation water vapor roughness length | z_{om}/7.4 | z_{om}/2.5 |

z_{om,bs} [m] | Bare soil momentum roughness length | 0.015 | |

z_{ov,bs} [m] | Bare soil water vapor roughness length | z_{om}/10 | |

θ_{s} [-] | Saturated soil moisture | 0.53 | |

b [-] | Slope of the retention curve | 8 | |

k_{s} [m/s] | Saturated hydraulic conductivity | 5 × 10^{−6} | |

|ψ_{s}| [m] | Air entry suction head | 0.79 | |

d_{rz} [m] | Root zone depth | 0.19 |

#### 2.2.5. The Vegetation Dynamic Model

_{l}), stem (B

_{s}), living root (B

_{r}), and standing dead (B

_{d}). The VDM of grass only distinguishes three biomass states (green leaves, roots, and standing dead). The biomass [g DM m

^{−2}] components were simulated using the approach of Montaldo et al. [76], which consists of a balance between biomass production (related to photosynthesis for green leaves, stem, and roots biomass) and biomass destruction (respiration and senescence for green leaves, stem, and roots biomass), through ordinary differential equations, integrated numerically at a daily time step.

_{a}, a

_{s}and a

_{r}are the allocation (partitioning) coefficients for leaves, stem, and root states, respectively; R

_{l,μ}, R

_{s,μ}, and R

_{r,μ}are the maintenance respiration rates from leaves, stem, and root biomass, respectively; R

_{l,γ}, R

_{s,γ}, and R

_{r,γ}are the growth respiration rates from leaves, stem, and root biomass, respectively; S

_{g}, S

_{s}, and S

_{r}are the senescence rates of leaves, stem, and root biomass, respectively; L

_{a}is the litterfall. The model equations are given in Table 2 and the parameters are presented in Table 3.

_{l}is the specific leaf area of the green biomass. The VDM provides estimates of daily values of leaf biomass and, thus, the LAI of the tree and grass was used by the LSM to estimate evapotranspiration, energy flux, rainfall interception, carbon assimilation, and the soil water content at a half-hour time step [14]. The LSM provides soil moisture and aerodynamic resistances to the VDM. The coupled model was calibrated and validated in Montaldo et al. [14] and Montaldo et al. [57]. The details are given in Montaldo et al. [76], Montaldo et al. [14], and Montaldo et al. [57].

#### 2.2.6. The Ensemble Kalman Filter

_{j}, as follows:

_{e}, with N

_{e}the size of the ensemble) is predicted in parallel, using (13). The EnKF updates each ensemble member separately, using the $\overrightarrow{\delta}\left({t}_{j}\right)$ observation and the diagnosed state error covariance $\overrightarrow{{P}^{-}}\left({t}_{j}\right)$ (e.g., [39], Equation (6)). The superscripts “−” and “+” refer to the state estimates before and after the update at time t

_{j}, respectively. Ensemble members are updated using a combination of forecast model states and the observations [39], as follows:

_{s})—following Montaldo et al. [38]. The ensemble of soil moisture initial values is generated by altering a particular value of soil moisture through the addition of a normally distributed perturbation with a zero mean and SD

_{θ}(standard deviation). At each time step, the ensemble of precipitation is generated by multiplying the recorded precipitation value by a normally distributed random variable. An ensemble of saturated hydraulic conductivity values (${k}_{s}^{\zeta}$) is generated as being log

_{10}normally distributed with a mean of $log({\widehat{k}}_{s})$ (indicating with ${\widehat{k}}_{s}$ the base (i.e., best guess) value of the ${k}_{s}^{\zeta}$ ensemble) and the standard deviation of SD

_{logks}. In this way, an ensemble of ${\theta}^{\zeta}$, which includes model errors, is generated and evolves in time according to (5).

