1272 MONTHLY WEATHER REVIEW VOLUME 131

Variational Assimilation of Radiometric Surface and Reference-Level Micrometeorology into a Model of the Atmospheric Boundary Layer and Land Surface

STEVEN A. MARGULIS Department of Civil and Environmental Engineering, University of California, Los Angeles, Los Angeles, California

DARA ENTEKHABI Ralph M. Parsons Laboratory, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts

(Manuscript received 4 January 2002, in ®nal form 2 December 2002)

ABSTRACT Data assimilation provides a useful framework that allows us to combine measurements and models, by appropriately weighting the sources of error in both, to produce a statistically optimal and dynamically consistent estimate of the evolving state of the system. In this paper a variational approach is used to estimate regional land and atmospheric boundary layer states and ¯uxes via the assimilation of standard reference-level temperature and humidity and radiometric surface temperature measurements into a coupled land surface±atmospheric bound- ary layer model. Results from an application to a ®eld experiment site show that using both surface temperature and reference-level micrometeorology measurements allows for the accurate and robust estimation of land surface ¯uxes even during nonideal conditions, where the evaporation rate is atmospherically controlled and processes that are not parameterized in the model (i.e., advection) are important. The assimilation scheme is able to provide estimates of model errors, which has implications for being able to diagnose structural model errors that may be present due to missing process representation and/or poor or biased parameterizations. Because robust estimates are not always obtainable with either measurement type in isolation, these results illustrate possible synergism that may exist when using multiple observation types.

1. Introduction and background Many recent studies have outlined methods to esti- mate surface states and ¯uxes from potentially useful Diagnosing the regional surface water and energy data streams. Particularly, the advent of remote sensing budgets are important pursuits in many ®elds (e.g., hy- capabilities has taken hydrology, as well as many other drologic, atmospheric, and ecological science applica- ®elds in the geosciences, from a somewhat ``data poor'' tions). Whether the goal is to use antecedent soil mois- ture to predict possible ¯ooding, improve land surface situation, reliant mostly on sparse in situ measurements, forcing of numerical weather prediction or general cir- to a potentially ``data rich'' environment, by increasing culation models, or estimate necessary irrigation re- the types and numbers of observations that may be re- quirements for an agricultural ®eld, accurate estimates lated to the surface parameters we would like to esti- of the state of the land surface (moisture and temper- mate. For example, due to the strong sensitivity of bulk ature) and the ¯uxes between the surface and atmo- soil microwave dielectric properties to volumetric soil sphere are required. Unfortunately, these variables (most moisture, the feasibility of surface soil moisture esti- notably soil moisture and surface turbulent ¯uxes) are mation from microwave measurements has been dem- dif®cult, if not pragmatically impossible, to directly onstrated (e.g., Schmugge and Jackson 1994; Entekhabi measure over large scales. Therefore we are left with et al. 1994; Njoku and Entekhabi 1996; Jackson et al. the so-called inverse problem of trying to estimate these 1999; Reichle et al. 2001a; Margulis et al. 2002). While states and ¯uxes from environmental observations that there is some information content in higher frequencies are generally only indirectly related to the variable of (e.g., Lakshmi et al. 1997; Vinnikov et al. 1999), the interest. most promising results correspond to L-band (1.4 GHz) measurements, for which there currently is no opera- tional satellite. Corresponding author address: Dr. Steven A. Margulis, Dept. of As a result, other researchers have focused on using Civil and Environmental Engineering, 5732D Boelter Hall, UCLA, Los Angeles, CA 90095. currently available remote sensing measurements, no- E-mail: [email protected] tably radiometric surface temperature, to estimate sur-

᭧ 2003 American Meteorological Society

Unauthenticated | Downloaded 09/28/21 03:18 AM UTC JULY 2003 MARGULIS AND ENTEKHABI 1273 face soil moisture and/or ¯uxes (e.g., Carlson et al. tended to provide estimates in situations where one mea- 1981; Wetzel et al. 1984; Wetzel and Woodward 1987; surement type may be unavailable or insuf®ciently sen- Diak and Stewart 1989; McNider et al. 1994; Gillies sitive to the parameter of interest, (ii) unresolvable states and Carlson 1995; Anderson et al. 1997; Kustas et al. for a given measurement type may be resolved when 1999; Castelli et al. 1999; Norman et al. 2000; Lakshmi the information contained in other observations is in- 2000; French et al. 2000; Boni et al. 2001). These ap- cluded, (iii) trade-offs in sparseness (both in time and proaches generally either (i) use the physical connection space) between measurements may allow for better in- between surface moisture and the thermal inertia of the terpolation, (iv) the estimation algorithms may be gen- surface to infer soil moisture and/or (ii) use the depen- erally more robust. dence of sensible heat ¯ux on surface temperature to In this study, we present a variational framework to estimate the surface ¯uxes with radiometric surface tem- assimilate reference-level micrometeorological vari- perature and auxiliary data. While showing promising ables (temperature and humidity) and radiometric sur- results, these methods can sometimes fail due to heavy face temperature in order to estimate land surface and vegetation cover (which masks the ground surface), in- ABL states and surface ¯uxes. The data assimilation accuracies in the radiometric surface temperature mea- approach outlined below is appealing because it allows surements related to atmospheric corrections (i.e., at- us to combine measurements and a physical model of mospheric transmission and emission effects), and are the system, and by appropriately weighting the sources susceptible to errors related to cloud effects. They often of error in both, to ultimately produce a statistically represent underdetermined inverse problems and hence optimal and dynamically consistent estimate of the use empirical closure relations (e.g., ground heat ¯ux evolving state of the system. Most importantly, included as a fraction of net radiation). in the framework are estimates of surface ¯uxes and In addition to remotely sensed variables, standard time-varying structural model errors. The algorithm is ground-based observations may contain potentially use- also very ¯exible with respect to the inclusion of ad- ful information about land surface states and ¯uxes. ditional measurement types and therefore could easily Recent studies have investigated the use of reference- be extended for future research applications. In section level micrometeorological data to try to initialize sur- 2 we brie¯y present the coupled model used in this face soil moisture for mesoscale weather prediction or study. Section 3 describes the variational framework. A atmospheric boundary layer (ABL) models (e.g., Mah- synthetic test experiment and a case study application fouf 1991; Bouttier et al. 1993; Ruggiero et al. 1996; of this framework to the First International Satellite Callies et al. 1998; Bouyssel et al. 1999; Rhodin et al. Land Surface Climatology Project (ISLSCP) Field Ex- 1999; Hess 2001; Alapaty et al. 2001). These methods periment (FIFE) site is described in section 4, with con- rely on the sensitivity of reference-level temperature clusions discussed in section 5. and/or humidity to surface states primarily via the sur- face turbulent ¯uxes. However when the soil is suf®- 2. Forward model description ciently moist, land surface evaporation is not limited by soil moisture, but is instead limited by available energy The one-dimensional model we have developed for (i.e., ``controlled'' by atmospheric conditions). In these use in this study is composed of a land surface param- cases, the surface layer micrometeorology does not nec- eterization and an accompanying mixed-layer model essarily provide enough information for soil state esti- representation of the overlying boundary layer (see Fig. mation. Similar problems can be experienced for cases 1). It is a simple model meant to capture the regional with low surface radiative forcing, or during strong ad- coupled water and energy budgets of the land surface vection events. and lower atmosphere. The description of the model and For the most part, the studies mentioned above, as its testing is presented in detail in Margulis and Entek- well as many other land surface estimation schemes, habi (2001, hereafter ME01) and Margulis (2002) and have focused on using a single measurement type to try is summarized brie¯y below. While many land data as- to estimate surface parameters. These studies have gen- similation systems (LDAS) use an uncoupled land sur- erally tested the limits to which the estimation can be face model, the choice of a coupled model is useful as pushed when using a single observation type. While it allows for: (i) a signi®cant reduction in the auxiliary these results are quite useful, it is also necessary to begin data needed to the model (i.e., reference-level mi- to design frameworks that are ¯exible enough to take crometeorology and downwelling longwave radiation advantage of multiple data streams that are already are not needed since they are computed internally); and available or may be in the near future, and to appro- (ii) variables that are generally used as forcing for land priately each to provide an optimal estimate. surface models (i.e., reference-level temperature and hu- Also, signi®cant additional (synergistic) bene®ts may midity) are instead internal diagnostic variables that can exist when combining multiple types of measurements be assimilated as observations when available. The cou- in an estimation scheme and therefore is an area of pled land surface±boundary layer model described be- potentially fruitful research. These bene®ts may take low is a computationally ef®cient compromise between several forms: (i) the estimation scheme may be ex- an uncoupled model and a fully coupled single-column

