## 1 Introduction

The magnitude of available energy at the land surface and, particularly, its partitioning between sensible and latent heat fluxes directly affects land-atmosphere interactions and boundary layer formation [*Bateni and Entekhabi*, 2012]. Accurate estimates of surface turbulent heat fluxes are important in many fields such as meteorology, hydrology, and climate change studies [*Caparrini et al*., 2004a; *Rigden and Salvucci*, 2015].

In situ measurements of surface heat fluxes are difficult and expensive, and they are only available from a handful of sparse flux tower networks (e.g., FluxNet, AmeriFlux, and AsiaFlux) [*Baldocchi et al*., 2001]. Mapping regional heat fluxes from point measurements is hampered by strong spatial heterogeneity [*Semmens et al*., 2015], which is influenced by factors such as vegetation cover, soil moisture, and local landscape [*Caparrini et al*., 2004a]. Remote sensing techniques provide measurements of the land surface at various spatial and temporal scales and are a promising data source for surface heat flux estimation. Unfortunately, surface heat fluxes do not have a unique signature that can be detected directly by remote sensing instruments [*Sini et al*., 2008].

In previous studies, surface heat fluxes have mainly been estimated using four groups of methods. In the first group, empirical relationships are built between heat fluxes and local predictors such as land surface temperature (LST) and vegetation indices (VI) [*Gillies et al*., 1997; *Nagler et al*., 2005a, 2005b; *Tang et al*., 2010]. This group of methods is also referred to as the “triangle method” [*Carlson*, 2007]. In the second group, surface heat fluxes are estimated by solving the surface energy balance (SEB) [*Kustas et al*., 1996; *Anderson et al*., 1997; *Bastiaanssen et al*., 1998a, 1998b; *Jiang and Islam*, 2001; *Su*, 2002; *Timmermans et al*., 2007; *Allen et al*., 2007; *Anderson et al*., 2011]. The SEB models often utilize instantaneous LST observations to solve the instantaneous energy balance [*Kalma et al*., 2008]. Ancillary data such as surface roughness and leaf area index (LAI) are needed, and a closure assumption is often imposed [*Sini et al*., 2008]. A major assumption is that ground heat flux is a fraction of net radiation.

In a departure from the diagnostic methods, the third group estimates surface heat fluxes by assimilating data into land surface models (LSM) [*Oleson et al*., 2010; *Niu et al*., 2011] to update land surface states and parameters [*Yang et al*., 2007; *Li et al*., 2012; *Sawada and Koike*, 2014; *Han et al*., 2014]. Data assimilation is the technique of combining complementary information from model simulation and observations into an optimal estimate of the geophysical field of interest [*Reichle*, 2008]. The most commonly used methods are variational methods, ensemble methods, and particle methods. Variational data assimilation (VDA) methods merge model simulation with observations by constructing and minimizing a cost function derived from the forward model within a time window [*Alavi et al*., 2009]. A VDA method can be one-dimensional (1-D-Var), three-dimensional (3-D-Var), or four-dimensional (4-D-Var) depending on the spatial and temporal dimensions of the states. Ensemble methods are based on the Monte Carlo theory and estimate state variables by propagating and updating an ensemble of model states. They can be subdivided into sequential filtering (e.g., ensemble Kalman filter (EnKF)) [*Reichle et al*., 2010, 2013] and smoothing methods (e.g., ensemble Kalman smoother (EnKS)) [*Dunne and Entekhabi*, 2005, 2006; *Dunne et al*., 2007; *Bateni and Entekhabi*, 2012]. Particle methods have their origin in Bayesian estimation, and model states are estimated as the weighted sum of all particle estimates, where the particle weights are derived from a likelihood function [*Moradkhani et al*., 2012; *Yan et al*., 2015].

In the fourth group, LST time series are assimilated into heat transfer models to estimate two key parameters which characterize surface heat fluxes: neutral bulk heat transfer coefficient (*C _{HN}*) and evaporative fraction (EF).

*C*scales the sum of available energy, while EF governs the partitioning between sensible and latent heat fluxes. Sensible heat flux is calculated using

_{HN}*C*and meteorological data, and latent heat flux is estimated using sensible heat flux and EF. In consequence of the simple model structure, this group of methods greatly reduces the data demand compared to using LSMs. This group of methods has been applied using variational [

_{HN}*Castelli et al*., 1999;

*Boni et al*., 2001;

*Caparrini et al*., 2003, 2004a, 2004b;

*Sini et al*., 2008;

*Bateni and Liang*, 2012;

*Bateni et al*., 2013a, 2013b;

*Xu et al*., 2014, 2015] or ensemble smoothing [

*Bateni and Entekhabi*, 2012] methods. This study belongs to the fourth group, and a detailed description of the method can be found in section 2.4.

The PBS has recently been introduced separately by *Margulis et al*. [2015] to estimate snow water equivalent and by *Dong et al*. [2015] to estimate soil moisture from distributed temperature sensing. It can be seen as an extension of the particle filter (PF) [*Dong et al*., 2015] as both algorithms use similar marginal distribution to derive the particle weights in the updating process. The difference is that states and parameters within a window are updated in a batch using all available observations in that window in the PBS, while the PF assimilates observations sequentially. As the dimension of states of the PBS is usually higher than that of the PF, the PBS requires a large amount of particles to avoid particle degeneracy. Despite the computational demand, the PBS is unique in many aspects.