_{swin}), and the photosynthetically active radiation (PAR); and (iii) model parameters—the maintenance respiration coefficients for the aboveground biomass (m

_{a}) of grass and trees (Table 2). We chose m

_{a}as the VDM parameter for data assimilation after a sensitivity analysis of LAI to VDM parameters, which proved the high sensitivity of grass and trees LAI to m

_{a}[56]. The ensemble of LAI initial values was generated by altering a particular value of LAI through the addition of a normally distributed perturbation with a zero mean and SD

_{LAI}standard deviation. At each time step, the ensembles of R

_{swin}and PAR were generated by multiplying the recorded R

_{swin}and PAR values by normally distributed random variables. The ensembles of grass and tree maintenance respiration coefficients (${m}_{a,g}^{\zeta}$ for grass and ${m}_{a,t}^{\zeta}$ for trees), were generated as being normally distributed with means of ${\widehat{m}}_{a,g}$ and ${\widehat{m}}_{a,t}$ and standard deviations of SD

_{mag}and SD

_{mat}, respectively. In this way, ensembles of LAI

^{ζ}of grass and trees, which include model errors, were generated and evolved in time according to (8) and (12). The time steps of models and observations are shown in Figure 2.

_{j,θ}were obtained including the $\overrightarrow{{\u03f5}^{l}}$ random error in the ε observations derived from Sentinel 1 according to (14), where the operator $\overrightarrow{H}$ is the inverse of ${\mathsf{\Gamma}}_{\theta}$ in (4). Similarly, the NDVI observations available at time t

_{j,L}derived from Landsat 8 (or Sentinel 2) were altered randomly according to (14), where the operator $\overrightarrow{H}$ is the inverse of ${\mathsf{\Gamma}}_{L}$ in (1).

#### 2.2.7. The Updating of Model Parameters through the Assimilation

_{s}parameter of the LSM, and the grass and tree maintenance respiration coefficients of the VDM.

_{s}parameter is updated using the approach of Montaldo et al. [38], based on an expression derived by Montaldo and Albertson [81], that estimates the biased error in k

_{s}from analysis of the persistent-state variable bias (as defined by a longer time average). Each component of the ${k}_{s}^{\zeta}$ ensemble is updated over an appropriate averaging time interval (Δt

_{5}; Figure 2), which coincides with time steps t

_{ξ,θ}, through

_{3}is the radar observation time steps (Figure 2). The overbar in (17) and (18) provides an averaging in the Δt

_{5}time steps (≥Δt

_{3}) for capturing an estimate of the “persistent” moisture bias estimating the required change in the saturated conductivity. In this way, the biased model error can be removed after a learning (calibration) period, and the Kalman filter assumption, the zero mean model error, can be recovered.

_{a}based on observations of persistent bias in the modeled biomass (i.e., LAI). The proposed procedure derives the required m

_{a}adjustment from the conservation equation of the biomass (i.e., LAI), and we applied it for both grass and tree LAI. Each component of the ${m}_{a,g}^{\zeta}$ and ${m}_{a,t}^{\zeta}$ ensembles was updated over the Δt

_{6}time interval, which coincides with time steps t

_{ξ,L}(Appendix A), as follows:

_{4}is the NDVI observation time steps (Figure 2), and the overbar in (20) and (21) provides an averaging in the Δt

_{6}time steps (≥Δt

_{4}). In this way, an estimate of the “persistent” LAI bias is used for evaluating the necessary change in m

_{a}. Thereby, after a learning (calibration) period, the error of the model can be eliminated. We used the same solution for grass (${m}_{a,g}^{\zeta +}$) and tree (${m}_{a,t}^{\zeta +}$) maintenance respiration coefficients.