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terizing the advective ¯uxes in the model, they are left as time-varying model error terms in the potential tem- perature and humidity budget equations. As will be shown below, these model error terms can be included as unknowns in the estimation algorithm. The model also includes a parameterization of the daytime growth and evening collapse of the ABL. Mod- eling the evolution of the height of the mixed layer is important not only because it exhibits a strong diurnal cycle (which is a function of the heating of the ABL by the surface), but because it generally grows in con- junction with the entrainment of warm/dry air from the overlying free atmosphere. The entrainment ¯uxes serve as the main connection between the tightly coupled land surface±ABL system and the overlying free atmosphere. They not only directly affect the ABL energy and mois- ture budgets, but ultimately the surface budgets as well. The model remains relatively simple but includes the in¯uential processes in the land surface±ABL system. FIG. 1. Schematic diagram of coupled land surface±atmospheric Finally, by simultaneously taking into account the cou- boundary layer model. The system state variables are bounded by pled nature of the system, the forcing requirements for boxes. All symbols are de®ned in appendix C. the model are reduced to incoming solar radiation, large- scale wind speed, and lapse rates in temperature and humidity above the mixed layer. atmospheric model, making it well-suited for use as part of a stand-alone LDAS. 3. Variational data assimilation framework The core of the model involves solving the coupled energy and water budget equations of the land surface Data assimilation provides a framework that allows and overlying atmospheric boundary layer. The land sur- us to merge measurements and physical models under face component of the model consists of prognostic the supposition that both provide useful information equations for three temperature states (canopy temper- about the state of the system, while containing mea- ature, surface ground temperature, and deep ground tem- surement and model errors, respectively. By appropri- perature) and three soil moisture states, representing the ately weighting the sources of error in both, we hope soil moisture pro®le through the uppermost 1±2 m of to ultimately produce a statistically optimal and dynam- soil. Included in the land surface model is an explicit ically consistent estimate of the evolving state of the representation of vegetation and its impact on the sur- system. In this paper we use a variational approach to face energy and water budgets (via its effects on both assimilate radiometric surface temperature and standard radiative and turbulent ¯uxes). For the boundary layer, reference-level temperature and humidity measurements a slab model is used with prognostic equations for the into the coupled land surface±atmospheric boundary mixed-layer potential temperature and speci®c humidity layer model described above, with the ultimate aim of state variables (which are uniform with height within obtaining estimates of the evolving model states, ¯uxes, the mixed layer), which represent the ABL energy and and model error. Variational data assimilation tech- moisture budgets, respectively. In addition, a detailed niques have a well-established history in meteorology parameterization of the radiative feedback between the and (e.g., LeDimet and Talagrand 1986; surface and overlying atmosphere is included in the Talagrand and Courtier 1987; Thacker and Long 1988; model that accounts for the radiative ¯ux divergence in Li et al. 1993; Lu and Hsieh 1997; Bennett et al. 1998). the ABL energy budget. The model presented here does More recently these techniques have begun to be applied not include a parameterization of advective ¯ux diver- in land surface hydrological applications (e.g., Castelli gence of heat and moisture into the ABL. Advection et al. 1999; Boni et al. 2001; Reichle et al. 2001a; can be important as midlatitude regimes are often char- ME01). The reader is referred to these references for acterized by intermittent frontal systems that modify the detailed methodological developments. The key aspects local boundary layer temperature and humidity. As a of the technique are described below. result, the advective ¯ux divergence can also have a Consider the coupled model as de®ned generally in signi®cant impact on the micrometeorological obser- dynamical form: vations that are used in the assimilation. Therefore, to dy obtain optimal estimates when assimilating microme- ϭ F(y) ϩ ␻ y(t0) ϭ f(␤), (1) teorological observations, it is important to account for dt the possibility of advective ¯uxes. Instead of parame- where y(t) is the state vector, F is a nonlinear vector

