# Hydrologic Remote Sensing and Land Surface Data Assimilation

## Abstract

**:**

## 1. Introduction

## 2. Soil Moisture Observation

## 3. Snow Observations

- 1)
- The scanning Multichannel Microwave radiometer (SMMR), a 5 frequency radiometer providing observations from October 1978 to August 1987;
- 2)
- The Special Sensor Microwave Imager (SSM/I), providing observations from September 1987 until present; and
- 3)
- The Advanced Microwave Scanning Radiometer for the Earth Observing system (AMSR-E), providing observation from May 2002 until present.

## 4. Hydrologic Data Assimilation

#### 4.1. Sequential Bayesian Data Assimilation using Ensemble Filtering

_{t}as the quantity of interest to be estimated within the Bayesian framework. Due to stochastic nature of x

_{t}, the pertinent information about it at any time t can be extracted from the observation Y

_{t}= [y

_{1}, y

_{2}, … y

_{t}] through the recursive Bayes law:

_{t}/Y

_{t-1}) can be estimated via Chapman-Kolmogorov equation [33] assuming that x

_{t}follows the Markov property, therefore:

#### 4.1.1. Ensemble Kalman Filter

_{t,}, given a set of observation y

_{1:t}is presented by the conditional probability density function p(x

_{t}/y

_{1:t}). Ensemble methods can be used to calculate the sample approximation to this density function by generating the random replicates of model state variables. Following Jazwinski [33] the generic nonlinear dynamic system in earth sciences are written in discrete-time for both state and measurement equations as follows:

_{t,}is an n-dimensional vector of true but uncertain state variables, u

_{t}is a vector of uncertain true of model inputs, θ is vector of model parameters and w

_{t}represents the uncertainties due to errors in model formulation, y

_{t}is the measurement vector and v

_{t}is a vector of additive random measurement errors. The model and measurement errors are typically assumed to be Gaussian and independent random vectors with mean zero and covariances Q

_{t}and R

_{t}respectively. Two sequential estimation operations are discerned in filtering applications:

- 1)
- the forecasting step which is the transition of state variables from one observation time to the next represented through transition probability p(x
_{t}/x_{t-1}) in eq. (5), - 2)
- the analysis (updating) step which involves updating of the forecasted (propagated) states with the new observation.

^{i}and w

^{i}denote the i

^{th}sample and its weight before and after updating shown by minus and plus signs respectively. The random replicates and associated weights are generated through a variety of methods, one of which is the ensemble Kalman filter (EnKF).

_{t}|Y

_{t-1}) from the uncertainty in ${u}_{t}^{i}$ using p(u

_{t}) and in some applications by the uncertainty inherent in the parameters of the model through p(θ). For more details on the inclusion of parameter uncertainty in the filtering, see Moradkhani et al., [43, 44].

^{−}as the ensemble of forecasted model state (x

_{1}, x

_{2}, …, x

_{m}) at each time t, for each of the state variables having n-ensemble members, that is

_{t}at each time should be perturbed, usually using normal distribution with zero mean and variance

**R**. This creates an ensemble of perturbed observation which are used in eq. (15) to update the model ensemble members.

#### 4.1.2. Particle Filter

_{t}| y

_{1:}

_{t}

_{−1}) and we defined earlier that ${w}_{t}^{i-}=1/n$. By substituting eqs. (17) and (18) into eq. (9), we can obtain the update (posterior) probability distribution. In the case of Gaussian likelihood, the problem of degeneracy of particles (ensemble collapse to a single point) may be experienced as those particles that are closer to the measurement get higher weights while others are discarded. One solution is to use many particles which, in the case of a distributed model, may not be a cost effective solution. The second method is to implement the resampling technique to prevent the samples from degeneracy. Some of the sampling techniques used in particle filtering are the Sequential Importance sampling (SIS), Sequential Importance Resampling or Sampling Importance Resampling (SIR) and regularized sampling [2] as the most commonly used sampling procedures. Employing the proper sampling technique keeps the particles from dispersion due to stochastic behavior of the system or degeneracy. For detailed information on SIR-particle filter and sampling see Moradkhani et al. [44]. Through the SIR filter, the resampling is made with replacement n times. In fact, the probability of a selection of any sample of j is equal to ${w}_{t}^{j+}$. When resampling is over, the new ensemble of equally weighted particles of ${x}_{t}^{j+}$ with weights ${w}_{t}^{j+}=1/n$ is created. In this process the replicates with higher weights (probabilities) have a higher chance to be selected and the low weight replicates are more likely to be discarded.

#### 4.1.3. DA Experiment Setup through Observing System Simulation Experiment (OSSE)

## 5. Summary

## Acknowledgments

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**Figure 1.**Sequential Bayesian scheme for evolution of the conditional probability density of the state variables by assimilating observations from time t-1 to time t.

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**MDPI and ACS Style**

Moradkhani, H.
Hydrologic Remote Sensing and Land Surface Data Assimilation. *Sensors* **2008**, *8*, 2986-3004.
https://doi.org/10.3390/s8052986

**AMA Style**

Moradkhani H.
Hydrologic Remote Sensing and Land Surface Data Assimilation. *Sensors*. 2008; 8(5):2986-3004.
https://doi.org/10.3390/s8052986

**Chicago/Turabian Style**

Moradkhani, Hamid.
2008. "Hydrologic Remote Sensing and Land Surface Data Assimilation" *Sensors* 8, no. 5: 2986-3004.
https://doi.org/10.3390/s8052986