#### 2.2.8. The Multiscale Assimilation Approach

- A land surface model that predicts the ensemble of soil moisture states through (5) at the half-hourly timescale (Δt
_{1}); - A vegetation dynamic model that predicts the ensembles of grass and tree LAI through (8) and (12) at a daily timescale (Δt
_{2}); - EnKF filters of the ε observations (4), which are available every 6 days on average (Δt
_{3}); these account for moderate LSM errors and provide optimal updates of the ensemble of ${\theta}_{}^{\zeta -}\left({t}_{j}\right)$ through (15) to arrive at ${\theta}_{}^{\zeta +}$; - EnKF filters of the NDVI remote data (1) of grass and trees, available over the weekly timescale on average (Δt
_{4}), which optimally update the ensembles of $LA{I}^{\zeta -}$ of grass and trees through (15) to arrive at $LA{I}^{\zeta +}$; - An ensemble of the key LSM parameter, ${k}_{s}^{\zeta}$, which is updated through (16) over the weekly timescale (Δt
_{5}); - Finally, the ${m}_{a}^{\zeta}$ ensembles of grass and trees that are updated through (19) at > weekly (e.g., 3 weeks) timescale (Δt
_{6}).

_{s}and the grass and tree m

_{a}model parameters (compared with the calibrated values), comparing EnOL, EnKF, and EnKFdc performances.

#### 2.2.9. Application of the Assimilation Approach to the Case Study

^{0}

_{vv}backscattering coefficients were available. Data were collected, analyzed, and corrected to include the vegetation growth effect on the radar backscatter in Montaldo et al. [33], so that we used the ε time series produced by Montaldo et al. [33].

_{1}= −0.435, β

_{2}= 1.014 and β

_{3}= 0.4029 for grass in the autumn–winter period; β

_{1}= −0.141, β

_{2}= 1.720, and β

_{3}= 1.674 for grass in the spring–summer period; β

_{1}= 0, β

_{2}=5.392 and β

_{3}= 0.486 for trees).

_{1}), the VDM time step was one day (Δt

_{2}), the assimilation time step of grass and tree NDVI data (Δt

_{4}) was variable according to data availability, ranging from 2 days to 20 days with an average of 6 days, and the time step of the ${m}_{a,g}^{\zeta}$ and ${m}_{a,t}^{\zeta}$ updating was 3 weeks for both grass and trees (Δt

_{6}, Figure 2). Note that, because the VDM was applied distinctly for grass and tree in the Sardinian heterogeneous ecosystem, the grass and tree cells need to be selected in the field as representatives of the two main vegetation components for distinctly assimilating grass and tree NDVI, which were estimated in those cells in the VDM. The assimilation time step of the Sentinel 1 dielectric constant was ≤6 days (Δt

_{3}), and ${k}_{s}^{\zeta}$ was updated every 6 days (Δt

_{5}). In the EnKF, N

_{e}was 100, which is a sufficiently large number for accurate predictions [38,39,79].

_{g}) and tree (LAI

_{t}) from a Gaussian distribution with means of 0.5 and 5.5, respectively, intentionally different from the observations, and a standard deviation ${\sigma}_{LAI}$ of 0.2 for both grass and tree LAI. In the LSM, the ensembles of initial ${\theta}^{\zeta}$ were generated from a Gaussian distribution with a mean of 0.2; this was intentionally lower (20%) than the observed value, with a standard deviation of 0.05. At each time step, we generated the ensembles of the following: (i) the precipitation, by multiplying the recorded precipitation value by a normally distributed random variable with a zero mean and a standard deviation equal to 20%; (ii) the incoming solar radiation; (iii) the PAR by multiplying the measured values by a normally distributed random variable with mean zero and a standard deviation equal to 10%. It should be noted that the errors of the initial model states and parameters were uncorrelated.

^{−1}for grass; 0.0001, 0.0003, 0.0006, 0.001, 0.004, 0.006, and 0.01 d

^{−1}for trees) with the same SD

_{mag}and SD

_{mat}(5% of the initial value), and for five initial ${k}_{s}^{\zeta}$ at most, generated for five different initial ${\widehat{k}}_{s}$ (5 × 10

^{−8}, 5 × 10

^{−7}, 5 × 10

^{−6}, 5 × 10

^{−5}, and 5 × 10

^{−4}m/s) with the same SD

_{logks}of 0.98.