Unauthenticated | Downloaded 09/28/21 03:18 AM UTC JULY 2003 MARGULIS AND ENTEKHABI 1275 operator that is a function of the model states, and ␻(t) over the assimilation window. The scalar functional is represents a time-varying and unknown model error minimized by taking its ®rst variation (see ME01), vector. In a traditional open-loop forward modeling which yields the forward model [as shown in Eq. (1)] framework, this model would be integrated with prior as well as the adjoint model of the system: information only (i.e., best-guess initial conditions and MץF TTץ d␭ zero model error). In the assimilation framework the Ϫ1 ϭϪ ␭ Ϫ [␦]C␯ [Z Ϫ M(y)] y ΃ץy ΂ץunknown model initial condition y(t 0) is a function of dt ΂΃ a time-invariant random vector ␤, which is used so that bounded states like soil saturation can be mapped from ␭(tf ) ϭ 0, (4) the variable ␤, which varies over the entire real domain. where [␦] is the diagonal matrix of Dirac delta functions The random (uncertain) vectors ␤ and ␻(t) are initially and equations for the gradients in J with respect to ␤ characterized by their mean values [␤␻ and (t)] and and ␻ are their covariances [C␤ and C␻(t, tЈ)]. The mean values f T ץ Jץ represent our prior knowledge of the parameters (before ϭ (␤ Ϫ ␤)T CϪ1 Ϫ ␭ (5) 0 ␤ץ΂΃ ␤ ␤ץ -conditioning on observations) and the covariance ma trices represent the postulated uncertainty about the pri- J tfץ or values. The prior model error is generally assumed ϭ ␻(tЈ)CϪ1(tЈ, t) dtЈϪ␭(t). (6) ␻(t) ͵ ␻ץ -equal to zero. Posterior estimates are made by assimi t0 lating measurements. Development and testing of the accuracy of the adjoint The model states are related to the observations via model is described in ME01. Note that the adjoint model the measurement equation: must be integrated backward in time to obtain the adjoint Z ϭ M(y) ϩ ␯, (2) variables ␭(t). It depends on the state vector and is forced by the measurement±model mis®t. The adjoint where Z is the measurement vector that in this appli- model is a powerful tool because once the adjoint var- cation contains vectors of radiometric surface temper- iables are known, all of the gradients shown above can ature (TTobs ) and reference-level temperature ( obs ) and srbe computed and used in a gradient search algorithm to speci®c humidity (qobs ) measurements throughout the r minimize J. This procedure is done iteratively by (i) assimilation window. The measurement operator M(y) integrating the forward model with prior parameters, represents the diagnostic equations that map the model (ii) integrating the adjoint model backward in time about states onto T , T , and q (see appendix A). The mea- s r r the state trajectory, (iii) computing the gradients surements are assumed to include unbiased measure- and using them to update the parameters and model ment error, ␯, with a speci®ed covariance matrix C . ␯ error, and (iv) repeating steps (i)±(iii) until suf®cient For simplicity, measurement errors are assumed uncor- convergence is achieved. While any gradient search al- related in this application. Estimation of biases and pos- gorithm can be used, the particularly simple method terior covariance structures are possible and their fea- used here yields a set of update equations that are used sibility needs to be shown in future studies. in the above iterative procedure. The update equations To obtain an estimate that balances the uncertainties are shown in appendix B. due to measurement error and model error, the following Bayesian least squares performance index can be min- imized with respect to ␤ and ␻ over the assimilation 4. Application of the variational framework (or smoothing) window t ϭ [t , t ]: 0 f a. Synthetic data tests J ϭ [Z Ϫ M(y)]T CϪ1[Z Ϫ M(y)] ␯ Synthetic screening tests were ®rst performed to de- T Ϫ1 ϩ (␤ Ϫ ␤) C␤ (␤ Ϫ ␤) termine which parameters are identi®able when assim- ilating surface temperature and reference-level micro- ttff ϩ ␻(tЈ)T CϪ1(tЈ, tЉ)␻(tЉ) dtЈ dtЉ meteorology into the coupled model. This allows us to ͵͵ ␻ determine whether the estimation scheme can estimate tt00 the known ``truth.'' For a synthetic test, speci®ed initial tf dy conditions and model errors are used in the model to ϩ 2 ␭T Ϫ F(y) Ϫ ␻ dt. (3) generate half-hourly observations (to coincide with ͵ dt t0 [] FIFE measurement frequency) of the system that can The ®rst term in the performance index represents the then be used in the assimilation algorithm to attempt to mis®t between the data (Z) and model predictions M(y). recover the speci®ed parameters. Measurement noise is The next two terms penalize deviations from the prior also added to the observations. While not a guarantee values. All of these terms are normalized by their cor- that the estimator will work with real data, synthetic responding covariances. The ®nal term is the adjoined experiments yield important insight into the information system model constraint that is by de®nition zero, but content of the assimilated observations and lends some ensures that the estimates are dynamically consistent con®dence in the results from real applications.

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Based on explicit relationships contained in the mea- equations from the observations used here. In reality, surement operators and implicit dependence seen in off- due to the stiffness of these variables, sustained errors line sensitivity studies, the model states that are most that occur in these prognostic equations are most likely closely linked with the assimilated observations are the the result of errors in the longer memory states (i.e., canopy and surface ground temperature (Tc and Tg), soil moisture). Also, while surface temperature and ref- ABL potential temperature and speci®c humidity (␪ and erence-level micrometeorology were found to be suf- q), and rootzone soil moisture (W1 and W2). The ®rst ®ciently sensitive to the initial conditions in rootzone four variables are explicitly related to Ts, Tr, and qr via soil moisture (which determine the general soil moisture the measurement operators (see appendix A), while the trajectory), attempts to estimate time-varying errors (␻) soil moisture is implicit in the measurements due to its about this trajectory were not successful. In other words, role in the partitioning of net radiation at the surface. the observations are sensitive to the mean soil moisture

For initial condition speci®cation, ␪ 0, q 0, W10, and state, but are not sensitive enough to small perturbations

W20 are most important because of the long-term mem- (i.e., model error) about the mean trajectory to estimate ory associated with these variables, which allows errors soil moisture model error. The study by Reichle et al. to propagate for many days or weeks. Conversely, can- (2001b) has shown the potential to estimate soil mois- opy and ground temperature have very limited memory ture errors with microwave radio brightness measure- such that errors are generally dissipated within several ments (which has a stronger physical connection to soil hours. The other ABL states (i.e., temperature and hu- moisture), but reached similar conclusions to ours for midity inversions at the top of the ABL and ABL height) the canopy and ground temperature errors. In our initial are essentially reinitialized within the model each day synthetic experiments it was found that errors in the ␪ due to the collapse of the boundary layer in the early and q prognostic equations were identi®able from sur- evening. Therefore, for multiday assimilation windows face temperature and reference-level micrometeorology the speci®cation of the initial conditions of these vari- observations. It should be noted that without the use of ables is much less important. Finally the deep soil mois- reference-level observations (radiometric surface tem- ture and ground temperature do not affect the measured perature observations only) these errors were not iden- variables on the timescale of this study. As a result, for ti®able because the physical connection between ABL this application the estimated initial conditions are lim- advection and surface temperature is weak (the esti- ited to ␪ 0, q 0, W1200, and W . mation problem becomes too underdetermined). During The time-varying model error terms were conceptu- cases where signi®cant advection-related errors are pre- alized as ¯ux errors in the prognostic equations for Tc, sent, the combination of surface temperature and mi- Tg, ␪,q,W1, and W 2. Independent error time series for crometeorology observations allowed for estimation of each equation were synthetically generated using a lag- both land surface states and advection-related model 1 autoregressive model, which by de®nition has an ex- errors. The ability to estimate these errors is quite useful, 2 ponential covariance structure [C␻(t, tЈ) ϭ ␴ ␻ exp( | t Ϫ considering the lack of a parameterization of advective tЈ |/␶)]. This type of error covariance model is appro- ¯ux divergence in our model. priate for correlated errors that can be characterized by Based on these initial synthetic tests, we conclude an autoregressive process. In cases where the covariance that the control variables that can be estimated through structure of errors is poorly known, this type of model assimilation of surface temperature, and reference-level is often used (e.g., Reichle 2000). The errors were used temperature and humidity observations include: initial in the reference simulation and the assimilation scheme conditions in rootzone soil moisture (W1200and W ) and was used to try to estimate them. The standard deviation mixed-layer temperature and humidity (␪ 0 and q 0)as

