1. Introduction
The problem of determining a physically consistent and accurate snapshot of the atmosphere is a central theme in the atmospheric sciences. The motivation for obtaining a three-dimensional, meteorologically complete (i.e., winds and thermodynamic variables), and accurate picture of the atmosphere ranges from obtaining fields to be subsequently analyzed in order to increase scientific understanding of a particular set of atmospheric processes or to initialize a numerical model. Given the scope of the possible avenues of scientific and operational interest, it is not at all surprising that a wide variety of approaches have been developed to solve the problem of determining an accurate estimation of the state of the atmosphere. One common set of solutions is data assimilation where atmospheric measurements are combined with solutions from a numerical model. For detailed information on data assimilation, the reader is referred to reviews found in Bengtsson (1975), McPherson (1975), Bourke et al. (1985), Lorenc (1986), Hollingworth (1986), Ghil and Malanotte-Rizzoli (1991), and Harm et al. (1992). In general terms, data assimilation techniques can be divided into 1) intermittent data assimilation where the observations are objectively analyzed along with the output from a numerical model in order to improve the estimates in data-sparse regions or to improve estimates of poorly sampled variables and 2) continuous data assimilation where data is ingested into a numerical model during a simulation. For our purposes we will refer to continuous data assimilation with the commonly used term four-dimensional data assimilation (FDDA). A variety of FDDA techniques exist, including Newtonian relaxation where an extra term is added to the prediction equation(s) in order to keep the model solution close to the observations and variational techniques that iteratively adjust the model forecast to observations by changing the initial and boundary conditions.
The utility (and perhaps even the superiority) of deriving an accurate representation of the true state of the atmosphere by combining model output with observations through some form of data assimilation is evident by the adoption of such techniques at operational forecast centers. However, there are still classes of problems in the atmospheric sciences where it is scientifically and/or philosophically preferable to derive an estimate of the state of the atmosphere that is independent of any numerical model through undertaking objective analysis of measurements alone. One example of this class of problems is special field projects designed to address poorly understood and poorly forecasted processes. These projects include major field campaigns such as GATE [GARP (Global Atmospheric Research Project) Atlantic Tropical Experiment] and TOGA COARE (Tropical Oceans Global Atmosphere Coupled Ocean–Atmosphere Response Experiment). A description of the objective analysis of the special and routine measurements taken during GATE and TOGA COARE can be found in Ooyama (1987) and Lin and Johnson (1996), respectively. An insightful discussion of the errors associated with the objective analysis of special datasets can be found in Ooyama (1987). According to Ooyama, one of the primary problems with such an objective analysis is that each observation is influenced by the full spectra of scales inherent in the atmosphere, while the measurement network can only resolve certain scales of motion. The aliasing of the smaller scales onto the resolvable scales is difficult to remove and will often result in significant errors in the analysis.
Another application where an objective analysis of measurements may be preferred is the Atmospheric Radiation Measurement (ARM) Program. An overview of ARM can be found in Stokes and Schwartz (1994). The project was founded to improve the understanding of radiative processes in the atmosphere and, in particular, the influence of clouds and cloud-radiative feedbacks. These processes represent a large source of uncertainty in climate research and modeling (e.g., Ramanathan et al. 1989). This uncertainty partly arises since the general circulation models (GCMs), frequently used to investigate climate prediction and global change issues, have extremely coarse horizontal grid spacing (∼200 km × 200 km), relative to the scale of moist vertical transports (∼1 km). Hence, clouds and their radiative feedbacks must be parameterized in the horizontal as a subgrid-scale process. In some cases, such as for thin cirrus clouds, the vertical grid spacings are also coarse relative to observed cloud features. Overview discussions on the parameterization of clouds and cloud-related processes in GCMs can be found in a number of studies including Slingo (1987), Sundqvist (1988), Teidtke (1988), and Hack (1994). A recent overview of the current priority issues in the parameterization of cloud-related processes in GCMs can be found in Browning (1994).
The observational strategy of ARM is to take long-term measurements that could be used to advance the understanding of cloud and radiative processes and to develop and/or test the parameterization of these processes in GCMs. The first measurement site deployed for ARM is the Cloud and Radiation Testbed (CART) located in Oklahoma and Kansas. As described in Stokes and Schwartz (1994), the CART domain covers an area roughly the size of a GCM grid box and a wide variety of variables are measured at the CART site, primarily at the CART central facility. One aspect of this measurement strategy is to use the special observations and other more routine measurements to obtain boundary conditions to the CART domain and to use these boundary conditions to drive numerical models. The model predictions could then be compared against the observed conditions within the CART domain in order to either verify or discount the parameterizations employed. The models employed in these tests range from cloud-resolving models to a “single-column model (SCM)” of a GCM.
Directly testing parameterizations over a limited area using observations for the boundary conditions has the philosophical advantage of the tests being independent of model behavior outside of this region. However, the primary difficulty with predicting conditions over such a limited area is that the quality of the solutions within the domain is highly dependent on the accuracy of the boundary conditions. Thus, although it may seem that the idea of providing measurements to test the parameterization of cloud and radiation processes is relatively straightforward, the accuracy requirements make the ARM goals extremely ambitious. Slingo (1987) also discussed the difficulties in attempting to use diagnostic techniques to develop prediction schemes for cloudiness based on large-scale variables and concluded that there was not an observational database available at the time that could be used to relate cloudiness to the large scale. The basic problem in this area for ARM is that the variables that could be used to predict cloud fields, such as relative humidity, vertical velocity, atmospheric stability, and surface fluxes, need to be representative of only those scales resolved by a GCM (scales of motion with wavelengths of many GCM grid points). In contrast, relative to the GCM-resolved scales, most available measurements are essentially point values, each containing a variety of temporal and spatial scales (e.g., Horst and Weil 1992). In addition, these point values are measured by an array of stations that can by design only resolve certain scales of motion (e.g., Ooyama 1987). In the case of the ARM project, most variables are not even measured over an area consistent with the spatial scales representative of the scales of motion present in GCMs.
In recognition of this problem, Mace and Ackerman (1996) attempted to quantify the errors associated with a “pure” objective analysis of the observations for ARM. A unique aspect of their study dictated by the goals of ARM was that they were interested in determining the mean conditions for and the boundary conditions to a very limited area that covered only 200 km × 250 km. In contrast, the TOGA COARE outer sounding array covered 20° latitude and 40° longitude. Another difference is that the TOGA COARE and GATE studies dealt with the Tropics, while Mace and Ackerman were concerned with the middle latitudes where there are far stronger advective changes and a stronger balance constraint between the mass and wind fields. The Mace and Ackerman (1996) study found that the errors due to unresolved scales of motion are often unacceptably large for the goals of the ARM project.
According to their study, the errors in the objective analysis are largest when strong spatial gradients or rapid evolution occur within the GCM grid cell, such as with convective or mesoscale organization. These conditions frequently occur over the southern Great Plains given the large number of articles in the journals discussing squall lines, severe weather, drylines, and frontal systems in this region. As noted by Mace and Ackerman (1996), these difficult conditions are often precisely the instances of primary interest for ARM. Other difficulties that ARM must overcome to meet their measurement objective include 1) some variables (i.e., vertical motion, upper-level relative humidity) are difficult to measure and/or include significant measurement errors; 2) to verify that the subgrid-scale effects predicted by the parameterization are correct, an accurate characterization is needed of the subgrid-scale processes of interest taking place over the area of interest; 3) understanding and verifying parameterizations for radiative transfers in a cloudy environment may also require microphysical observations, which are generally not routinely available; and 4) due to different measurement strategies and observational errors, it is difficult to maintain the dynamical and physical relationship between variables. In summary, despite ARM’s reliance on modern remote and in situ sensing technology (Stokes and Schwartz 1994), “the ARM observations will still be imperfect, incomplete, redundant, indirect, and unrepresentative, despite the best efforts at equipping the CART site with the best instruments” (Louis 1993).
The size of the errors in the Mace and Ackerman study led them to suggest that data assimilation be explored as a possible method to obtain an accurate and meteorologically complete estimate of the state of the atmosphere over these limited areas of interest to ARM. The use of FDDA systems for ARM goals has a number of advantages, including that it can maintain the physical and dynamic relationships between variables and derive a “complete set” of meteorological variables. This latter characteristic can help overcome the limitations associated with poorly sampled variables. FDDA techniques can also be used to keep the model solution close to the observations, thereby reducing potential model errors and biases. In view of these points it is not surprising that ARM has supported a variety of data assimilation efforts (Stokes and Schwartz 1994) and that current SCM efforts for ARM are using datasets produced from both FDDA systems and objective analysis that only use observations.