## 3. Results

^{−4}m/s, 0.12 d

^{−1}, and 0.01 d

^{−1}, respectively), which were greatly higher than the corresponding calibrated values (Figure 3). The use of the EnKF was not enough for guiding the models, because grass LAI was still underpredicted during the growing seasons, similar to the results with the EnOL configuration; RMSE values were still high, up to 0.8; and observed grass LAI was higher than 1 (Figure 3d,g). The predicted tree LAI using the EnKF was even worse, with values lower than 0.3, while the observed tree LAI from remote data was around 4, and RMSE became close to 4 after 8 months of simulation when the initial model conditions were lost (Figure 3e,h). The results for θ predictions using the EnKF configuration were slightly better when compared with those using the EnOL configuration (Figure 3f,i); however, the model was still underpredicting the soil moisture during the wet months (up to 50% in autumn 2017) when the hydraulic conductivity greatly affected the soil moisture budget predictions due to the prescribed errors in k

_{s}. The EnKFdc approach dynamically calibrated the three key model parameters, which converged to values close to the calibrated values after 8–13 months (Figure 3), becoming the coupled model recalibrated for the 2017 and 2018 predictions. In 2016, the model was not yet recalibrated and the RMSE was still high, because the approach requires time for capturing and correcting the persistent model errors. Thanks to the model parameter updating, after almost one year, the RMSE of tree LAI became almost negligible (Figure 3h), and the RMSE of grass LAI decreased to values lower than 0.3 (Figure 3g). Soil moisture was better predicted using the EnKFdc configuration (Figure 3f), with values close to the radar-observed soil moisture during the wet months due to the correction of the hydraulic conductivity.

^{−8}m/s, 0.0032 d

^{−1}, and 0.0001 d

^{−1}, respectively). These were greatly lower than the corresponding calibrated values (Figure 4). Again, while the EnKF configuration was not enough to guide the model, the use of the EnKFdc approach updated the three model parameters, which reached values close to the corresponding calibrated values after almost one year (Figure 4a–c). Using the EnKFdc approach, the errors of grass and tree LAI predictions when compared to satellite observations were almost negligible (RMSE < 0.15 for grass LAI and RMSE < 0.1 for tree LAI in the 2017 and 2018 years; Figure 4g,h). These also decreased for the soil moisture predictions, especially during the unusually wet 2018 (Figure 4i).

_{s}, m

_{ag}, and m

_{at}(e.g., Figure 3).

_{c})—which are strictly related to soil moisture and vegetation growth. We used the eddy-covariance-based ET and F

_{c}observations to evaluate the model. Although it improved the ET predictions when compared with the predictions using the EnOL configuration, the use of the EnKF was not enough to guide the model and predict the ET well for the full range of parameter combinations: total ET was underpredicted by up to 70% and was overpredicted by up to 10% for high and low ${\widehat{m}}_{a,g}$ and ${\widehat{m}}_{a,t}$, respectively (Figure 8b). The use of the proposed EnKFdc allowed us to remove the model bias, and the ET was accurately predicted for the full range of the parameters (misprediction of the total ET was lower than 4%, see Figure 8c).

_{c}predictions. Indeed, the use of the EnKF approach did not remove the model bias in F

_{c}predictions when extremely high and low ${\widehat{m}}_{a,g}$ and ${\widehat{m}}_{a,t}$ values were initially assumed: underprediction and overprediction of the total observed F

_{c}occurred by up to 80% and 70%, respectively (Figure 8e). The dynamic calibration of the key model parameters using the EnKFdc approach allowed us to remove the model bias and predict the F

_{c}well for the full ranges of parameters (misprediction of the total F

_{c}was lower than 15%, see Figure 8f).