(␴␻) and decorrelation timescale (␶) of the model error well as model errors in the ␪ and q prognostic equations for each prognostic variable mentioned above were giv- (␻␪ and ␻q), which are attributed to synoptic (advection) en reasonable values of 30 W mϪ2 and 6 h, respectively. forcing. Therefore in this application the model error We also need to specify the initial condition error co- vector shown in Eq. (1) consists of nonzero elements variance C␤. For simplicity, it is specifed as a diagonal in the ABL budget equations only. matrix with error standard deviations of 20% for the To illustrate these results, the assimilation algorithm soil moisture saturation initial conditions, and 2 K and was run for a 5-day synthetic experiment with half- 2gkgϪ1, respectively, for the initial conditions of hourly observations generated by integrating the model mixed-layer potential temperature and speci®c humid- with synthetically generated model errors (in the ABL ity. The standard deviation in the observation errors are budget prognostic equations only, using the covariance 0.5 K for the radiometric surface and reference-level model discussed above) and adding random measure- temperature and 0.5 g kgϪ1 in reference-level speci®c ment error to the model outputs. Results from the ex- humidity. periment, are shown in Table 1 and Figs. 2±4. Table 1 Results from our initial synthetic tests (not shown shows that the true initial conditions were almost per- here) indicate that because of the limited memory in fectly recovered by the estimation algorithm and that canopy and ground temperature, the estimation algo- the objective function is reduced to near the true value. rithm has no skill in estimating errors in these prognostic Note that the truth is not perfectly recovered because

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TABLE 1. Comparison of prior and posterior (estimated) initial con- ditions to true values from synthetic experiment. Reduction in ob- jective function is shown in last column.

q0 Ϫ1 ␪0 (K) (kg kg ) W10 (Ϫ) W20 (Ϫ) J (Ϫ) Prior 300.0 18.0e-3 0.65 0.65 24 500 Posterior 297.0 16.7e-3 0.42 0.48 1100 True 297.0 16.7e-3 0.40 0.50 950 of the presence of observation and model error. Figure 2 shows the comparison between the prior, estimated, and actual model errors in ␪ and q. The model error estimates capture the primary features and low-fre- quency variability of the actual model error. The fact that the estimated model error is much smoother than the true error is consistent with other recent studies (e.g., Reichle et al. 2001). For comparison purposes, a 12-h moving average of the true error is also shown. Based on the general agreement between the estimate and the moving average, it appears that the estimate captures errors that have an integrated impact on a half-day time- scale. This is the case because the high-frequency error

FIG. 3. Comparison of prior (dashed line), estimated posterior (solid line), and observed (dots) for (a) surface temperature, (b) reference- level temperature, and (c) reference-level humidity for the synthetic experiment.

variability generally has negligible impact on the ob- servations and therefore cannot be captured by the es- timate. Additionally, because of the strong diurnal var- iability of the land surface ¯uxes, the relative impor- tance of the errors varies strongly with time. When er- rors are small relative to the magnitude of other ¯uxes (i.e., especially near midday) they have minimal impact on the observations and therefore cannot be estimated well. However, because the relative magnitude of the errors are small at those times, their estimation is not as important. Figure 3 shows the signi®cant improve- ment in ®t between the assimilated observations and posterior predictions compared to the priors. The time series of the primary ABL and land surface state esti- mates are shown in comparison to the true values in Fig. 4. The estimation algorithm performs quite well in removing large biases that were present when using the prior values. Finally, by reproducing the states, the land surface ¯uxes are also reproduced by the estimation algorithm (not shown here). FIG. 2. Comparison of prior (dash±dot line), estimated posterior (solid line), and true (dotted line) model errors in the (a) mixed-layer potential temperature budget, and (b) mixed-layer speci®c humidity b. Application to the FIFE site budget for the synthetic experiment. Note that the prior error is by de®nition zero and that the estimated model errors closely resemble To test the methodology presented above with real a 12-h moving average (dashed line) of the true error. (Beginning of data, we applied the data assimilation framework to the each day on ®gure corresponds to midnight local time.) dataset obtained at the FIFE site in Kansas in the sum-

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size of the control vector smaller, which aids in con- vergence, and it is desirable in operational applications. In an operational setting, single-day or forecast cycle windows could be used. The drawback of sequential windows is the proper reinitialization of prior covari- ances, which is discussed further below. The ®rst 10 days of the study are marked by several moderate storms, atmospherically controlled evapora- tion (wet soil), and a large advection event (on days 193±195), which is most noticeable in the speci®c hu- midity time series. The second 10 days consist of a long dry down event where the soil exerts signi®cant control on the surface turbulent ¯uxes such that the reference- level micrometeorology is strongly coupled to the sur- face. There also appears to be a small advection event on days 201±203. The FIFE site, as well as many mid- latitude regimes, is marked by periodic frontal systems that can advect moisture and energy of different char- acteristics into the region. During the course of the sum- mer, nine identi®able events occurred (approximately one every 10 days) at the FIFE site. The advection events seen in Fig. 5 coincide with high sys- tems moving into and out of the region.

1) RESULTS FOR DAYS 190±200 We ®rst applied the assimilation scheme to days 190± 200. This particular scenario provides a very signi®cant FIG. 4. Comparison of prior (dashed line), estimated posterior (solid line), and true (dots) model states for the synthetic experiment for test of the scheme because there appears to be a large (a) potential temperature, (b) speci®c humidity, (c) layer/soil satu- advection event that signi®cantly affects the reference- ration, (d) layer-z soil saturation, (e) canopy temperature, and (f) level micrometerology variables, concurrent with con- surface ground temperature. ditions where evaporation is near the potential rate (when micrometeorology is relatively insensitive to soil moisture). The model is initialized by setting the ABL mer of 1987 (see Sellers et al. 1992). The FIFE dataset potential temperature and humidity equal to the ob- (Betts and Ball 1998) provides the needed forcing var- served reference-level micrometeorology, and the root- iables for the model, surface radiometric temperature zone soil moisture to relatively dry values that were and micrometeorological measurements for the assim- thought to be an intentionally poor prior guess (see Table ilation scheme, as well as ¯ux observations for vali- 2). The assimilation scheme is applied iteratively, as dation. The forward model described above was tested described above, until convergence in the initial con- at the FIFE site and shown to perform well (ME01). In dition and model error estimates are achieved. The pos- this paper we present results from an application to a terior (estimated) initial conditions as well, as the re- 20-day window in July (days of year 190±210; hereafter duction in the objective function are shown in Table 2. designated days). This particular period was chosen be- The ABL temperature and humidity initial conditions cause it contains a variety of environmental conditions, do not change by a large amount, which is to be expected some of which can confound estimation algorithms (i.e., since the micrometerological values should be repre- advection, precipitation, atmospherically controlled sentative of those in the ABL. However a large change evaporation regimes). The key surface meteorological is shown in the rootzone soil saturation (ϳ20%). For variables at the site are shown in Fig. 5. An open loop the grassland vegetation at the FIFE site, the soil-con- forward simulation, which by de®nition uses all prior trolled evaporation regime corresponds to soil moisture values (including zero model error), was run for the 20- values below 45% saturation. Thus, while the prior ini- day period that will be used for comparison to the as- tial rootzone soil moisture is within the soil-controlled similation results below. The estimation was performed evaporation regime, the estimate is within the potential over two consecutive assimilation windows (from days evaporation (atmospherically controlled) regime. 190±200 and 200±210), with model and observation The estimated time-varying model errors for days error statistics equal to those described in the previous 190±200 are shown in Fig. 6. While the error estimate section. The reason for using sequential assimilation can represent a combination of factors, it is attributed windows is mostly for practical reasons. It keeps the here mostly to advection processes that are not repre-