In this study, we will explore the advantages and disadvantages of data assimilation in determining conditions over a limited area in terms of the goals of the ARM project. While the goals of the ARM project are closely associated with testing parameterizations for cloud and radiative processes in GCMs, our results are also directly relevant to other meteorological applications associated with mesoscale modeling. We will use the approach of observing system simulation experiments (OSSEs). A unique aspect of these experiments is the use of high-resolution simulations in the OSSEs in environments with very large local departures in the flow due to features such as orography, a sharp cold front, and deep convection. The high-resolution datasets created in these environments also allow us to extend the results of the Mace and Ackerman investigation into the nature of the errors in objective analysis of the observations. A limitation of their study was that it dealt with errors due to scales of motion not resolved by the network but used observations with a horizontal resolution of 60 km. We also investigated their hypothesis that the magnitude of the mean error relative to an independent data point may be used to estimate the error in the spatial gradient terms.
The underlying theme of our work is to explore the applicability of one type of data assimilation to the estimate of conditions over a very limited mesoscale area and contrast it to objective analysis. Harm et al. (1992) note “data assimilation is relatively new on the meso-alpha and sub-alpha scales and needs considerable research with improved data sets.” One of many strategies for these scales is to use the previously mentioned Newtonian relaxation (often termed nudging) technique at finer and finer horizontal resolutions. For example, Kuo and Guo (1989) conducted OSSEs from sampling an 80-km-resolution simulation and conducting assimilation tests with grids ranging from 80 to 360 km, Kuo et al. (1993) conducted simulations using real data on a 40-km grid, while Stauffer and Seaman (1994) used a nested model in a real data test with a 10-km inner grid. According to the later study, nudging the simulation toward an analysis on the outer grids and toward individual observations on the inner grid improved the replication of the mesoscale flows on the inner grids. However, no network density experiments were carried out by Stauffer and Seaman as they used all the available observations.
While future experiments will deal with both real data tests and variational assimilation techniques, the relaxation technique (often termed nudging) is used in this current OSSE study. One motivation for this work is that despite the success of the Stauffer and Seaman study, a number of questions need to be addressed, including how this technique would work in the presence of extremely strong nonlinear gradients in space and time where the timescale of the relaxation is likely to be far slower than the timescale of the evolving flow. Under these conditions, one might expect the nudging to have small benefits or even degrade the accuracy of the solution. Another potential problem that arises with large gradients is data representativity and how the observations are assimilated in space and time. In this context, there may be distinct advantages in using a finer-scale model that better resolves the details of the flow and could therefore better match the observed flow. Other currently unknown issues include how the results of nudging in a nonhydrostatic model with active convection depend on boundary conditions, the values of the nudging coefficients, and network spatial resolution and extent. In this study we address these issues by using a nonhydrostatic model in the presence of active convection.
2. Model description
The model used in this study is the Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) mesoscale model. This model was described by Anthes and Warner (1978) and Anthes et al. (1987). The fifth-generation mesoscale model (MM5) is the primary model version used in this study, although some early simulations were conducted with the fourth-generation model (MM4). A more complete description of MM5 and how this model differs from MM4 can be found in Dudhia (1993) and Grell et al. (1994). Perhaps the most important modification is that MM5 is a fully compressible, nonhydrostatic model. The vertical coordinate with the nonhydrostatic option is a sigma reference-pressure coordinate. The basic model governing equations in this coordinate system are presented in Dudhia (1993) and include predictive equations for the three wind components, the pressure perturbation from a reference state, temperature, specific humidity, and the microphysical variables. The microphysical scheme includes predictive equations for cloud water, cloud ice, rainwater, and graupel. The scheme is based on the original MM4 bulk microphysical scheme described in Hsie (1984) and as modified by Dudhia (1989). The scheme has many similarities to the well-known Rutledge and Hobbs (1983) parameterizations.
The model includes an option for a radiative upper boundary condition based on the work of Klemp and Durran (1983) and Bougealt (1983). It also employs a surface energy budget described in Zhang and Anthes (1982) that uses land-use categories. The surface fluxes are treated using a revised version of Blackadar’s planetary boundary layer formulation (Blackadar 1976; Zhang and Anthes 1982). The model also has an option for simple longwave and shortwave atmospheric radiation packages based on broadband emissivity and single-stream integration, respectively (Chen and Cotton 1983; Dudhia 1989).
In these simulations, the model design includes a nesting capability with inner grids at increasingly finer horizontal resolution. The smallest horizontal grid used in this study was 5 km. These nested grids may be allowed to interact in a two-way fashion, but only one-way nesting was used in the tests presented in this study. At higher resolution, the model is designed to explicitly resolve convective structures, while on larger grids, the Grell (1993) cumulus parameterization scheme was used. A nonhydrostatic, high-resolution model with a fine inner grid was selected in this effort to avoid the direct dependence of the assimilation products on the parameterization of deep convection within the GCM area. Such a dependence obviously would weaken the potential for using these assimilation datasets in tests of GCM parameterizations for clouds as such a test would simply be a comparison between parameterization schemes in GCMs and mesoscale models. Convective parameterizations were also avoided since, despite significant effort and considerable ingenuity, there is still considerable room for improvement in parameterizing clouds and convection in numerical models as evidenced by the difficulties models have in predicting warm-season convective systems (Uccellini et al. 1994). However, despite the high resolution, the nested model configuration is not entirely free of convective parameterization on the finer grid as the boundary conditions to the finer grid would be impacted by convective parameterization if convection occurs in the outer grids. There are also model uncertainties due to other parameterizations such as for microphysical processes.
3. Data assimilation technique
The FDDA concept was introduced with the pioneering work of Charney et al. (1969). The motivation for the Charney et al. study was to reconstruct unobserved variables from inserting a set of observed variables into a numerical model during the simulation. Modern remote sensing instruments, such as the wind profilers, radiometers, and radio acoustic sounding systems (RASS) associated with the ARM observing network, are particularly suited for applications of this technique since they provide high temporal resolution measurements of a single meteorological variable. In this study we use the Newtonian relaxation or nudging approach as our FDDA technique. Nudging is described in Anthes (1974), Hoke and Anthes (1976), Kuo and Guo (1989), and Stauffer and Seaman (1990) and has also been applied recently in Kuo et al. (1993) and Stauffer and Seaman (1994). The basic idea of Newtonian nudging is to insert observations where and when available into a numerical model during a simulation to keep the simulation close to the true state of the atmosphere. The model can be nudged directly with the observations, often termed station nudging, or a three-dimensional analysis of the observations, termed analysis nudging. In either case, the nudging of the model is meant to account for slight inadequacies in model physics, initialization, or boundary conditions. A strength of this method for direct tests of parameterization schemes is that it allows for an imperfect model and attempts to keep the solution close to the observations. Model biases and drifts tend to be reduced using this technique in a limited-area model with prescribed boundary conditions. There is a dynamical aspect to nudging as well. As has been shown by Kuo and Guo (1989) and will be shown here, the model’s dynamic balance allows for correction of the mass field by reducing errors in the wind field. Furthermore, integrating the model forward results in observational information being carried downstream into data-sparse areas. Hence these techniques can be regarded as “dynamical analysis” methods.
The equations for the nudging technique are given in the appendix. From these equations, a number of inferences can be drawn concerning Newtonian nudging. For example, the method is relatively easy to implement and, in contrast to variational techniques such as adjoint methods or Kalman filters, does not carry a large computational burden. In addition, nudging can be applied to any predictive variable measured by remote or in situ sensors. In this study, the impact of nudging the horizontal wind components, temperature, and humidity will be investigated. From these equations, one can also see some potential limitations of nudging, such as the technique not being easily amenable to the incorporation of measurements of variables not directly predicted in the numerical model (e.g., radiances, radar reflectivities, rainfall totals, and refractive indices). In contrast, variational techniques can use these indirect measurements as constraints.
