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Accurate, reliable and skillful forecasting of key environmental variables such as soil moisture and snow are of paramount importance due to their strong influence on many water resources applications including flood control, agricultural production and effective water resources management which collectively control the behavior of the climate system. Soil moisture is a key state variable in land surface–atmosphere interactions affecting surface energy fluxes, runoff and the radiation balance. Snow processes also have a large influence on land-atmosphere energy exchanges due to snow high albedo, low thermal conductivity and considerable spatial and temporal variability resulting in the dramatic change on surface and ground temperature. Measurement of these two variables is possible through variety of methods using ground-based and remote sensing procedures. Remote sensing, however, holds great promise for soil moisture and snow measurements which have considerable spatial and temporal variability. Merging these measurements with hydrologic model outputs in a systematic and effective way results in an improvement of land surface model prediction. Data Assimilation provides a mechanism to combine these two sources of estimation. Much success has been attained in recent years in using data from passive microwave sensors and assimilating them into the models. This paper provides an overview of the remote sensing measurement techniques for soil moisture and snow data and describes the advances in data assimilation techniques through the ensemble filtering, mainly Ensemble Kalman filter (EnKF) and Particle filter (PF), for improving the model prediction and reducing the uncertainties involved in prediction process. It is believed that PF provides a complete representation of the probability distribution of state variables of interests (according to sequential Bayes law) and could be a strong alternative to EnKF which is subject to some limitations including the linear updating rule and assumption of jointly normal distribution of errors in state variables and observation.

Hydrologic modeling has greatly benefited from observation of land surface water, energy, and carbon conditions which are of critical importance owing to their profound impacts on real world water resources applications such as flood control, weather and climate prediction, agricultural production and water resources management which collectively control the behavior of the climate system.

Many studies have demonstrated that initial and boundary conditions of state variables such as soil moisture, soil temperature or vegetation water content at different temporal and spatial scales exercise strong controls on climate, weather and hydrologic processes. [

Remote sensing has shown great promise for providing an abundance of data and information that were lacking with the in-situ observations. It has also been a valuable tool in many hydrologic modeling applications due to its capability of providing unrestricted collection of information with wide spatial coverage and temporal repeat [

Soil moisture plays a key role in the terrestrial water cycle and is responsible for the partitioning of precipitation between surface water (runoff) and storage through infiltration. Surface and root zone soil moisture control the redistribution of incoming radiation (available energy) on the land surface into sensible and latent heat (evaporative) fluxes. Understanding soil moisture is pivotal in various fields such as agriculture, ecology, hydrology and even geotechnical engineering. Furthermore, root zone soil moisture carries memory from weekly to monthly timescales; therefore its accurate initialization may contribute to enhanced prediction of summer precipitation [

Snowpack is a major component of seasonal water supply in many middle to high latitude alpine catchments and it contributes a considerable percentage (for example, 70-80 percent in the northwestern US) to the total annual runoff in these regions. Snow has large influence on land-atmosphere energy exchanges due to its high albedo, low thermal conductivity and considerable spatial and temporal variability resulting in the dramatic change of surface and ground temperature. Accurate estimation of the amount and timing of snowmelt coupled with proper monitoring of snow properties including snow extent and snow water equivalent (SWE) are vital to estimating a more accurate water supply forecast as required for water resources management. Analyses of snowpack observations collected over the past decade indicate that packs are melting earlier in the year and SWE data collected from snow course sites display negative trends over the period of record from 1950-2000. Therefore, understanding the evolution of snow energy and mass balance processes is imperative for a complete description of the hydrological cycle at basin and regional scales to accurately characterize and estimate snow properties in the snow dominated regions for flood and avalanche warnings, environmental compliance, and water geochemistry for use by water resources managers and planers.

Soil moisture information may be obtained in two ways: 1) it may be derived by running a land surface model through which the meteorological forcing observation is propagated; 2) it may be retrieved from in-situ measurement or from low-frequency passive and active microwave data. It has long been recognized that reliable, robust and automated methods for the measurement of soil moisture content could be extremely useful, if not essential, in hydrologic, environmental and agricultural applications. Despite the availability of various methods in retrieving soil moisture at a single location there are currently no networks of in-situ sensors that provide regional or global data sets. Considering that such networks are expensive and impracticable, attention has gone to remote sensing data, which are able to provide large-scale information suitable for regional and global applications. Platforms for supporting remote sensing instruments have varied from ground-based supports to aircraft and satellites. Ground-based systems can be mounted on trucks or on special structures such as rails to allow for movement of the sensor. The advantage of these ground-based systems is the relatively small footprint of the sensor providing easy control during the measurement period. The main disadvantage is the small coverage of large areas. The aircraft mounted systems can overcome some of these limitations while mapping the larger area and can serve as prototypes for future satellite sensors. However, satellite remote sensing offers the optimal solution owing to their capability of monitoring large areas with long term repetitive coverage.

Satellite observations alone are not sufficient because of the temporal and spatial gaps in their coverage. Also the deeper soil moistures cannot be observed directly from space. Therefore, the best possible system would integrate the benefits of land surface models, in-situ and satellite observations to assess global soil moisture conditions. This can be done through Data Assimilation (DA) as a means of merging observation with model output to improve upon the accuracy of the estimation. This will be explained in detail in section 4.