## 4. Discussion

_{a}parameters can guide the VDM, preserving the biomass balance of grass and tree species. The proposed EnKFdc approach for both LSM and VDM was the only approach to predict not only the soil moisture and the tree and grass LAI well, but was also the only approach to predict the main outputs of the coupled model, the evapotranspiration, and the carbon exchanges (Figure 8), which are strictly related to soil moisture and vegetation dynamics [57,90]. The use of the EnKF alone was not enough to obtain good predictions of the two key land surface fluxes, with errors up 70% in ET predictions, when lower m

_{a}values were prescribed; furthermore, the errors were up to 120% in F

_{c}predictions when high initial m

_{a}values were prescribed. Such high model bias in ET predictions affects soil–water balance predictions in these semiarid ecosystems because ET is the main loss term of the soil water budget, with a yearly magnitude that may be equal to the precipitation [6,91,92]. ET and F

_{c}predictions were good for the full range of model parameters when the proposed EnKFdc approach was used, showing the high performance of the approach (errors less than 4% and 15% in ET and F

_{c}predictions, respectively).

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{a}based on observations of persistent bias in the modeled biomass (i.e., LAI). Substituting (12) in (8), the biomass balance for the modeled “m” state variables is:

_{a}at each time the LAI is updated through NDVI, from knowledge of the change in $\Delta LA{I}^{o,m}$ since the last update. By averaging Equation (A10) over an appropriate time interval (Δt

_{6}, e.g., 3 weeks, Figure 2) to capture a reliable estimate of the “persistent” LAI bias, the required change in the maintenance respiration coefficient can be estimated with

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**Figure 1.**Representation of the Sardinian heterogeneous ecosystem: (

**a**) the position of the tower (red cross) in Sardinia; (

**b**) aerial photography of summer 2016 with the position of the eddy covariance tower (red cross); (

**c**) the NDVI map from a Landsat 8 image with the position of the eddy covariance tower (red cross), and the representative grass and tree cells, which NDVI evolutions in time during the 2016 year are reported in the inset; (

**d**) a picture of the landscape on a dry summer day.

**Figure 2.**The scheme of the multiscale assimilation approach: soil moisture (θ) and leaf area index (LAI) are the two assimilated and predicted variables by the coupled land surface model (LSM) and vegetation dynamic model (VDM); the saturated hydraulic conductivity (k

_{s}) and the maintenance respiration coefficient (m

_{a}) are the dynamically updated parameters through the assimilation approach; EnKF is the ensemble Kalman filter applied to both LSM and VDM. The timescales of the models, ENKF, and parameter updating are reported. The equations for the θ, LAI, k

_{s}, and m

_{a}adjustments are referenced within parenthesis.

**Figure 3.**Assimilation results for soil moisture, grass, and tree LAI predictions at the Sardinian site using initial high ${\widehat{k}}_{s}$, ${\widehat{m}}_{a,g}$ (for grass), and ${\widehat{m}}_{a,t}$ (for trees) values of 5 × 10

^{−4}m/s, 0.12 d

^{−1}, and 0.01 d

^{−1}, respectively: (

**a**–

**c**) show the evolutions of the ${k}_{s}^{\zeta}$, ${m}_{a,g}^{\zeta}$, and ${m}_{a,t}^{\zeta}$ ensembles, respectively, using the EnKFdc approach (the means of the ensembles are in black thick lines; for reference, the calibrated values of k

_{s}, m

_{a,g}, and m

_{a,t}are reported in dotted horizontal lines); (

**d**–

**f**) show the comparison between LAI and soil moisture observations derived from assimilated remote data (obs.) and the ensemble means predicted using the EnOL, EnKF, and EnKFdc approaches, respectively; (

**g**–

**i**) show the evolutions of the RMSE of the ensemble mean of soil moisture and LAI predicted using the EnOL, EnKF, and EnKFdc approaches concerning the observed soil moisture and LAI (derived from remote sensing data) using a 60-day window, translated in 10-day increments.

**Figure 4.**Same as Figure 3 but for initial low ${\widehat{k}}_{s}$, ${\widehat{m}}_{a,g}$ (for grass), and ${\widehat{m}}_{a,t}$ (for trees) values of 5 × 10

^{−8}m/s, 0.0032 d

^{−1}, and 0.0001 d

^{−1}, respectively.