As described in Stokes and Schwartz (1994), the ARM CART site provides a variety of remote and in situ measurements that may be assimilated into the model. An important aspect of this site is that in addition to instrumentation deployed by ARM, the region is already well sampled by existing measurement facilities such as the Oklahoma and Kansas surface mesonets, NEXRAD (Next Generation Weather Radar) Doppler radars, National Weather Service Automated Surface Observing Systems, and the Wind Profiler Demonstration Network. The combination of ARM-deployed instrumentation with the existing measurement facilities in the area makes the measurement density extremely high over the area. The density of the measurement facilities and the occurrence of a mix of well- (e.g., wind) and poorly sampled (e.g., temperature and humidity) variables are partial motivations for the approach of using data assimilation for ARM. Another motivation is the shortcoming in the accuracy of objective analysis when the observed circulations are poorly resolved. It is worth noting that some datasets either currently or soon to be available in the ARM domain are taken by instruments, such as wind profilers, RASS, and Raman lidars, that were developed or refined within the last decade. This point illustrates the recent advancements in measurement technology (e.g., Dabberdt and Hardesty 1990) and how this study, by using modern remote sensors and high-resolution numerical modeling, takes advantage of two revolutionary changes in the atmospheric sciences.
From the nudging equations shown in the appendix, we can also see that in order to implement Newtonian nudging, a number of parameters, such as the nudging coefficient, radius of influence, and the time window must be specified a priori. Unfortunately, there is no general guideline for setting these parameters and very little practical experience (e.g., Stauffer and Seaman 1994) in using nudging in a nonhydrostatic model at high spatial resolution. The values of these parameters are crucial for the implementation of the nudging technique as one would like to have these parameters as large as possible so that the simulation closely follows the trends within the measurements, yet not too large so that the physical and dynamical relationship between these variables is destroyed. A strong departure from this relationship will ultimately produce spurious effects in the model fields that will limit the usefulness of the assimilation products. A timescale of the order of 1 h for the nudging coefficient was used for all these simulations based on preliminary experiments with the model (Kuo and Guo 1992).
In many ways, our high-resolution approach of obtaining a research dataset for the ARM characterization of the conditions associated with a limited measurement area is analogous to the previously mentioned use of data assimilation to create global datasets from the forecast centers to study global climate issues. Recent efforts in this area have centered on global datasets, including projects at the European Centre for Medium-Range Weather Forecasts (Bengtsson and Shukla 1988), the National Meteorological Center (recently renamed the National Centers for Environmental Prediction; Kalnay and Jenne 1991), and the Goddard Space Flight Center (Schubert et al. 1993). These efforts are aimed at providing datasets that are useful to global climate studies rather than concentrating on the ARM objectives of characterizing the cloud and radiative processes and their associated forcing parameters over a limited measurement area. Through our investigation into this problem, we also examine the benefits and drawbacks of using Newtonian nudging as a practical data assimilation technique for mesobeta scales.
4. Results from observing system simulation experiments
As mentioned in the introduction, a number of questions arise in implementing nudging in a nonhydrostatic simulation. One technique that can be employed to help answer these questions is to assimilate “observations” obtained from other independent numerical simulations. Ingesting output from one model simulation into another simulation is termed an observing system simulation experiment. The method we will adopt in these OSSEs begins with first generating the so-called observations with the model and treating this simulation as “truth.” The data were selected for assimilation from this truth simulation to imitate different observing strategies, such as different station densities and different types of measurements (i.e., wind profilers or rawinsondes). In some cases we used extremely dense and perhaps unrealistic networks. These networks were included since they clearly illustrate the limitations of the assimilation techniques as opposed to the limitation of not adequately resolving the observed flow. The data were then assimilated using nudging into independent simulations on the same or coarser grid sizes with different boundary conditions. By comparing these independent simulations against the truth simulation, one can estimate how much improvement we can hope to have with a particular observing system strategy (Kuo and Guo 1989; Kuo et al. 1993).
The results from OSSEs are generally thought to be optimistic in comparison to assimilating actual measurements for a number of reasons including that the observations taken from the model are dynamically balanced and free of measurement error. Also, both the truth simulation creating the observations and independent simulations used in the assimilation often have generally the same or similar model numerics and physics. In addition, the OSSEs implicitly assume that the model is sufficiently accurate to well represent the observed flow. One should keep these points in mind in the following discussions. In this study, we created “independent” simulations by using different model grid spacings and different initial and boundary conditions. The degree of independence will be subsequently discussed in the following sections.
In these OSSEs, three methods of verification (rms errors, biases in the mean, and precipitation threat scores) were calculated using the truth simulation against the FDDA simulations and the control simulation (without FDDA). When the simulations were conducted at different resolutions, the results were degraded to the coarser grid by averaging, and the rms errors and the threat scores were calculated over the domain of the finer grid. For example, sets of hourly fields with 20-km grid spacing were typically generated for the truth, the control, and the data assimilation simulations. The root-mean-square error and mean error of the fields in the domain were compared to the truth after the fields had been interpolated to pressure surfaces at 200, 500, and 800 hPa. The fields assimilated and tested included the two components of the horizontal wind, the temperature, and the water vapor mixing ratio. The rms errors are both a measure of the ability of data assimilation to represent features that would be subgrid scale in a GCM and any changes in the GCM-resolved scale flow. In order to investigate the ability of an assimilation system to estimate the GCM grid box mean, the results from the FDDA and control simulations were averaged over the area of the finer-mesh domain and compared to the mean for the truth simulation over the same area. This difference was calculated for the same fields as were undertaken for determining the rms errors.
Since ARM obviously has a strong interest in cloud fields, we also tested the ability of the simulation to improve precipitation estimates. This test is stringent because the truth simulation used the nonhydrostatic model at 5-km, which generated finescale structures in the precipitation amounts that would be difficult to reproduce using a 20-km grid. The precipitation threat score was calculated taking a threshold of precipitation, 1 mm for instance, over a time period (3 h) to define an area that exceeds the threshold. Three areas are calculated: (A) where the truth exceeds the threshold, but the test simulation does not; (B) where both truth and test exceed the threshold; and (C) where the test exceeds the threshold, but the truth does not. The threat score is then given by the conjunction area (B) divided by the union (A + B + C), so that 1 corresponds to a perfect forecast and 0 corresponds to a completely missed forecast.
Observational nudging will be used here as it is more appropriate to the smaller scales and asynoptic data involved in the special networks we are considering (Stauffer and Seaman 1994). Two cases were chosen for these tests. One was a wintertime cold frontal passage in Colorado, and the other a springtime severe storm outbreak in Oklahoma. The OSSEs based on these cases cover similar areas of interest as the ARM CART domain with similar possible observing strategies. These two OSSEs represent diverse conditions, both in terrain and in the scale and types of the dynamics involved. The tests are relatively stringent in that the situations contain intense circulations that would be subgrid scale in a GCM. In contrast, the results of Mace and Ackerman (1996) were not able to address the impact of small-scale variability in their experiments since their work used a 60-km horizontal grid.
a. Springtime severe storms, 10–11 April 1979, Oklahoma
1) Experiments
The first OSSE deals with a squall line that formed over the southern Great Plains and was associated with a severe tornado outbreak during SESAME (Severe Environmental Storms and Mesoscale Experiment) in 1979. This outbreak included a devastating tornado in Wichita Falls, Texas, that caused 42 deaths around 1800 CST 10 April. Kuo et al. (1993) give an overview of this case and describe experiments that assimilated column-integrated estimates of water vapor into the model. To help establish some meteorological perspective on this case, the modeled accumulation rainfall for 0900–1200 UTC 11 April is shown in Fig. 1a. To generate truth simulations at 20- and 5-km grid size for this experiment, a hydrostatic simulation from Kuo et al. was used as a basis. Their simulation used a 40-km horizontal grid and included data assimilation of all available observations including special soundings from the SESAME program. A 20-km nested truth simulation, with 49 × 57 grid points, was conducted over the same period (1200 UTC 10 April–1200 UTC 11 April), taking hourly boundary conditions from this 40-km simulation and using the nonhydrostatic option. Finally, a nested truth simulation at 5 km was also derived that covered a 500-km square centered on Oklahoma and was initialized at 1800 UTC 10 April using the 20-km simulation for initial and hourly boundary conditions. These simulations produced results that compared well with the rainfall observations.
