# Assimilation of MODIS Snow Cover Area Data in a Distributed Hydrological Model Using the Particle Filter

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{2}). Section 2 provides a brief review of the particle filter, while Section 3 describes the MODIS data, the methods used to deal with cloud cover, and the SDC used to convert the model SWE into SCA. In Section 4, the LISFLOOD model is discussed and the study area and the different sources of errors are described. The results of the experiments on the upstream Kromericz basin (8, 000 km

^{2}, located within the Morava River basin), and the results for the complete Morava basin are respectively presented in Sections 5 and 6. In Section 7 the results of all the different experiments are discussed. Finally, the conclusions are presented in Section 8.

## 2. Data Assimilation

**x**(t) be a vector containing the model prognostic variables and

**y**(t) a vector containing the available observations, at time t. The vector

**y**(t) is defined as:

^{o}(t) is the error on observations at time t, defining the observation error covariance matrix

**R**(t) (

**R**(t) = E(∊

^{o}(t)∊

^{oT}(t))),

^{T}is the transpose of a matrix, and

**H**is the measurement operator transforming the model state to the observation space. The

**R**matrix has been defined in this study as a percentage of the squared observations. Several percentages values of the observation error have been used to study its impact on the efficiency of the particle filter: 5, 10, 15, 20, 30 and 40%. These values were squared to obtain the coefficient applied to

**R**. Applied to our work, it becomes clear that

**y**(t) represents the SCA observations at time t, and

**x**(t) represents the SWE simulated by the LISFLOOD model at time t. These two variables do not belong to the same state space, so the measurement operator

**H**has to be defined. This is done in Section 3.3 with the definition of the SDC.

**x**(0 : t − 1)|

**y**(1 : t − 1)) is the prior, p(

**y**(t)|

**x**(t)) is the likelihood, p(

**x**(t)|

**x**(t − 1)) is the transition probability, and p(

**y**(t)|

**y**(1 : t − 1)) is the normalization factor. According to [37], the particles drawn from the posterior distribution at time t are used to map the integrals to discrete sums by the following empirical approximation:

^{(n)}(t) is the normalized weight of the particle n at time t and δ() is the Dirac delta function. To update the particle weights during the assimilation procedure, importance sampling is performed using a proposal distribution. In order to avoid that the entire historical trajectory of the particle needs to be stored, we apply a commonly-used simplification as outlined in [37], where the proposal distribution is modified such that q(

**x**(t)|

**x**(0 : t − 1),

**y**(1 : t)) = q(

**x**(t)|

**x**(t − 1),

**y**(t)). Thus, Equation (3) can be simplified as:

^{*(n)}(t)) with the sum of the importance weights of all the particles:

## 3. Satellite Snow Cover Area

#### 3.1. The Data

#### 3.2. Preprocessing of SCA Data

#### 3.3. Conversion from SWE to SCA

**H**in Equation (1).

_{SCA=1}defines the minimum SWE required for full snow cover for a specific land use and τ is a shape parameter relating the total amount of snow to the snow cover fraction in a pixel. In this study, we follow [6] and assign SWE

_{SCA=1}values ranging from 13mm for bare soil, 20mm for sparse forest and 40mm for full forest coverage. τ was set to 4, as in [6]. Since the SWE values are directly given by the output of LISFLOOD, all parameters in Equation (7) are known and the corresponding SCA value can be easily calculated. Clearly, more complex approaches to derive SWE from SCA could be employed, depending on the availability of additional input data. However, a detailed assessment of the advantages and disadvantages of the different approaches is beyond the scope of this manuscript.