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FIG. 5. Meteorological conditions at the FIFE site during days 190± 210 in 1987. Here R is incoming solar radiation, u is reference-level FIG. 6. Estimated posterior model error (solid lines) in (a) ABL s r potential temperature budget and (b) ABL speci®c humidity budget wind speed, P is precipitation, Ps is surface pressure, and Tr and qr are the reference-level air temperature and speci®c humidity, re- at FIFE during days 190±200. Note prior model error (dash±dot lines) spectively. is equal to zero by convention.

drier ABL (which captures less radiation from the sur- sented in the one-dimensional model. The estimate in- face) rather than on cooler air entering the region. These dicates signi®cant coherent advection of cool/dry air model error estimate results imply that it is possible to into the region during days 192±195, followed by a less estimate processes that are not explicitly parameterized pronounced front of warm/moist air during days 195± (or perhaps poorly parameterized) in the model. Another

198 (note time rate of change of Tr and qr in Fig. 5 with important point is that, for this particular case (with the values of model error estimates in Fig. 6). Also, the large model errors), the ability to estimate both the ad- advection of moisture plays a much larger role than vection-related model error and the soil moisture initial advection of heat, with a peak ¯ux of moisture equiv- conditions depend on using both surface temperature alent to approximately 80 W mϪ2, compared to less than and micrometeorological observations in the assimila- 20WmϪ2 in the ABL energy budget. In fact, from tion scheme. For example, we found that when using of¯ine model runs (not shown) it is apparent that the only surface radiometric observations, the advection decrease in ABL potential temperature between days model error cannot be estimated because the problem 192 and 194 is mostly due to the radiative effect of a becomes too underdetermined. Conversely, when using

TABLE 2. Prior and posterior (estimated) initial conditions for each 10-day assimilation window (for days 200±210: * corresponds to results using reinitialized prior error covariance and ² corresponds to results using approximated prior error covariance). Reduction in objective function is shown in last column.

Ϫ1 ␪0 (K) q0 (kg kg ) W10 (Ϫ) W20 (Ϫ) J (Ϫ) Days 190±200 Prior 297.0 17.0e-3 0.39 0.41 87 000 Posterior 297.6 16.7e-3 0.59 0.62 7200 Days 200±210 Prior* 300.2 13.2e-3 0.33 0.35 2 ϫ 107 Posterior* 298.7 16.5e-3 0.46 0.52 8200 Prior² 298.0 17.5e-3 0.48 0.49 15 000 Posterior² 298.6 16.6e-3 0.46 0.51 7900

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FIG. 7. Comparison of prior (dashed line), estimated posterior (solid FIG. 8. Comparison of prior (dashed line), estimated posterior (solid line), and observed (dots) for (a) surface temperature, (b) reference-level line), and observed (dots) for (a) surface sensible heat ¯ux and (b) temperature, and (c) reference-level humidity during days 190±200. surface latent heat ¯ux during days 190±200. only reference-level observations, soil moisture cannot be reliably estimated because, during atmospherically part of the window and, because of the overly dry soil, controlled conditions, reference-level observations are is too dry by the end of the window. On the other hand, not suf®ciently sensitive to soil moisture. Thus the com- the posterior estimate captures the overall trend quite bination of both observation types leads to a robust well. estimate of the land and ABL states. This is an example As a result of the application of the assimilation of the synergism that may exist when using multiple scheme, it is expected that model predictions will ®t the types of observations in the assimilation scheme. assimilated observations well. A more important goal The ®t of the prior and estimated radiometric surface is the estimation of unknown and dif®cult to measure temperature and reference-level micrometeorology to states (i.e., soil moisture) and ¯uxes (i.e., surface tur- the assimilated observations for days 190±200 is shown bulent ¯uxes). While there are no spatially averaged in in Fig. 7. The improvement between the prior (open situ ground-truth soil moisture observations at FIFE, loop) and posterior estimate is quite clear. The diurnal Colello et al. (1998) show some point measurements of amplitude of the open loop radiometric surface tem- rootzone soil moisture at one location within the FIFE perature is generally too large due to drier than actual site. Their observed rootzone soil saturation on day 190 soil moisture. Due to the dominance of the advection is ϳ64%, which compares well to our depth-averaged signal in the micrometeorological data, it is the surface estimate of 62%. This result should be viewed with temperature that is largely responsible for resolving the caution however due to the high spatial variability ex- initial soil moisture condition. The reference-level tem- pected in rootzone soil moisture over large scales. A perature is certainly improved due to the advection es- better measure of the estimation algorithm is the com- timate, but appears to show a cold bias compared to the parison of the regional surface turbulent ¯uxes (which observations near midday. While the error estimates are are strong functions of soil moisture) and can therefore able to capture the large low-frequency errors (see Fig. be used to infer that the assimilation algorithm is pro- 2), they are not able to also resolve the diurnal errors viding consistent estimates of large-scale soil moisture. that may be present. The most pronounced difference The comparison of the prior and estimated ¯uxes to between prior and posterior estimates is in the reference- observations taken at the FIFE site are shown in Fig. level humidity. The prior is too humid during the middle 8. The results clearly indicate the improved estimation