**Figure 5.**The root mean square error (rmse) of soil moisture (θ) predictions using the (

**a**) EnOL, EnKF, and EnKFdc approaches with the assimilation of the radar backscatter (related to θ) only (

**b**,

**e**), the assimilation of grass and tree NDVI (related to LAI) only (

**c**,

**f**), and the assimilation of both radar backscatter and grass and tree NDVI (

**d**,

**g**), while varying the initial ${\widehat{m}}_{a,g}$, ${\widehat{m}}_{a,t}$, and ${\widehat{k}}_{s}$ values (low m

_{a,g}and m

_{a,t}values correspond to 0.0032 d

^{−1}and 0.0001 d

^{−1}, respectively, while high m

_{a,g}and m

_{a,t}values correspond to 0.12 d

^{−1}and 0.01 d

^{−1}, respectively; cal—calibrated values of Table 3).

**Figure 6.**Same as Figure 5 but (

**a**–

**g**) for the root mean square error (RMSE) of the grass leaf area index (LAI), and (

**h**–

**n**) the root mean square error (RMSE) of the tree leaf area index (LAI).

**Figure 7.**Comparison of seasonal errors in soil moisture (θ) in (

**a**), tree, and grass LAI (

**b**,

**c**) predictions using the EnOL, EnKF, and EnKFdc approaches and assimilating (in EnKF and EnKFdc configurations) radar backscatter and grass and tree NDVI; with varying initial ${\widehat{m}}_{a,g}$, ${\widehat{m}}_{a,t}$, and ${\widehat{k}}_{s}$ values. The statistics of all runs obtained varying values, and the initial ${\widehat{m}}_{a,g}$, ${\widehat{m}}_{a,t}$, and ${\widehat{k}}_{s}$ are in each estimation box (investigated period: from October 2016 to September 2018). Diamonds indicate the means; black lines indicate the median; the box and whisker plots represent the quartiles; outliers are depicted individually.

**Figure 8.**The errors in total evapotranspiration (ET) and carbon exchange (F

_{c}) predictions using the EnOL (

**a**,

**d**), EnKF (

**b**,

**e**), and EnKFdc (

**c**,

**f**) approaches, with varied initial ${\widehat{m}}_{a,g}$, ${\widehat{m}}_{a,t}$, and ${\widehat{k}}_{s}$ values (low m

_{a,g}and m

_{a,t}values correspond to 0.0032 d

^{−1}and 0.0001 d

^{−1}, respectively, while high m

_{a,g}and m

_{a,t}values correspond to 0.12 d

^{−1}and 0.01 d

^{−1}, respectively; cal—calibrated values), compared with ET and F

_{c}observations from the eddy-covariance-based tower.

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## Share and Cite

**MDPI and ACS Style**

Montaldo, N.; Gaspa, A.; Corona, R.
Multiscale Assimilation of Sentinel and Landsat Data for Soil Moisture and Leaf Area Index Predictions Using an Ensemble-Kalman-Filter-Based Assimilation Approach in a Heterogeneous Ecosystem. *Remote Sens.* **2022**, *14*, 3458.
https://doi.org/10.3390/rs14143458

**AMA Style**

Montaldo N, Gaspa A, Corona R.
Multiscale Assimilation of Sentinel and Landsat Data for Soil Moisture and Leaf Area Index Predictions Using an Ensemble-Kalman-Filter-Based Assimilation Approach in a Heterogeneous Ecosystem. *Remote Sensing*. 2022; 14(14):3458.
https://doi.org/10.3390/rs14143458

**Chicago/Turabian Style**

Montaldo, Nicola, Andrea Gaspa, and Roberto Corona.
2022. "Multiscale Assimilation of Sentinel and Landsat Data for Soil Moisture and Leaf Area Index Predictions Using an Ensemble-Kalman-Filter-Based Assimilation Approach in a Heterogeneous Ecosystem" *Remote Sensing* 14, no. 14: 3458.
https://doi.org/10.3390/rs14143458