## 4. Hydrological Model and Setting Up of the Assimilation System

#### 4.1. The LISFLOOD Model

^{−1}· day

^{−1}), R is the daily rainfall (mm) and T is the daily temperature (°C). As LISFLOOD uses large grid cells (5 × 5 km), the sub-grid cell heterogeneity in snow accumulation and snow-melt is taken into account by modeling these processes for three separate elevation zones. For this, a normal distribution was used to split the average grid cell height into three equal parts. Furthermore, a sinusoidal function was used inside the model in order to modify the snow-melt coefficient according to the season [60]. This assumption relies on the fact that because of the snow albedo and the solar radiation modifications throughout the year, the snow-melt rate is more important during the summer. Finally, an additional snow-melt process was added for periods from 15 June to 15 September in order to mimic glaciers melt (see LISFLOOD User Manual, [61]). The hydrological simulations are computed at a daily time-step.

#### 4.2. Case Study

^{2}with an elevation ranging from 120 to almost 1,500 m. Daily discharge data were available for our study area at seven gauge stations in the Morava River basin. A smaller part of this basin (8, 000 km

^{2}), upstream of the Kromericz gauging station, has also been used for preliminary experiments.

**x**and

**y**vectors in Equation (1) are 1-dimensional. The basin upstream of the Kromericz gauging station has an area of 8, 000 km

^{2}only, which allows assuming that the snow conditions are more homogeneous than in the complete Morava River basin (26, 000 km

^{2}). This size can be considered as very large, but at the scale of Europe, and for the European Flood Awareness System especially, it represents a rather small river basin. The aim of the experiments realized in this area was to consider a simple case where multi-dimensionality would not be a problem in the data assimilation algorithm. Moreover, this particular river basin is almost not influenced by human interaction.

#### 4.3. Sources of Errors

^{−1}· day

^{−1}). During summer, there is no data assimilation performed, the perturbed meteorological fields have been replaced by the observed fields, and particles have not been resampled (in the case of the particle filter).

#### 4.4. Definition of the Scores Used

**z**(t), that contains for each day of the period, even for the experiments running at a lower frequency than the daily one, the average of all the particles of the variable of interest (i.e., SCA or discharge) over the particles. The observation is given by

**o**(t) and N represents the number of days of the vectors. The Ratio-RMSE is the classical RMSE divided by the average of the observation:

**ō**is defined as:

## 5. Assimilation of the Area Upstream of the Kromericz Station

#### 5.1. Synthetic Experiments

#### 5.2. Real Experiments

## 6. Assimilation on the Whole Morava Basin

^{2}), compared with the much smaller Kromericz basin (8, 000 km

^{2}). In such a case, it is very difficult to make the model SCA to correspond with MODIS SCA observations on a large area. This justifies the division of the Morava basin in seven more homogeneous zones in the following. The experiments described in this Section 6 are summed up in Table 3 and linked to every subsection.

#### 6.1. Synthetic Experiments

#### 6.2. Real Experiments

## 7. Discussions

## 8. Conclusions

^{2}), assimilation results were positive for discharges improvement. However, dealing with spatial heterogeneities seems more difficult, which makes the algorithm poorly efficient (even if one could consider that as already being interesting) or even counter-productive when applied on a larger area (the entire Morava River, 26, 000 km

^{2}).

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The differents steps for merging MODIS maps. The Snow Cover Area (i.e., SCA) is shown here, from 0 for no snow to 1 for pixels entirely covered by snow. Details are given in the text in Section 3.

**Figure 3.**Map of the 7 different zones used for the assimilation experiments on the Morava basin (the black part was not taken into account for the particle filter). The data assimilation experiments on the area upstream of the Kromericz station only include the zones 1, 2, 3 and 4.

**Figure 4.**SCA ratio-RMSE (

**left**) and Nash (

**right**) for the synthetic SCA particle filter assimilation experiments on the area upstream of the Kromericz station with 50 particles. Impact of the magnitude of the

**R**matrix and of the frequency of assimilation is shown.

**Figure 5.**SCA ratio-RMSE (

**left**) and Nash (

**right**) for the real MODIS SCA particle filter experiments on the area upstream of the Kromericz station with 50 particles. The SCA proxy (not plotted) has a ratio-RMSE of 0.51 compared with the composite MODIS and 0.50 compared with the daily MODIS; the proxy discharges have a Nash of 0.85.