Degraded boundary conditions for the independent control simulation came from a different 40-km simulation by Kuo et al. (1993) that did not use data assimilation and, instead, relied upon only standard observations and 12-h boundary conditions from analyses. In a procedure analogous to the truth simulations, a nested 20-km control simulation was created that took initial and hourly boundary conditions from the degraded simulation. Subsequently, a nested 5-km control simulation was initialized from the 20-km control at 1800 UTC 10 April. In other simulations, data from the truth simulation was assimilated into the control simulation for the FDDA tests. Table 1 summarizes the various simulations. In Table 1, the IC, BC, and FDDA columns denote whether a simulation, analysis, or observation was used to provide the initial condition, boundary conditions, or FDDA data, respectively. Also, C, D, and E denote series of assimilation experiments with different amounts of data assimilated or with the assimilation parameters varied. For series C and D the control simulation is B2, and for E the control is B3. The results from these series were verified against the truth simulations, using threat scores for rainfall, rms, and mean errors on selected pressure levels. Unless stated otherwise, the data assimilated were hourly wind, temperature, and moisture. The nudging coefficient was kept the same for all of these experiments. The following section is a summary of the principal results.
2) Results
The first series (C) consisted of experiments in which data from the 20-km truth was assimilated into “independent” 20-km FDDA runs. Two idealized profile networks, with a 5 × 5 and a 10 × 10 regular square grid with spacings of 200 and 100 km, respectively, were defined to cover the entire 20-km domain defined in Fig. 1a. For the sake of brevity, we will only show the rms errors for the u component of wind and the temperature in this section. Since the magnitudes for the u component are larger than for the υ component, one should consider the u results to be a “worse-case scenario.”
The general improvement in the simulations with data assimilation, as is evident by the surface rainfall accumulations, can be seen in Fig. 1. In this figure, the control simulation without data assimilation (Fig. 1b) clearly underestimates the intensity of the convective system in the truth simulation (Fig. 1a). The general envelope of precipitation is more closely replicated when data from the 5 × 5 (Fig. 1c) and 10 × 10 (Fig. 1d) networks were assimilated. Another general conclusion that can be drawn from these surface rainfall distributions is that although the simulations using assimilation more closely resemble the overall distribution for the truth simulation, there are differences, particularly in the location of heavy rainfall areas within this envelope. This behavior can be interpreted to say that the convective system is correctly replicated but that differences in initial and particularly the boundary conditions make it extremely difficult to predict the exact locations of individual active convective cells within that envelope. One would expect this behavior given the highly nonlinear behavior of convection.
The rms errors for series C are shown in Figs. 2a and 2b for the 5 × 5 and 10 × 10 networks, respectively, where both wind and temperature profiles were assimilated. In this case, the rms errors were zero at the initial time since the two simulations were initialized from the same analysis. For later assimilation experiments, the initial conditions will also be independent. However, since the two solutions diverged with time as evidenced by the rms errors increasing throughout the experiment, one can conclude that the boundary conditions were sufficiently independent and that small differences in the boundary and initial conditions can lead to largely divergent solutions in this convective environment. Another result is that a marked improvement was found with the finer 10 × 10 network, with the rms wind error often decreasing twice as much compared to the control experiment without data assimilation.
Two additional simulations series C experiments were carried out on the 10 × 10 network with just wind or temperature assimilated at all the stations. The results of Kuo and Guo (1989) that were confirmed in that assimilating wind alone helped improve temperature rms errors (Fig. 3b). In fact the improvement was almost as much as assimilating temperature alone, but assimilating both wind and temperature was clearly better. Conversely, temperature assimilation had almost no improvement on the winds’ rms errors (Fig. 3a). An understanding of this behavior can be obtained from knowledge of geostrophic adjustment and how nudging is implemented. The assimilation of data into a model can be thought of as imposing a disturbance on the solution constrained by model physics. How the model reacts to a disturbance depends on the temporal and spatial scales of the disturbance. In this formulation, the nudging coefficient imposes an inverse timescale on the forcing, and as stated earlier this coefficient is defined to be small enough so that the incorporation of data into the model will not generate spurious gravity waves. Hence, when data are incorporated into the model, the energy associated with the disturbance will not be dissipated by gravity waves and the model will adjust to a new solution. Assimilating wind improves the pressure and thermal patterns by causing vertical circulations that bring the thermal field, and hence the pressure, into better geostrophic agreement with the winds.
From the results of temperature assimilation it appears that the reverse correction is rather less effective. The temperature is assimilated into the thermodynamic energy equation. The temperature changes should impact the wind field through the hydrostatic pressure via the momentum equations. Assimilating the correct temperature may improve the estimate of the vertical wind shear, but since the temperature does not completely define the pressure (mass) field and its spatial gradients, the information is not sufficient to improve the total estimate of the wind. A full specification of the mass field requires other parameters, primarily knowledge of the surface pressure. We would therefore speculate that assimilation of the surface pressure in addition to thermal fields would improve assimilating the wind field.
Series (D) tested the capabilities of assimilating data from a concentrated network in a limited area and thus used only the 5-km truth run to generate the data for assimilation into the 20-km simulation. In contrast to the 20-km simulation that employed a convective parameterization scheme, this high-resolution simulation resolved broad areas of ascent in the 20-km simulation into separate updrafts and was conducted without a convective parameterization on the 5-km grid. The difference in physics between these two simulations makes this assimilation experiment particularly challenging, as the solutions should be more independent. Data networks for this set consisted of 2 × 2, 3 × 3, 4 × 4, and 5 × 5 square arrays spread over 500 km (Fig. 1e), corresponding to spacings between 250 and 100 km. An additional simulation was undertaken on the 3 × 3 network assimilating only 3-h data instead of hourly. The simulations were initialized at 1200 UTC 10 April, and data assimilation did not begin until 1800 UTC 10 April when the 5-km simulation started. Therefore only results from 6 to 24 h will be shown.
The threat scores for the various precipitation thresholds and verification times for the 5-km domain are shown in Table 2. The mean threat scores calculated through averaging over time for each of the threshold levels from these experiments is shown in Fig. 4. These results indicate improvement in threat scores that is clearly evident when more data are assimilated into the model. The greatest improvements are noted at higher thresholds where the skill was initially far lower in the control experiment. The lower skill at higher thresholds is partly due to the small area covered by higher rain rates. In addition, the lower skill at higher rainfall rates in the assimilation experiments is consistent with Fig. 1 where data assimilation improved prediction of the rainfall envelope, but replicating the exact location of heavy rainfall events is more difficult. Evidently for this experiment the assimilation does have some impact on the forecast of where the stronger convective cells are most likely to occur. The improvement in the threat scores increases with time, which indicates that the assimilation is effective in reducing the drift of the model solution from the truth simulation. For example, the improvement at 24 h was the result of the data assimilation correcting the strength of a squall line near the eastern boundary of the domain, which was too weak in the control simulation without assimilation. The improvement that results from assimilation is also evident as hourly ingestion of data resulted in higher threat scores than simulations where data was assimilated only every 3 h (D333).
The threat scores in these experiments were comparable but slightly higher than for operational models, particularly at the lower thresholds for the higher scores with data assimilation. For example, the simulations presented in Kuo et al. (1997, manuscript submitted to Wea. Forecasting) had threat scores below 0.50. While the assimilation process generally improves the simulation, it should be noted that a careful examination of the threat scores in Table 2 reveals instances where assimilating additional data was a detriment. For example, some times can be found where the hourly assimilation experiments had lower scores than the 3-h insertions and where higher density networks had lower scores than less dense networks. Also, the mean values shown in Fig. 4 indicate a lower mean threat score for the 5 × 5 network than for the 4 × 4 network, despite threat scores that generally increase with network density and insertion frequency. The possibility of this behavior occurring with nudging in convective environments due to data representativity was raised in the introduction. These tests suggest that although this problem does occur, the general effect of assimilation on predicting the precipitation field is favorable.