- Compared to the variational methods [
*Caparrini et al*., 2003, 2004a, 2004b;*Sini et al*., 2008;*Bateni and Liang*, 2012;*Bateni et al*., 2013a, 2013b], the PBS requires no computation of model adjoint or background error covariance; hence, it is much easier to implement. - Compared to the ensemble (e.g., EnKF and EnKS) methods [
*Bateni and Entekhabi*, 2012], the PBS makes no assumptions about the prior distributions, which is theoretically more accurate for hydrological applications in which the prior distributions are often non-Gaussian (a demonstration of the prior distribution of model states from this study is provided in Supporting Information) and the performance of ensemble methods are often suboptimal [*Moradkhani et al*., 2005;*Dong et al*., 2015;*Yan and Moradkhani*, 2016]. It is also better suited to parameter estimation [*Dong et al*., 2016a], as the PBS tracks the entire prior distribution of parameters using Monte Carlo sampling, which performs more robustly when the Gaussian error assumption is violated [*DeChant and Moradkhani*, 2012]. - Compared to the PF, the PBS utilizes information contained not only in every single observation but also in the temporal evolution of observations, as all available observations in the window are assimilated in a batch. This makes the PBS preferable in estimating surface heat fluxes from LST time series.

In this study, the PBS is used to estimate surface heat fluxes by assimilating LST observations into a heat transfer model through a joint state-parameter assimilation strategy in the first experiment (section 2.4.1). This is the first study that adopts the PBS to estimate surface heat fluxes by assimilating LST data.

Although LST time series contains information about surface energy partitioning, many studies have demonstrated that the assimilation strategy performs poorly on wet or densely vegetated surfaces [*Caparrini et al*., 2004a; *Crow and Kustas*, 2005; *Bateni and Entekhabi*, 2012; *Xu et al*., 2014]. This happens because under these conditions, the surface energy partitioning becomes more energy-limited, which weakens the constraint of LST on surface energy balance [*Caparrini et al*., 2004a]. *Sini et al*. [2008] demonstrated that using soil wetness information to constrain EF could improve flux estimation under these conditions. Soil moisture controls the partitioning of available energy into sensible and latent heat fluxes through its influence on evapotranspiration [*Entekhabi et al*., 1996; *Margulis et al*., 2002; *Koster et al*., 2004; *Entekhabi et al*., 2010; *Seneviratne et al*., 2010; *Crow et al*., 2015]. Many studies have demonstrated a positive correlation between EF and soil moisture at different depths [*Kustas et al*., 1993; *Lhomme and Elguero*, 1999; *Dirmeyer et al*., 2000; *Basara and Crawford*, 2002; *Wang et al*., 2006; *Gentine et al*., 2007; *Santanello et al*., 2011]. Here we investigate for the first time in depth the potential value of joint soil moisture and LST assimilation through comparative experiments. In the second experiment (section 2.4.2), a simple soil water transfer scheme is introduced and coupled to the heat transfer model, and soil moisture observations are assimilated simultaneously with LST observations. To provide an additional constraint on EF, a soil wetness-EF relationship is adopted.

A potentially interesting extension of this study is to use LST and soil moisture observations from remote sensing. This would open the path to estimating surface heat fluxes consistently at larger scales. Soil moisture products are available from L-band microwave remote sensing from missions such as the Soil Moisture and Ocean Salinity (SMOS) mission and the Soil Moisture Active Passive (SMAP) mission, which provide global soil moisture observations every 2–3 days [*Kerr et al*., 2001; *Entekhabi et al*., 2010]. However, the estimation robustness may be affected by the number of available LST observations, in addition to the influence of spatial resolution and data accuracy, among others. Potential sources for LST observations include the Advanced Very High Resolution Radiometer (AVHRR), the Moderate Resolution Imaging Spectroradiometer (MODIS), and the Geostationary Operational Environmental Satellites (GOES), among others. Typically, the same area is observed no more than twice each day by polar-orbiting satellites, and the observations may fall outside the nominal assimilation window. For geostationary satellites, cloudy-sky conditions which represent more than half of the day-to-day weather [*Jin*, 2000] can dramatically reduce the number of available observations. A simulation test is conducted to assess the influence of LST data availability on flux estimates.

Based on the discussion above, there are three objectives of this study: (1) to investigate the performance of the PBS on the assimilation of LST observations for surface heat flux estimation; (2) to introduce a soil moisture transfer scheme to constrain EF and jointly assimilate LST and soil moisture observations to improve the poor performance on wet and densely vegetated surfaces; (3) to explore the influence of LST data availability on flux estimation in the potential application with remote sensing data. We aim to answer the following three science questions: (1) What are the effects of assimilating LST observations to estimate surface heat fluxes with the PBS? (2) Can the estimation be improved if soil moisture observations are assimilated simultaneously with LST observations, particularly on wet or densely vegetated surfaces? (3) Given the data availability issue in potential applications with remote sensing data, will the conclusions hold when the number of LST observations is limited?

This paper is organized as follows: Section 2 describes the PBS algorithm, the models used, the experiment design and the data sets. Section 3 provides the assimilation results and the discussion. Finally, the conclusions are drawn in section 4.