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TABLE 3. Bias and rmse between observed and estimated sensible (H) and latent (LE) heat ¯uxes for the two assimilation windows. Results from days 200±210 correspond to the reinitialized prior case. The statistics based on the prior estimates are given in parentheses. H LE (W mϪ2) (W mϪ2) Days 190±200 Bias 27 (50) Ϫ35 (Ϫ83) Rmse 30 (56) 38 (88) Days 200±210 Bias 18 (96) Ϫ23 (Ϫ129) Rmse 21 (112) 25 (148)

of ¯uxes due to the assimilation of surface temperature and micrometerological observations. The bias and root- mean-square error (rmse) between the observations and estimates of surface turbulent ¯uxes (averaged over the daytime to remove the overly optimistic statistics ob- tained when comparing variables with diurnal cycles) are given in Table 3. The magnitude of the bias and rmse between the prior and posterior estimates for both the average daily surface sensible and latent heat ¯uxes are reduced by approximately half via the assimilation. However the posterior estimates still contain a positive bias in the sensible heat ¯ux estimates and a negative bias in the latent heat ¯ux estimates. This is most likely due to biases in the model. Including estimation of mod- el biases is notoriously dif®cult and it is widely rec- ognized to have adverse impacts on data assimilation. FIG. 9. Comparison of prior (dashed line), estimated posterior (solid The estimation of these biases remains a topic of re- line), and observed (open circles) mixed layer: (a) potential temper- search. ature, (b) speci®c humidity, and (c) height during days 190±193. In addition to the surface ¯uxes, a comparison be- tween prior, estimated, and observed boundary layer 190±200. Instead, for a more fair comparison between states is possible during days 190±193 when radiosonde assimilation results and those from only forward sim- data were available for the site. These results are shown ulation, we ®rst applied the estimation algorithm using in Fig. 9. For mixed-layer potential temperature, hu- initial conditions taken from the 20-day open loop sim- midity, and height, the estimated states are signi®cantly ulation discussed above with the same prior error var- closer to the observations. Thus the assimilation of ob- iances used for days 190±200. (Results using initial con- servations is clearly improving the estimates of bound- ditions obtained from the ®rst assimilation window will ary layer states. Additionally, the cooler, moister, and be discussed brie¯y below.) Given these priors for the shallower boundary layer are all consistent with a moist- second window, the assimilation scheme is iteratively er land surface condition (consistent with assimilation applied using the observations from days 200±210. The result). This lends further credibility to our inference resulting initial condition estimates are shown in Table that the estimated soil moisture state is much improved 2. The estimated ABL potential temperature and speci®c over the open loop. humidity initial conditions change by about 2 K and 3 gkgϪ1, respectively, while the initial soil saturation in the two rootzone layers change by 13% and 16%, re- 2) RESULTS FOR DAYS 200±210 spectively. Again, the depth-averaged rootzone soil Next we applied the assimilation algorithm to days moisture estimate (52%) agrees well with the point ob- 200±210. A primary difference between the ®rst and servations of Colello et al. (1998), who observed a root- second assimilation windows is the issue of how to ini- zone saturation of 51%. As expected from the results tialize the priors. The prior initial condition values for shown for days 190±200, these results con®rm that the this assimilation window could be taken as the terminal open loop results were too hot and dry. The extremely conditions of the estimated (posterior) states from the large prior value for the objective functional is primarily ®rst window. In fact, this is the exact procedure that a result of a large bias in the reference-level speci®c would be used in an operation context. As a result, the humidity observations, which is removed by the assim- estimates would necessarily be much better prior esti- ilation algorithm (discussed below). mates than those used in the ®rst window, since they The advection-related model error estimates for the would be conditioned on the observations during days day 200±210 window are shown in Fig. 10. In this case

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FIG. 11. Comparison of open loop (dashed line), estimated (solid FIG. 10. Comparison of prior model error (dash±dot lines) to es- line), and observed (dots) of (a) surface temperature, (b) reference- timated posterior model error using the reinitialized prior error co- level temperature, and (c) reference-level humidity during days 200± variance (solid lines) and approximated prior error covariance (dashed 210. line) for (a) ABL potential temperature budget and (b) ABL speci®c humidity budget at FIFE during days 200±210. rithm, with differences in peak ¯ux values of up to 150± 200WmϪ2. This is largely due to the better initialization the model error estimates for both the ABL energy and of rootzone soil moisture. The statistics for daily average moisture budgets are less than 10 W mϪ2. Except during ¯ux estimates are compared to the FIFE observations nighttime conditions, the magnitude of these errors are in Table 3. The bias and rmse for both sensible and generally much smaller than those of other ¯uxes in the latent heat ¯uxes are reduced roughly by a factor of 5. model, indicating a much less important role than the Just as for days 190±200, there remains a slight positive error terms for days 190±200. These results are con- bias in the sensible heat ¯ux estimates and a negative sistent with the ®eld experiment measurements (Fig. 5), bias in the latent heat ¯ux estimates. Note that both the which do not seem to indicate signi®cant advection bias and rmse are signi®cantly smaller for the day 200± events during this second time window. The comparison 210 window. Assuming the FIFE ¯ux observation error of the prior and posterior estimates of radiometric sur- is stationary in time, this is most likely due to the fact face temperature and micrometeorology to the obser- that the advection-related model errors are smaller dur- vations are shown in Fig. 11. It is clear that the open ing days 200±210, indicating a stronger land±atmo- loop ABL speci®c humidity initial condition is too dry sphere coupling (i.e., more information about surface (Fig. 11c), which leads to a large bias in the reference- ¯uxes in the observations). Overall, the results indicate level humidity. Also, it can be reasonably inferred that signi®cant value in assimilating observations to estimate the open loop soil moisture initial condition is too dry land surface states and ¯uxes. based on the larger diurnal amplitude in radiometric In addition to the two independent assimilation win- surface and reference-level temperature (Figs. 11a,b) dows (reinitialization of prior initial conditions and their and increased drying of the mixed layer (Fig. 11c) com- error covariance), we applied the algorithm to days 200± pared to observations over the 10-day window. 210 using the terminal conditions of the states from days A comparison of the open loop and estimated surface 190±200 as the prior initial condition values. This is the turbulent ¯uxes is shown in Fig. 12. These results show approach one would take in an operational context. The a very large improvement in the surface turbulent ¯uxes primary complicating factor is the speci®cation of the as a result of the application of the assimilation algo- prior covariance for these initial conditions, because it

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Given these approximated priors for the second win- dow, the assimilation scheme was again applied using the observations from days 200±210. The prior and es- timated (posterior) initial condition estimates are shown in Table 2. Note that in comparison to the reintialized case discussed above, the priors are much closer to the ®nal estimates. This is explained by the fact that the prior initial condition estimates have already been con- ditioned on the observations from days 190±200. The change in the initial conditions between the prior and posterior estimates re¯ect the new information intro- duced by the observations during the day 200±210 win- dow. Also note that the posterior estimates are quite close to those obtained in the reinitialized case. There- fore the choice of prior (reinitialization vs approxima- tion) does not seem to make much difference. This is also true in the model error estimates, which are shown in Fig. 10. The difference between the model error es- timates from the two different methods is relatively small, indicating that the algorithm is robust. As a result of the similar initial condition and model error estimates, the posterior surface ¯ux estimates are almost identical to those shown in Fig. 12.