The corresponding rms scores for series D (Fig. 5) calculated over the inner 200 km2 of the 5-km domain also show the relatively large improvements for 2 × 2, 3 × 3, and 4 × 4 networks for wind and temperature, with a comparatively small added improvement for 5 × 5. The latter point, which is consistent with the mean threat scores shown in Fig. 4, is particularly true of the 200-hPa temperature verification. The magnitude of the errors for this limited area experiment were generally similar to the larger-scale experiments in series C. The mean differences between the truth and the simulations in experiment series D calculated through averaging conditions over the entire 5-km domain are shown in Figs. 6a and 6b. Assimilation of the data from the 5-km truth simulation generally improved the estimates of the means with similar tendencies as for the rms errors. In a finding similar to the rms results, the largest mean errors were found in the 200-hPa wind fields. However, comparison of Figs. 5 and 6 indicates that there are instances where the rms errors are dominated by the mean error. For example, at later times, the rms field for u at 200 hPa seems to be dominated by a mean error, which we found to be associated with an underestimation of the impact of the outflow of the storm’s updrafts over a large area of the domain. It is also evident that at lower layers the mean errors can be quite small with relatively large rms errors (e.g., see the u fields at lower levels). This type of situation again reinforces the hypothesis that the model often may be unable to produce the correct local details of cellular convection at a given time and place. The general mean errors, with the exception of the winds at 200 hPa, are quite reasonable with temperature errors of 1 K or less and wind errors of about 1 m s−1. These magnitudes are similar to measurement errors in the atmosphere.
In series D, data from a 5-km simulation was assimilated into a 20-km grid, with considerable differences in physics due to convection being resolved directly on the finer grid. Series E, which consisted of simulations at 5-km grid size that assimilated data from the 5-km truth run, the same as used in series D, allows one to determine the relative importance of this difference in physics. The boundary conditions came from the control run in series D unless otherwise stated. Again, these simulations covered the 18 h of the truth run. The rms errors for this test series are shown in Fig. 7 and are evaluated in the central 200-km square of the 500-km domain. The striking finding was that errors appeared to be comparable or even less improved by data assimilation than for the corresponding series D undertaken on the 20-km resolution grid. We hypothesize that this result was due to differences in the 5-km domain’s boundaries between the series D and E tests (see Table 1). The boundary conditions in the 5-km control and truth simulations used in series E were specified from a control simulation. In the 20-km simulations used in series D, the radius of influence employed in the nudging extended the influence of the assimilation beyond the borders of the 5-km truth simulation (the radius of influence varied between 100 km at low levels and 200 km at high levels). As confirmation of this hypothesis, the rms errors in series E were found to decrease as one included more of the boundary zone (not shown). This result suggests that the accuracy of boundary conditions are crucial for an accurate simulation over limited domain such as the ARM CART site.
As with series C, tests were performed with wind nudging only and temperature nudging only to determine the extent to which the nonnudged fields were retrievable. This time temperature nudging alone had little positive impact on the wind field (Fig. 7a) especially at high levels, while wind nudging alone had no significant improvement on the temperature (Fig. 7b). These results indicate that for wind nudging to be beneficial, the domain and data coverage need to encompass a broad enough scale. The 500-km domain is too small compared to the Rossby radius (of order of 1000 km for deep modes) to produce secondary circulations that retrieve the correct balance.
A further test that produced an informative result was to apply no nudging but to use boundary conditions that were identical to the truth simulation and the initial conditions the same as in the control and the assimilation tests. After 7–8 h of the 18-h simulation, this simulation started to significantly outperform the rms scores of the FDDA experiments, and the improvement continued until rms errors were less than half those of the FDDA simulations at 18 h (Fig. 8). The rainfall pattern at the end of the simulation was in almost perfect agreement with the truth simulation. This result again shows how deterministic the simulation is with respect to its boundary conditions, as even dense data coverage over the area does not overcome the lack of quality boundary conditions. Thus a multiscale data assimilation approach is desirable over local data assimilation, supporting the conclusion of Stauffer and Seaman (1994). This result is not surprising in view of the short advection timescale across the domain (∼5 h) compared to the simulation length of 24 h.
Further tests were done on the assimilation parameters: the nudging coefficient, the radius of influence, and the time window. The latter two had little sensitivity around the values chosen, and both can be adequately estimated from the spatial and temporal density of the observations. The nudging coefficient represents an inverse relaxation timescale and its optimal value is not so obvious. The test simulation increased its value by a factor of 10 from 0.0003 to 0.003 s−1. However, while the test run showed reduced rms scores for the upper-level winds, the winds of other levels and the temperature were degraded (Figs. 8a and 8b). The results also showed unacceptable levels of noise in the rainfall pattern and other fields. A similar result was found by Kuo and Guo (1992). This finding shows that nudging with a timescale that is too short will generate high-frequency modes, gravity waves, that have unrealistic amplitude. Clearly the nudging’s forcing frequency must not overlap these modes and should be more consistent with the frequencies of balanced circulations. Hence, one would not expect that nudging will produce satisfactory results in reproducing the details of finescale rapidly evolving circulations except for instances where nudging can produce improved replication of mesoscale circulations that may lead to the model more accurately generating these finescale circulations itself.
b. Wintertime front, 13 February 1990, Colorado
The second case investigated is a frontal case described in more detail by Warner et al. (1992) and Rasmussen et al. (1995) in a study using the PSU–NCAR Mesoscale Model at 5-km grid size to represent a cold frontal passage through Colorado. Later this front was associated with the Valentine’s Day storm, a heavy snowfall event that occurred across much of the Midwest. The event took place during the Winter Icing and Storms Program (WISP) of 1990 (Rasmussen et al. 1992). A sharp cold front formed during the simulation period between 0000 and 1200 UTC 13 February 1990 as cold air advanced from the northeast (Fig. 9a), decreasing the temperature 20–25 K in a 12-h period. This cold pool was shallow and its advance was slowed by the Front Range of the Colorado Rocky Mountains. From the discussion in Warner et al. (1992) it is evident that the 5-km simulation captured mesoscale features associated with the interaction of the front with severe orography and the front’s observed “collapse” in spatial scale over the plains.
The “truth” simulation for these experiments was a simulation with a 5-km horizontal grid spacing that was generated in three stages. First, a hydrostatic simulation at 40 km was conducted for 48 h beginning at 0000 UTC 12 February. The initialization and assimilation of this simulation only used the National Weather Service (NWS) standard observations. The domain covered the 48 contiguous United States with boundary conditions derived from 12-h analyses. In the second step of generating data for this OSSE, this 40-km hydrostatic simulation provided initial and hourly boundary conditions to a nested nonhydrostatic simulation with a 20-km grid starting at 1200 UTC 12 February and ending at 1200 UTC 13 February. This simulation covered the area 1900 × 1600 km shown in Fig. 9a. Finally, nested within this 20-km simulation was a 5-km nonhydrostatic simulation initialized with hourly boundary conditions from the 20-km grid, starting at 0000 UTC and ending at 1200 UTC 13 February. The 5-km domain covered most of Colorado (Fig. 9a) and resolved the front more finely than the previous coarser simulations (Fig. 9b). This 5-km simulation was taken as truth for the purposes of the OSSEs with datasets generated every hour. The hourly “data” taken from this simulation were assimilated into the subsequent independent simulations. Table 3 summarizes these simulations.
The control simulation without FDDA and assimilation tests for the OSSEs in this case were hydrostatic simulations at 20-km grid size. These simulations were initialized from an analysis based on standard observations at 0000 UTC 13 February. The boundary conditions were based only on linear interpolation of the 0000 and 1200 UTC analyses, while the hourly boundary conditions for the truth simulation were obtained from the 40-km simulation. Thus these simulations are largely independent of the 20-km truth simulation mentioned above, as the standard analysis and boundary conditions provide a very smooth structure at 0000 UTC compared to the 12-h forecast from the previous 20-km run.
To test the sensitivity of the assimilated fields to the station spacing, three regular networks of data profiles were defined in the truth simulation assuming 3 × 3, 4 × 4, and 5 × 5 square arrays uniformly spaced at 167, 125, and 100 km, respectively; for example, see Fig. 9c. These spacings were within the range of the previous set of OSSEs. Each of these profiles sampled the truth simulation’s wind, temperature, and moisture at a single grid column each hour. Since all three primary meteorological variables are sampled, the rawinsonde would be the sensor analogy to this OSSE strategy, although a collection of remote sensors could also provide this data.