Some of the most commonly used remote sensing instruments for soil moisture observation are the Multi-Spectral Scanner (MSS), Thematic Mapper (TM), thermal infra-red line scanner, Synthetic Aperture Radar (SAR), and microwave radiometer. Although numerous remote sensing systems are in existence and have been utilized for soil moisture measurement, the most appropriate is microwave remote sensing. Microwave remote sensing provides a means of direct measurement of soil moisture for a range of vegetation cover conditions. Such remote measurement provides the opportunity of observing frequent, global sampling of soil moisture with large spatial resolution. The main advantage of microwave measurements is that they are not affected by cloud cover and variable solar illumination; however, the accuracy in soil moisture estimation is limited to regions with either bare soil or low to moderate amounts of vegetation [

The two approaches used in microwave soil moisture measurement are active and passive [

The Advanced Microwave Scanning Radiometer (AMSR-E) of the Earth Observing System (EOS) is currently used for the global soil moisture mapping [

During calibration and validation field campaigns of the Soil Moisture EXperiments in 2002-2004 (SMEX02, SMEX03, and SMEX04) [

Fractional snow cover area (SCA) observations can typically be obtained from visible or infrared satellite sensors providing high spatial resolution observations [

The scanning Multichannel Microwave radiometer (SMMR), a 5 frequency radiometer providing observations from October 1978 to August 1987;

The Special Sensor Microwave Imager (SSM/I), providing observations from September 1987 until present; and

The Advanced Microwave Scanning Radiometer for the Earth Observing system (AMSR-E), providing observation from May 2002 until present.

The AMSR-E operational snow mapping algorithm employs an empirical relationship to estimate SWE from surface brightness temperature while providing SWE estimate at a spatial resolution of 25 km. The microwave sensors measure snow mass under cloudy and nighttime conditions, however, dense vegetation cover and water bodies cause large retrieval errors [

Snow Cover Area (SCA) among many other land surface features is available since 1999 as a 500-meter daily gridded product from the Moderate Resolution Imaging Spectroradiometer (MODIS) sensor flown on board the Terra Earth Observing System (EOS) platform [

An explosion of activities has been witnessed over the past two decades on the development and application of data assimilation systems. Data assimilation is a way to integrate the data from variety of sources with different resolutions and accuracies with model prediction to improve deterministic model accuracy [

One of the data assimilation techniques that have been used in hydrologic application is the variational method [see

Among many other data assimilation techniques, the sequential assimilation algorithms using filtering have garnered the attention of hydrologists due to flexibility in handling all sources of uncertainties and as well as the possibility of ingesting the data sequentially as it becomes available. One of the early hydrologic data assimilation methods is the application of linear Kalman filter [e.g., 34, 35]. In the case of nonlinear, hydrological model the data is rendered in state-space form and by assuming that the model state variables are differentiable the Extended Kalman filter (EKF) can be used. By using this method the model error at the time of observation can be estimated by propagating the covariance matrix of model errors. As reported by Evensen [

Another approach to data assimilation is Monte Carlo (ensemble) methods. These methods have received considerable attention by hydrologists in recent years as they are easy to implement and the computational burden is less of an issue with the increased computing power nowadays. To cope with the drawbacks of the EKF, a Monte Carlo-based Kalman filter called ensemble Kalman filter (EnKF) was introduced by Evensen [

The EnKF uses a Monte Carlo approach to approximate the conditional second-order moments of variables of interest using a finite number of randomly generated model replicates. However, the major limitation of all the filtering techniques rooted in Kalman filtering is their closure at the second-order moments implying that the filter evolution is characterized by their model state covariance. Also, the EnKF is limited to the linear updating rule with considerable simplification while using a highly nonlinear hydrologic model. This has encouraged the hydrologists to look into other filtering techniques, such as Particle filter (PF) [

The mathematical framework of estimation theory provides the tools required to approach variety of data assimilation problems. The basic objective of data assimilation is to characterize the state of an environmental system at some future time based on the knowledge of the initial system state. Bayesian inference provides a mechanism to combine the quantitative (hydrologic data) and qualitative data (prior information obtained by the experience of experts in the field) to yield the posteriori as more informative probability distribution of variable of interest. Bayesian formulation allows hydrologists to estimate the uncertainty about model prediction in a systematic way and can be accomplished without resort to calibration which is sometimes problematic in certain applications.

In a Bayesian formulation, the solution to an inverse problem is given by posterior probability distribution

where the denominator,

With the assumption that the model residuals are mutually independent (normally distributed) with constant variance (i.i.d.) the likelihood function can be computed using [

In the absence of an explicit mathematical expression for

The original Bayes law explained above (

Let's consider the state variable _{t}_{t}_{t}_{1}_{2}, … _{t}

As seen in the schematic of recursive Bayes law (see _{t}_{t-1})_{t}

The main complication in using the recursive Bayes law remains in the multidimensional integration of forecast density as shown in

In sequential filtering, the uncertain state of a hydrological system _{t,}_{1:t}_{t}_{1:t})

where _{t,}_{t}_{t}_{t}_{t}_{t}_{t}

the forecasting step which is the transition of state variables from one observation time to the next represented through transition probability _{t}_{t-1})

the analysis (updating) step which involves updating of the forecasted (propagated) states with the new observation.