**Figure 6.**SCA ratio-RMSE (

**left**) and Nash (

**right**) for the real MODIS SCA particle filter experiments on the area upstream of the Kromericz station with 200 particles. The SCA proxy (not plotted) has a ratio-RMSE of 0.51 compared with the composite MODIS and 0.50 compared with the daily MODIS; the proxy discharges have a Nash of 0.85.

**Figure 7.**SCA ratio-RMSE (

**left**) and Nash (

**right**) for the synthetic SCA particle filter experiments on the whole Morava basin with 50 particles.

**Figure 8.**SCA ratio-RMSE (

**left**) and Nash (

**right**) for the synthetic SCA particle filter experiments on the whole Morava basin with 200 particles.

**Figure 9.**SCA ratio-RMSE (

**left**) and Nash (

**right**) for the real MODIS SCA particle filter experiments on the whole Morava basin with 50 particles. The SCA proxy (not plotted) has a ratio-RMSE of 0.52 compared with the composite MODIS and 0.55 compared with the daily MODIS; the proxy discharges have a Nash of 0.76.

**Figure 10.**SCA ratio-RMSE and Nash for the real MODIS SCA particle filter experiments on the area upstream of the Kromericz station. The frequency of assimilation (freq) and the number of particles (nb part) are indicated for every panel on the y-axis. The boxplots are drawn for 10 similar-condition experiments. The corresponding no-assimilation experiment is represented with a line.

**Figure 11.**SCA ratio-RMSE and Nash for the real MODIS SCA particle filter experiments on the whole Morava basin. The frequency of assimilation (freq) and the number of particles (nb part) are indicated for every panel on the y-axis. The boxplots are drawn for 10 similar-condition experiments. The corresponding no-assimilation experiment is represented with a line.

**Table 1.**Snow categories in the MODIS classification (

**left part**) and as used for the preprocessing (

**right part**).

MODIS Classification | Used Classification | ||
---|---|---|---|

Value | Data | Value | Data |

0 | Missing | ||

1 | No decision | ||

11 | Night | Missing value | “we don’t know” |

50 | Cloud obscured | ||

254 | Detector saturated | ||

255 | Fill | ||

25 | Snow-free land | 0 | No snow |

37 | Lake or inland water | ||

39 | Open water (ocean) | ||

100 | Snow-covered lake ice | 1 | Snow |

200 | Snow |

**Table 2.**Percentage of cloud coverage, averaged over the period 10 January 2004–10 December 2006, for the different improvement methods.

Aqua+Terra | +Day-1 | +Days-2/-3 | +Days-4/-5 | +Days-6/-7 |
---|---|---|---|---|

48.6 | 33.2 | 16.8 | 9.0 | 4.9 |

Area | Method | Observations | Obs. Error Range | Frequency | |
---|---|---|---|---|---|

Section 5.1 | Upstr. Kromericz | Synthetic | |||

Section 5.2 | Particle filter | MODIS | From 5% to 40% | 1,2,3,7 days | |

Section 6.1 | Morava basin | Synthetic | |||

Section 6.2 | MODIS |

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## Share and Cite

**MDPI and ACS Style**

Thirel, G.; Salamon, P.; Burek, P.; Kalas, M.
Assimilation of MODIS Snow Cover Area Data in a Distributed Hydrological Model Using the Particle Filter. *Remote Sens.* **2013**, *5*, 5825-5850.
https://doi.org/10.3390/rs5115825

**AMA Style**

Thirel G, Salamon P, Burek P, Kalas M.
Assimilation of MODIS Snow Cover Area Data in a Distributed Hydrological Model Using the Particle Filter. *Remote Sensing*. 2013; 5(11):5825-5850.
https://doi.org/10.3390/rs5115825

**Chicago/Turabian Style**

Thirel, Guillaume, Peter Salamon, Peter Burek, and Milan Kalas.
2013. "Assimilation of MODIS Snow Cover Area Data in a Distributed Hydrological Model Using the Particle Filter" *Remote Sensing* 5, no. 11: 5825-5850.
https://doi.org/10.3390/rs5115825