The rms errors for wind, temperature, and water vapor mixing ratio are shown in Fig. 10. The errors at the initial time show that the simulations employed in the OSSEs are sufficiently independent as the maximum errors initially ranged up to nearly 8 m s−1, over the 3°C, and over 1 g kg−1, respectively. From the rms errors for the u component of the wind (Fig. 10a), it is evident that even the coarsest data assimilation run (3 × 3) gave a significant improvement over the control simulation as errors were generally decreased by half through data assimilation. Little or no improvement in the rms errors was gained in increasing the network resolution; a 5 × 5 network produced errors only marginally less than the 3 × 3 network. The general trend for an improvement in the quality of the simulation through assimilation was independent of height and generally resulted from correcting the control simulation’s positioning of the front, which was too far south.
The rms scores for the temperature field (Fig. 10b) also support a general improvement in the model estimates through employing assimilation. However, these scores also reveal a potential problem in assimilating data into models with coarse horizontal resolution, such as those with a 20-km grid size, as data assimilation shows marginal improvement, if any, in the temperature field over the control at 200 hPa, which is in the lower stratosphere. The reason for this behavior is a fundamental problem in data assimilation, namely, data representativity. In the truth simulation, structures in the flow field respond to sharp meteorological features, such as the cold front, and terrain features in the Rocky Mountains that are resolved with a 5-km grid but are poorly represented at 20-km grids. Sampling the high-resolution simulation and forcing a coarser-resolution run with this data can sometimes degrade the results, because the resulting higher-resolution flow is inconsistent with the coarser grid and smoother terrain. In particular, gravity waves aloft will be aliased by the sampling, and the forcing may not lead to the correct balanced state in the assimilation runs.
The problem of representativity arises not only in OSSEs, but also in real data applications of data assimilation. Often models cannot resolve features such as mountains, valleys, or small lakes that may have an impact on an observation. If this observation is assimilated, it will be spread over an area where the measurement becomes unrepresentative. For this reason preprocessing of data is an important step in assimilation. The data must be checked against that at neighboring sites and thrown out if found to be unrepresentative of model-resolved scales with respect to a given tolerance. Stauffer and Seaman (1994) describe the method used in the PSU–NCAR Mesoscale Model to prevent near-surface data from being spread too far in mountain-valley regions. However, as seen from the 200-hPa temperature error, high-level effects of terrain or fronts also may lead to assimilation problems, and these are not as predictable, so in practice the use of a neighbor check may be the only way to filter or remove unrepresentative data values. This finding also argues for the use of a high-resolution model in data assimilation to make full use of datasets by resolving small-scale features that may affect measurements locally.
The impact of the assimilation on the water vapor mixing ratio can be seen in Fig. 10c. Here the mixing ratio is normalized by typical saturated values at each level and the error is expressed as a percentage. In general, the rms errors show a similar result to the wind, temperature, and humidity fields in that data assimilation from a modest network significantly reduces the errors.
The corresponding mean differences between the truth simulation and the control and FDDA experiments are shown in Figs. 11a–c. These results in general show an improvement in the estimation of the mean that can be obtained through assimilating data into the simulation and that this improvement can come about through modest observing networks. The one primary exception is again the behavior of the temperature field at high levels. Thus, the behavior of the mean errors is qualitatively similar to that of the rms errors. From comparison of Figs. 10 and 11, one can conclude that there are both instances where the mean error contributes substantially, such as with 200-hPa temperature, and times where the mean errors are small with relatively large rms errors, suggesting that smaller-scale errors are primarily contributing to the total rms errors. It is instructive, however, to compare the magnitude of these biases to measurement error. In this controlled experiment, the wind biases are of order 1 m s−1, which is the same magnitude as the experimental errors in most rawinsonde methods and for wind profilers. The temperature bias in the simulation is often of order 1 K, which exceeds the 0.5-K error that is often quoted for rawinsonde accuracy when factors such as instrument error and time constant are considered. The bias in humidity in this case is small, but it often exceeds the rawinsonde accuracy of approximately 4% in relative humidity and the additional errors of about 10% can be observed due to data representativity (Paukkunen 1995).
From Fig. 9b, the precipitation is primarily on the west slope of the Rockies. It can be seen from the third verification method using threat scores that the skill generally improves both with the amount of data assimilated and with time (see Table 4). The average over time and threshold amount for the control and the three data assimilation experiments indicates a clear increase in skill with the amount of data assimilated (Fig. 12). This finding supports the general trend observed in the rms errors that a modest observational network makes a significant improvement in the quality of the simulation. For high thresholds, skill is low in part due to the small areas involved in this particular event.
5. Discussion
The previous experiments investigated the behavior of applying nudging in hydrostatic and nonhydrostatic mesoscale models and the dependence of these results on network density, measurement frequency, network scale, and the values of the free parameters in the nudging process (i.e., radius of influence, nudging coefficients). In the following subsection, we will begin by summarizing these idealized data assimilation tests and then discussing them in the context of the measurement goals of the ARM project. In the subsequent subsection, we will conduct a series of simple experiments using model output to test the hypothesis that the needs of the ARM project could instead be met through a simple objective analysis of the observations. It is hoped that these objective analysis and data assimilation experiments will illustrate the strengths and shortcomings of applying these methods to meet the measurement needs of the ARM project. The final subsection briefly describes a field project carried out at the CART facilities in order to further address these issues.
a. Data assimilation
From these OSSEs we can infer a number of points concerning the use of nudging in a nonhydrostatic mesoscale model. The first point is that nudging can improve the quality of the simulation and help keep the model closer to the observed state of the atmosphere even in a nonhydrostatic model with fine grid scales. The improvements occur in the model estimates of the wind, temperature, and moisture fields. Improvements in threat scores indicate direct improvements in the prediction of the location and intensity of precipitating cloud systems. Although threat scores have been used in testing assimilation schemes (e.g., Stauffer and Seaman 1990), our testing the dependence of precipitation prediction on assimilation at these scales is relatively unique. These improvements occur even in the presence of significant small-scale variability through sharp fronts, convective clouds, and mountainous terrain and at 5-km model grid spacing. One might instead expect the nudging technique to be affected by data representativity problems resulting from spreading the influence of local observations in nonlinear conditions over considerable distance and time in a high-resolution model. While we did find these problems to occur in convection and when the flow interacts with orography, the negative impacts were overwhelmed by the benefits.
The general positive results at grid spacings as small as 5 km extend the findings of earlier OSSE studies on far larger grids. The more recent Stauffer and Seaman (1994) study used observations on a 10-km horizontal grid. Since their study used all the available observations, Stauffer and Seaman were unable to investigate the dependence of their findings in network density. Our tests of varying the network density indicated that a significant portion of the benefit from the assimilation can come from a few observing sites. As expected from past studies, there are positive effects of nudging just the wind field, but we found these effects occur only when observational network covers a sufficiently large area. The later result supports the nested approach to improving a mesobeta-scale forecast as proposed by Stauffer and Seaman (1994).
A number of issues concerning the implementation of nudging on the mesobeta scale were also clarified by this study. For example, the results were found not to be strongly dependent on radius of influence or the time window. In our study, we generally used the same value as Stauffer and Seaman (1994) in their mesoscale assimilation experiments. Our tests showed that the dependence on the value of the nudging coefficient was rather strong with degraded results due to high-frequency modes occurring with coefficients that were set too high (i.e., at 0.003 s−1). This result illustrates a fundamental limitation of nudging in that the nudging coefficient must be small enough so adjustments to the simulations occur through near-balanced circulations. The limitation in how strong a model can be nudged has been well known since the problem of generating spurious small-scale motions arose from the direct insertion of observations into coarse-grid prediction models. However, the current result illustrates that even if these small scales may be properly resolved by the model and the observational network, the nudging coefficient must be small enough so that a near balance is maintained when the model is nudged. We suspect that the limitation of nudging maintaining a balance also arises in the issue of how increasing the observational density can produce diminishing improvements and even occasional detrimental effects.
As expected for such a limited area, the estimate of conditions within an area similar in size to a GCM grid box is extremely sensitive to knowledge of the boundary conditions. From these idealized experiments, it appears that even modest errors in the boundary conditions, especially for highly nonlinear situations such as deep convection, may produce divergent solutions, especially in the rms field. The large rms errors are likely due to data representativity problems with the nudging technique and predictability problems coupled with the fundamental problem of nudging having a slow adjustment time. For example, always predicting the right convective cell in the right location at the right time obviously would be unlikely given the strongly nonlinear behavior of convective systems, which cannot be corrected by nudging. In this regard, the rms errors in divergence were not significantly improved by data assimilation. Despite the large local rms errors in these situations, the general precipitation envelope as evidenced by the threat scores and many mean effects of convection over the region of interest, such as the correcting the sign and approximate magnitude of mean divergence error, appear to be correctly simulated. New information regarding the implementation of nudging in mountainous terrain was also discussed.