Ensemble procedures present a practical alternative to an exact Bayesian solution by relying on discrete estimation of forecast (priori) and analysis (posteriori) densities through a set of random variables and corresponding weights:

These are the empirical approximations of forecast and analysis (update) densities by summation of ^{i}^{i}^{th}

The forecasting step in the EnKF where the evolution of the model for each ensemble member is equally weighted,

Thereby, the forecast density in (8) will become:

It is noted that in this process, the generation of random input samples of
_{t}_{t-1}) from the uncertainty in
_{t}

If the dynamical system, including states and measurement equations, are linear and all sources of uncertainty are normally distributed the celebrated Kalman filter provides the optimal recursive solution to the state updating problem. If the system is nonlinear, as is the case for most of the hydrologic systems, the linearization of the system might be considered. Developed from the early work using state-space filtering, Georgakakos et al. [

Let denote ^{−} as the ensemble of forecasted model state (_{1}_{2}_{m}

If the priori error in the forecasted ensemble members is shown by

Then the model error covariance is calculated directly from the ensemble as follows:

Knowing the priori error covariance of model states from (14), the state updates can be obtained by (15):

where

In the EnKF implementation, the observation _{t}

Similar to Ensemble Kalman filter, the sequential Bayesian algorithm can be used to derive the particle filter. Various names are associated with the particle filters such as bootstrap filter, the condensation algorithm, sequential Monte Carlo sampling, interacting particle approximations, and survival of the fittest [

Where
_{t}_{1:}_{t}_{−1}) and we defined earlier that

After the resampling step, the posterior distribution will be presented as:

The new resampled replicates are taken and forwarded in time and procedure through

In general, OSSE is designed to enable the modeler to examine the performance of data assimilation procedures and even to obtain the sensitivity of the procedure to different models with different parameterizations and physical representations. OSSE typically consists of at least four components: (1) A simulated data sets of land model states, (2) a forward model to link the states with observations, (3) a model to degrade these observations accuracy and spatial resolution, (4) integrating the degraded observation into a prediction model. A schematic of OSSE is shown in

For such synthetic experiments, “truth” is defined when the model is integrated (run) for a set of meteorological, soil, vegetation and initial conditions. Considering that the observation from microwave remotely sensed data for soil moisture are in the form of brightness temperature, the model output (soil moisture) should be converted to brightness temperature to make them ready to be ingested in the updating step of the filtering process. Therefore, the modeled soil moisture is taken to a radiative transfer model (RTM) [

This paper provides a review of the most commonly used remotely sensed land surface measurements, mainly microwave remote sensing products for both soil moisture and snow (SWE and SCA) to be used in a data assimilation framework to improve upon the land surface model prediction. Passive microwave remote sensing provides a means of direct measurement of soil moisture for a range of vegetation cover conditions. Such remote measurement provides the opportunity of observing frequent, global sampling of soil moisture with large spatial resolution.

Fractional snow cover area (SCA) observations can typically be obtained from visible or infrared satellite sensors providing high spatial resolution observations [

The concept of data assimilation was discussed and two of the advanced techniques, mainly EnKF and PF were explained in detail. Considering that DA techniques are used mainly for reducing the uncertainty, there is still a lack of consensus in hydrologic community on the selection and implementation of a suitable land DA method to meet this need. It needs to be realized that contemporary DA methods are used for estimating the state variables (here, soil moisture, SWE or SCA), and uncertainties associated with them, however, not all sources of uncertainties are addressed in the assimilation process. These include uncertainty in model parameters and also model structure which are ignored in DA implementations. Although in this paper we did not intend to provide a comprehensive review of all the data assimilation methods, we focused on two of emerging techniques as reported by few studies in section 4. It was mentioned that the ensemble filtering using PF results to full representation of prognostic variable and even parameter probability distributions. The EnKF is limited to the linear updating rule as in the original Kalman filter and also assumption of Gaussian distribution of errors in observation and model. Considering that the soil moisture and snow water equivalent probability distribution significantly change over time and are often non-normal, the existing assumptions in EnKF limit its application in strongly nonlinear hydrologic models. Knowing the potentials of PFs, further implementation of PF as an alternative procedure for operational data assimilation is suggested. The synthetic study through OSSE design as seen in section 4.1.3 is an appropriate procedure to judge about the merits of a certain technique for land data assimilation and the method can be used to adequately quantify and minimize the hydrologic predictive uncertainty while including model parameter uncertainty in the whole scheme.

Author would like to thank anonymous reviewers for their constructive comments resulting in an improvement of the paper. The partial financial support for this study was provided by NOAA grant NA070AR4310203.

Sequential Bayesian scheme for evolution of the conditional probability density of the state variables by assimilating observations from time

Schematic of Observing System Simulation Experiment (OSSE).