3) ROBUSTNESS OF MODEL ERROR ESTIMATES As a ®nal test of the estimation algorithm, we are interested in the dependence of the model error estimates FIG. 12. Comparison of open loop (dashed line), estimated (solid line), and observed (dots) for (a) surface sensible heat ¯ux and (b) on the particular window chosen. This test is also related surface latent heat ¯ux during days 200±210. to determining whether the objective function minimi- zation during days 190±200 and 200±210 has converged to disparate local minima or nearly global minimum. should be conditioned on the results from the previous Accordingly, we use an overlapping assimilation win- assimilation cycle. In variational schemes (as compared dow from day 195 to 205 to examine how the model to sequential ®ltering schemes) there is no explicit com- error estimates compare to those from the other two putation of the evolving error covariance. While this windows. The results are shown in Fig. 13. Overall the leads to a signi®cant computational reduction in the al- posterior estimates seem to be robust, with the overlap- gorithm, it is a drawback if the methodology is applied ping estimate closely matching trends in the other es- in an operational environment with sequential assimi- timates. The largest discrepancy occurs near the end of lation windows. Instead of explicit knowledge of the the overlapping window. This is due to the fact that the evolving covariance, approximations must be used. In model error is by de®nition equal to a convolution of our test, the prior variances for the states on day 200 the adjoint variables (see appendix B), which are equal were obtained using the same method employed by to zero at the end of the assimilation window. The result Rhodin et al. (1999). Based on their work, after the is that the model error estimate is necessarily close to posterior initial condition estimates are obtained for zero at the end of the assimilation window. year-day 190, the posterior error covariance matrix (for the estimated initial conditions) is approximated by the 2 2 Ϫ1 5. Conclusions y0) ], which can be computedץ/J ץ)] inverse Hessian from ®nite-difference perturbation forward model runs In this study we show the ability to estimate land and would be exactly equal to the true error covariance surface and ABL states and ¯uxes from readily available if the model were linear (Tarantola 1987). An additional surface radiometric temperature and climate station ref- error (variance) is then added to the posterior initial erence-level micrometeorology. The variational data as- condition covariance to represent the uncertainty ac- similation framework provides an effective way to com- crued over the assimilation window (due to error prop- bine these measurements with a physical model of the agation via model dynamics) from day 190 to 200. This system to provide optimal estimates that take into ac- error can be estimated from of¯ine perturbation model count both measurement and model error. A key aspect runs. For further details on the method the reader is of this work is the use of a coupled model, which not referred to Rhodin et al. (1999). only reduces the needed auxiliary forcing data required

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increase the size and complexity of the problem, but may actually eliminate the need for advection-related error estimates since they would be explicitly computed by the model. However the issue of model error and its estimation as well as the possibility of mean bias and adaptive prior statistics estimation needs to be explored in follow-on investigations.

Acknowledgments. This research was sponsored by the NASA Earth System Science Fellowship and NASA Grant NAG5-11602. The authors would like to thank three anonymous reviewers for their helpful comments.

APPENDIX A

Measurement Operators

For the measurement operators used in the variational framework we need to construct the diagnostic equa- tions that map the model states to the measurements

used in this study (Ts, Tr, and qr). The radiometric sur- face temperature equation is described ®rst. Previous studies (e.g., Anderson et al. 1997) generally relate the radiometric temperature to canopy and ground temper- ature via a function of the form:

nn1/n Tsccϭ [VT ϩ (1 Ϫ V cg)T ], (A1) FIG. 13. Comparison of model error posterior estimate for over- lapping assimilation window over days 195±205 (dashed lines) to where V is a fractional vegetation cover (which can those from windows over days 190±200 and 200±210 (solid lines) c of (a) ABL potential temperature budget and (b) ABL speci®c hu- depend on view angle), and the exponent n is usually midity budget. equal to 4, so that Ts represents the effective temperature for the thermal radiation seen by the sensor. For ease of computation, and with generally little loss of accuracy by an uncoupled model, but allows for the inclusion of (fourth power nearly linear shape over the limited range observations (in this case micrometeorology) that could of encountered), the following simpli®ed not otherwise be assimilated. expression is often used: The results shown here illustrate the robust estimates that can be obtained when using multiple observation Tsccϭ VT ϩ (1 Ϫ V cg)T , (A2) types. Using both surface temperature and reference- level micrometeorology measurements allows for the which is the expression used in this study. Including accurate estimation of land surface ¯uxes even during (A1) is not a problem for future studies. The radiometric nonideal conditions, where large advection-related mod- surface temperature is thus simply an average of canopy el errors are present and/or when atmospherically con- and ground temperature weighted by the fractional veg- trolled evaporation is occurring. These conditions can etation cover. At the FIFE site the fractional vegetation confound the estimation algorithm when using only one cover is 80%. of the observation types. Furthermore the assimilation The diagnostic relationships between temperature and scheme is able to provide estimates of model errors. humidity at a given reference height (zr) and the model This allows for diagnosis of structural model errors that states is more complicated. The turbulent ¯uxes in the may be present due to missing process representation forward model are generally expressed as a gradient (in and/or poor or biased parameterizations. either temperature or vapor pressure) between two levels Future research directions should include an explo- divided by a resistance term. We assume that there is ration of other observation types and combinations (e.g., no major storage of heat or moisture within the surface radiosondes, microwave radio brightness, etc.) that may air layer so that the sensible and latent heat ¯uxes be- have signi®cant information content for land data as- tween the canopy (at height z1) and the reference level similation. An extension to a spatially distributed do- (at height zr) are equal to the ¯uxes between the ref- main is another area of future work. Using a two- or erence level and the bottom of the mixed layer (at height three-dimensional atmospheric model would obviously zm):

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1/2 (TarϪ T )(T rmϪ T ) 1 Fmr(z ) z mϪ dz rϪ d ␳cppϭ ␳c and (A3) ra1 ϭ ln ln . rra1 a2 F (z )␬2UF(z ) zzϪ d hr m[] mm ΂΃΂΃0H 1 ␳c (e Ϫ e )(␳c e Ϫ e ) pparϭ rm, (A4) (A9) ␥ ra1 ␥ ra2 Using the series resistance rule, the resistance ra2 is the where ra1 and ra2 are the aerodynamic resistances be- difference between ra and ra1, which yields the expres- tween the canopy and reference level and reference level sion and mixed layer, respectively (see appendix C for list  of symbols). By solving these two equations for T and zmmϪ dzϪ dz rϪ d r ln ln ln e , the following two diagnostic relationships are ob- 1/2 r ΂΃΂΃zz01Ϫ dzF (z ) ΂΃ 1Ϫ d tained: r ϭϪmr . a2 ␬2UF(z ) F (z ) F (z ) mhmmmhr[] rra2 a1 rra2 a1 Tramramϭ T ϩ Teϭ e ϩ e , (A5) (A10) rraa rr aa Based on these expressions the weighting terms in Eq. where ra is the aerodynamic resistance between the can- (A5) can be written as opy air and bottom of the mixed layer (and via the series resistance rule is equal to the sum of r and r ). There- z Ϫ d a1 a2 ln r fore the reference-level values of temperature and vapor 1/2 rF(z ) F (z ) ΂΃z1 Ϫ d pressure are a weighted averaged of the values within a2 ϭ 1 Ϫ mr hm and (A11) the canopy air and at the bottom of the mixed layer. rFammhr(z ) F (z ) [][]zm Ϫ d Note that the reference-level speci®c humidity (q ), ln r z Ϫ d which is the measured variable in this study is simply ΂΃1 equal to (⑀/Ps)er. In the model Ta and ea are themselves z Ϫ d ln r diagnostic equations (see Margulis 2002), where Ta is 1/2 ΂΃z1 Ϫ d a weighted average of T , T , and T , and e is a weight- rFa1 mr(z ) F hm(z ) c g m a , (A12) ed average of e (T ), e (T ), and e . For details about ϭ c g m rFammhr(z ) F (z ) * * [][]zm Ϫ d these equations the reader is referred to Margulis (2002). ln The speci®c expressions for the in the above ΂΃z1 Ϫ d equations can be derived as follows: The total aero- so that the diagnostic equations for Tr and er can simply dynamic resistance is given by be written as