From these OSSEs we can also infer a number of points concerning the use of nudging in a mesoscale model in relationship to the ARM goals of testing and developing parameterizations. First, it is evident that data assimilation can produce significant improvement in the accuracy of a numerical simulation. The tests were rather stringent in that these cases contained significant circulations that would be subgrid scale in a GCM using simulations that were fairly independent. The improvement is such that the mean error over a GCM grid box approaches measurement error and at times the magnitude of this degree of improvement extends to the rms errors. The accuracy of a simulation extending over limited spatial areas should also show strong dependence on the boundary conditions for coarser grid models, since the magnitude of physical processes and the timescale for an air parcel to traverse across the domain in the atmosphere are essentially independent of the model grid spacing. Hence, we suspect that other proposed ARM strategies for testing parameterizations, such as SCM tests where the boundary conditions are prescribed by observations (Stokes and Schwartz 1994; Randall et al. 1996), would also show a similar strong dependence of the accuracy of the result upon the accuracy of the boundary conditions. The problem of accurately predicting the boundary conditions for a limited area such as a GCM grid cell is a fundamental difficulty that ARM must overcome if parameterization tests are to be successful, especially given the findings of Mace and Ackerman (1996) and our tests to be discussed in the next subsection.
From these assimilations and past experiments, when the measurements take place over larger areas, it is evident that a significant improvement can be obtained from wind measurements alone. This point is important since the ARM CART site is located within the center of the current U.S. demonstration network of more than 30 wind profilers (U.S. Department of Commerce 1994). These profilers report hourly winds and have a spacing of approximately 250 km. The usefulness of wind profilers in intermittent data assimilation has been shown in Smith and Benjamin (1993). Other wind measurements are routinely available over this area from ACARS (ARINC Communication, Addressing, and Reporting System), which produces flight-level wind measurements from commercial aircraft at about 8000 measurements per day (U.S. Department of Commerce 1994). These and other data are intermittently assimilated into the MAPS (Mesoscale Analysis and Prediction System) (Bleck and Benjamin 1993) on a 60-km grid. In contrast, our OSSEs used only a fraction of the available data so that in practice greater improvements may be possible through nudging the model with more data (or MAPS output) or through using assimilation schemes that can utilize types of measurements that are not predictive variables in the model. According to our experiments, the location of the CART site within this high density of wind measurements has the potential for greatly increasing the accuracy of data assimilation for the CART domain through improving the boundary conditions. The current plan to supplement these profiler sites near the CART domain with RASS measuring virtual temperature is consistent with the need for regional measurements of wind and temperature as the maximum potential for improvement occurs from assimilating both temperature and wind estimates into the model.
The primary unknown for the use of data assimilation systems in providing input to testing cloud and radiation parameterizations is accuracy, particularly whether continuous data assimilation could reduce any model bias to a significant degree. In these experiments, the results were mixed as the mean errors over an area consistent with the CART site were often small, but the errors at some times and heights did point to the data representativity issue, a shortcoming of nudging. However, in the winter case, the accuracy of the assimilated fields even as measured by the rms errors actually approached measurement error. We are especially encouraged by this result since we believe that these two OSSEs were relatively challenging tests of the use of data assimilation as the meteorological situations contained large horizontal gradients, significant small-scale variability, and rapid evolution due to active deep convection, fronts, and orography. Another difficulty of these tests is that they were often using only local data assimilation and were faced with the difficulties in overcoming errors in boundary conditions over very limited areas. In addition, the simulations did seem to be sufficiently independent to test the effectiveness of data assimilation, although judging the independence of the simulations relative to operational errors is not very meaningful since operational forecasts are currently conducted at far larger grids.
b. Objective analysis
FDDA is only one strategy for determining the state of the atmosphere for testing parameterizations. An alternate strategy is to simply use the observations or an objective analysis of the observations to characterize the atmosphere. This hypothesis was explored for ARM activities by Mace and Ackerman (1996). Their conclusion was that the fields diagnosed from objective analysis most likely would be inaccurate when strong spatial and temporal gradients are present, which unfortunately are generally the conditions of interest. The Mace and Ackerman (1996) study was somewhat optimistic in that the influence of small-scale phenomena on the accuracy was not explicitly addressed, as their OSSE data used a 60-km grid. The data from our OSSEs is well suited to examine this influence, since simulations were conducted with a 5-km grid in the presence of significant small-scale variability, especially for the OSSEs based on the 10–11 April 1979 convective system. Hence, we will further explore whether observations alone or an objective analysis can meet the ARM goals of characterizing the conditions over domains, such as the CART site, and contrast these results with the insights obtained about FDDA from the OSSEs.
Perhaps the simplest way of determining the mean wind, temperature, and humidity over the CART site and the spatial gradients of those variables (i.e., divergence and advection) would be to use a few rawinsonde sounding sites near the boundaries and calculate these quantifies directly. This method, which can be classified as a simple form of objective analysis, can be tested taking data from a 2 × 2 network constructed using the 5-km truth simulation for the 10–11 April 1979 OSSEs. The separation between observations for this 2 × 2 network is 200 km. The current ARM strategy of collecting data for testing parameterizations is to conduct periodic intensive observation periods (IOPs) using rawinsonde ascents from a central site and four sites near the boundaries to the CART domain. This OSSE is similar to the ARM IOP rawinsonde strategy, especially if the observations at the central site are not used in the objective analysis but instead are reserved for verification. In this experiment creating data from a 5-km model grid would in general underestimate the impact that small-scale variability may have on a point measurement (e.g., rawinsonde) but somewhat overestimate the small-scale variability that might be present in an hourly average from a remote sensor, such as a wind profiler. With present technology, it is extremely difficult and costly to obtain accurate vertical profiles of water vapor from an array of remote sensors so that the incorporation of rawinsondes into the measurement strategy is necessary if one wishes to only use observations.
The results of the simple tests of the 2 × 2 network to capture the mean wind, temperature, divergence, and temperature advection for the domain are shown in Fig. 13. Examination of Fig. 13 leads us to conclude that this simple technique does fairly well at characterizing the mean u (Fig. 13a) and T (Fig. 13b) over the domain with the general time tendency well represented and the estimates of the means generally being within a few meters per second and 1°C of the truth, respectively. For comparison of this simple method to data assimilation, the reader is referred to the mean errors shown in Figs. 6a and 6b. In many regards these objective analysis fields are quite promising, especially when compared to the model wind estimates at 200 hPa. The domain mean estimates of the divergence of the horizontal wind (Fig. 13c) and the advection of temperature by the horizontal wind (Fig. 13d) would appear to be acceptable in a qualitative sense but are significantly less promising for quantitative purposes as the errors are large and the fields sometimes have the wrong sign. Our results are quantitatively similar to the general trends observed and predicted by Mace and Ackerman (1996) in that the mean fields were well represented, but the higher-order terms contain significant errors.
The error magnitude can be quantitatively illustrated through examining the rms errors in the temperature advection and the derived vertical motions as a function of height (Fig. 14). From this calculation it is evident that the rms errors in the temperature advection is on the order of a few kelvins per day (Fig. 14a). The magnitude is even more concerning given that this error is independent of any measurement errors. In order to judge the impact that measurement errors could have on the estimates of the mean, these calculations were repeated with the addition of random errors. For example in Fig. 13e, we have repeated the calculation of the mean temperature advection for a number of iterations where random errors that ranged up to 1 m s−1 and 1°C were added. As expected, the measurement error adds further additional uncertainty, and this uncertainty in many instances is as large as the actual signal. Also, depending upon the stability, errors in the vertical motions could also directly result in errors in the temperature prediction within the domain for a dry atmosphere or far larger errors in a moist situation if convection was erroneously initiated or suppressed.