1 z Ϫ dzϪ d Tramramϭ (1 Ϫ ␣)T ϩ ␣Teϭ (1 Ϫ ␣)e ϩ ␣e , r ϭ lnmm ln , (A6) a 2 (A13) Fhm(z )␬ Uz m ΂΃΂΃0H z1 Ϫ d where where the stability correction factor for heat (Fh)isa function of the bulk Richardson number and is taken zr Ϫ d from Louis et al. (1982). Here z0H is the scalar roughness ln 1/2 length scale. The wind speed at the bottom of the mixed F (z ) F (z ) ΂΃z1 Ϫ d ␣ ϭ mr hm . (A14) layer (Um) is related to the reference-level forcing wind Fmm(z ) F hr(z ) [][]zm Ϫ d speed (Ur) via ln ΂΃z1 Ϫ d z Ϫ d ln m 1/2 ΂΃z0H APPENDIX B Fmr(z ) Umrϭ U , (A7) Fmm(z ) []zr Ϫ d ln Update Equations ΂΃z0H For the gradient search algorithm in this study we use a simple steepest descent method. There are cer- where the stability correction factor for momentum (Fm) is also provided by Louis et al. (1982) and is of similar tainly more ef®cient algorithms, but this technique was found to work for this application and yields a simple form to Fh. The expression for the resistance between set of update equations. Future work could use a more z1 and zr(ra1) is analogous to that of ra and is given by ef®cient technique like the conjugate gradient method. 1 z Ϫ dzϪ d In the steepest descent algorithm, a parameter (u)at r ϭ lnrr ln . (A8) a1 2 iteration k ϩ 1 is given by Fhr(z )␬ Uz r ΂΃΂΃0H z1 Ϫ d J kץ This can be rewritten by substituting the expression re- ukϩ1 ϭ uk Ϫ␧ , (B1) uץlating Um and Ur to yield ΂΃

Unauthenticated | Downloaded 09/28/21 03:18 AM UTC 1286 MONTHLY WEATHER REVIEW VOLUME 131 where ␧ is a scalar step size. For the unknown param- q Mixed-layer speci®c humidity kg kgϪ1 Ϫ1 eters in this study (␤ and ␻) we specify the step size qr Reference-level speci®c hu- kg kg ␧ to be proportional to the corresponding uncertainty midity obs Ϫ1 (covariance) for each variable (i.e., ␧i ϭ ␩iCi). Substi- qr Reference-level speci®c hu- kg kg tuting this along with the gradient expressions shown midity observations Ϫ2 in section 3, update equations for each parameter are RAd Downwelling longwave radia- Wm given by tion from within the mixed layer T f R longwave radiation WmϪ2 ץ ␤kϩ1 ϭ ␤kkϪ ␩ (␤ Ϫ ␤) ϩ ␩ C ␭k , (B2) Au from within the mixed layer 0 ␤ץ΂΃ ␤␤␤ R Downwelling longwave radia- WmϪ2 t ad f tion from above the mixed ␻ kϩ1(t) ϭ (1 Ϫ ␩ )␻ kk(t) ϩ ␩ C (t, tЈ)␭ (tЈ) dtЈ. ␻␻͵ ␻ layer t 0 Ϫ2 (B3) Rcu Upwelling longwave radiation Wm from canopy into the mixed The scalar step sizes are speci®ed to aid in convergence. layer Ϫ2 Note that update equation for the model error involves Rgu Upwelling longwave radiation Wm a convolution integral, which is evaluated most ef®- from ground into the mixed ciently using a fast Fourier transform (FFT). layer Ϫ1 ra Aerodynamic resistance be- sm APPENDIX C tween canopy air and mixed- layer bottom List of Symbols Ϫ1 ra1 Aerodynamic resistance be- sm tween canopy air and reference level Ϫ1 Ϫ1 Ϫ1 cp Dry air speci®c heat J kg K ra2 Aerodynamic resistance be- sm C␤ Prior initial condition error co- tween reference-level and variance mixed-layer bottom

C␯ Measurement error covariance t Time s C␻(t, tЈ) Prior model error covariance t0 Initial time s d Zero-plane displacement m tf Terminal time s height Ta Canopy air temperature K Ϫ2 Ϫ1 Eg Evaporation from ground kg m s Tc Canopy temperature K Ϫ2 Ϫ1 Ec Transpiration from canopy kg m s Td Deep soil temperature K Ϫ2 Ϫ1 Etop Dry air entrainment kg m s Tg Surface ground temperature K e (T) Saturated vapor pressure at Pa T Temperature at bottom of K * m temperature T mixed layer ea Canopy air vapor pressure Pa Tr Reference-level temperature K obs em Mixed-layer vapor pressure Pa T r Reference-level temperature K er Reference-level air vapor pres- Pa observations sure Ts Radiometric surface tempera- K F Nonlinear vector operator in ture obs dynamical model T s Surface temperature observa- K Fh Stability correction for heat (-) tions Ϫ1 Fm Stability correction for mo- (-) Um Wind speed at bottom of mixed ms mentum layer Ϫ2 Ϫ1 H Total sensible heat ¯ux W m Ur Wind speed at reference level m s Ϫ2 Hc Sensible heat ¯ux from canopy W m Vc Fractional vegetation cover (-) Ϫ2 Hg Sensible heat ¯ux from ground W m W1 Layer 1 soil saturation (-) Ϫ2 Htop Sensible heat ¯ux entrainment W m W10 Initial condition of layer 1 soil (-) h Mixed-layer height m saturation

J Objective functional W2 Layer 2 soil saturation (-) Ϫ2 L␷ E Total latent heat ¯ux W m W20 Initial condition of layer 2 soil (-) M Model measurement operator saturation

vector W3 Layer 3 soil saturation (-) Ps Surface pressure Pa y Model state vector Ϫ1 q 0 Initial condition of mixed-lay- kg kg z0H Aerodynamic scalar roughness m er speci®c humidity length

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