The induced thermal changes due to these errors appear to be a similar magnitude as or even likely to exceed the thermal changes due to radiation in an SCM. Hence, it appears safe to conclude that the errors in these gradient terms are unacceptable if one is quantitatively testing predictions of radiational cooling/heating rates or trying to calculate such parameters as cloud fraction or hydrometeor content, especially given the sensitivity of the solution to lateral boundary conditions. Given the differences between the tropical and middle-latitude atmospheres, one would expect the advective terms and errors to be less important when applying this technique to the Tropics due to the lack of both strong baroclinicity and larger Rossby radii. Thus it should be far easier to conduct SCM simulations for the Tropics than for data taken at the CART site. The corresponding errors in the vertical motion (Fig. 14b), while important, seem to be less of an issue as these errors are reduced from knowing the boundary conditions in the integration of the divergence field to 1–2 cm s−1. On another optimistic note, it is possible that time filtering may reduce the advective errors to tolerable proportions. Time filtering for SCM simulations is also desirable, since the lateral boundary conditions would be more consistent with scales represented in a GCM. Current work in ARM focuses more in detail on this issue. Examination of our data suggests that time filtering would in some cases reduce the errors as the objective analysis error varies in time around the true estimate (e.g., temperature advection at 200 mb in Fig. 13d prior to 17 h), while time filtering will not help as much in other cases since the sign of the error stays consistent throughout the simulation (e.g., divergence at 500 mb in Fig. 13c). Time filtering is also not beneficial with low temporal frequency, such as the 3-h rawinsonde ascents used in the ARM IOPs that currently must be depended upon for a complete depiction of the thermodynamics.
The large magnitude of the errors may be surprising given that they occur even for perfect observations where aliasing is the primary error source. However, the importance of aliasing in objective analysis has been noted previously (i.e., Ooyama 1987). One problem with the objective analysis approach is that the presence of aliasing is difficult to detect, except for instances where local effects clearly dominate, such as when a rawinsonde ascends through a convective cell. Mace and Ackerman (1996) have proposed that differences in the winds measured at a central site and a velocity averaged over four boundary sites could be used as a guide as to when to reject the higher-order fields. If their hypothesis was true, then one would expect some strong correlation between the errors in the divergence estimate and differences between the mean wind speed and the wind at some central point. A casual examination of the relationship between these variables as a function of time at 19 levels (Fig. 15) clearly shows that such a correlation is not very strong at best so that their proposed error check is not likely to be useful in this situation and a linear regression between these two variables only accounts for 8% of the total variance. In some ways one could expect that the observed behavior we identified as adding random noise to a function (or simply superimposing two functions of a similar magnitude) can result in instances where the functions have the same or similar values but extremely different derivatives. Hence, while this approach will work at times, it is evident from Figs. 13 and 15 that there are clearly instances where the mean fields are accurate while the gradients are poorly estimated. While the proposed relationship does not hold in this test, it is possible that tests with far larger statistical samples might find their method to be more useful. It is also possible that smoothing the higher-order terms in time could also improve this relationship.
A more accurate use of the observations might be to use a denser array of stations and perform an objective analysis of the fields. In order to compare FDDA versus an objective analysis, we again used the OSSE simulations of the 10–11 April 1979 convective system. The particular OSSE employed in this case was series D (Table 1). In this test, observations were taken from the 5-km truth simulation with networks varying from 2 × 2 to 5 × 5. The station spacing for these tests was between 250 and 100 km. The objective analysis was performed on a 20-km grid. The objective analysis package is the multiquadratic technique described in Nuss and Titley (1994). The “first guess” background field was obtained from the control simulation. At very high station densities, the fields produced by objective analysis would be based primarily on observations. At lower station densities, the fields would be more similar to an intermittent data assimilation where the observations are being used to correct either initial fields derived from a model or large-scale analysis.
For brevity we will show only the divergence fields from this test for FDDA (Fig. 16a) and for objective analysis (Fig. 16b). Other derived higher-order fields that were examined behaved in a qualitatively similar manner and, again, with the mean fields were more closely repeated. From a comparison of the two plots it is clearly evident that objective analysis has a far greater dependence on station density than FDDA. An examination of Fig. 16 also suggests that in a relative sense, the objective analysis performs poorly relative to FDDA in estimating the derivative terms for the coarser networks (primarily the 2 × 2 and 3 × 3 arrays) but is quite accurate for the denser arrays. For example, at coarse station densities, the estimates due to objective analysis are very noisy and sometimes of the wrong sign. This result indicates that as the station density approaches a 100-km spacing, the circulations for this large mesoscale convective system are becoming well resolved and objective analysis performs quite well. For other convective events and meteorological situations, the scale at which the event can be resolved will vary. Some caution is urged with this point, however, as the aliasing from small scales has been reduced when the 5-km results are averaged to 20 km and since we did not include measurement error in this test. These influences would similarly decrease with increasing station density.
c. Description of a field test at the CART site
Results presented here are rather idealized but are also intended to give guidelines for practical applications of FDDA and objective analysis for ARM. Our experiments show that assimilating measurements from a modest observing network will improve the quality of the predicted fields. The question that naturally arises is how close an approximation are these fields to the true state of the atmosphere. Unfortunately, it is not possible to definitively answer this question through simply using OSSEs. In one sense, the results should be treated with caution, since OSSEs are thought to provide overly optimistic results for the reasons discussed earlier. However, it is equally valid to propose that the results may be overly pessimistic since the differences between initial fields in the independent simulations (at times several tens of meters per second) may be far larger than the errors in the fields used to initialize the model prediction. We make this statement due to the high density of observations, particularly in the wind field, present over this region. In order to answer this question, a practical test of this data assimilation concept for the ARM project began with a 10-day field experiment conducted from 16 to 26 June 1993. The experiment was conducted over northern Oklahoma and southern Kansas as a special IOP of the ARM project. In addition to these measurements, the standard observing network was supplemented with three additional Cross-chain Loran Atmospheric Sounding System (CLASS) sounding sites and two Integrated Sounding Systems (ISS) (Parsons et al. 1994) with 915-MHz wind profilers. The ISS were located several kilometers from the central ARM site near Lamont, Oklahoma. The locations of the instrumentation are also shown in Fig. 17. CLASS soundings were made every 3 h from the three CLASS sites and at either the ARM central facility or one of the nearby ISSs. In addition supplementary rawinsonde launches were taken every 6 h from the three neighboring NWS sites at Dodge City, Topeka, and Oklahoma City. During this 10-day experiment, a variety of meteorological conditions were observed, including frontal passages, deep convective activity, nocturnal low-level jets, and severe weather. The period of study corresponds with the beginning of heavy rainfall associated with the widespread flooding that took place over the central United States in the summer of 1993. A 10-day numerical experiment based on this dataset has been conducted. In these experiments, the data from the three CLASS sites are not assimilated but used as “truth” for simulations, with and without data assimilation, to quantify the potential improvement from applying data assimilation. Current work focuses on determining the accuracy of FDDA for real data situations and using these fields to test GCM parameterizations. These studies will be described in future work.
6. Conclusions
In this study we examined how Newtonian nudging using measurements from observational networks similar to the ARM CART domain could improve the accuracy of a nonhydrostatic, full physics numerical model. Our findings show that significant improvements could be realized through wind measurements over synoptic-scale areas and through wind and temperature measurements over limited areas. After an initial adjustment period, the data assimilation fields are in dynamic balance, which is an advantage over intermittent data assimilation techniques.
The ARM measurement strategy is well served by its location within the profiler demonstration network through providing hourly wind observations over broad areas of the central Midwest as the OSSEs point to the need for accurate boundary conditions. The special ARM measurements within the CART domain will provide some additional improvements in the model estimates of the true state of the atmosphere. The use of data assimilation as an ARM measurement strategy has a number of advantages over objective analysis including the following.
Providing a “complete set” of meteorological variables that is currently unavailable from objective analysis. For example, recent work by Grabowski et al. (1996) has proposed that neglecting hydrometeor advection into a domain, as may be done with single-column modeling, results in significant mean errors over areas larger than a GCM grid box.
Having reduced model biases and errors to the point where there is the potential to outperform objective analysis for higher-order fields, such as divergence, advection, and vorticity, when the data networks are sparse and there is significant small-scale variability present.
Being able to clearly detect when the solution does not fit the observed fields by keeping track of the model versus observational differences when the model is nudged.
Providing a smoothly varying, time continuous representation of the higher-order fields that are in physical balance.
Being able to use one observational field to derive another, as evident from the improvements in the derived temperature fields when wind measurements from a sufficiently large area are assimilated. In this regard the wind profiler network and ACARS winds are extremely useful.
Providing gridded fields at temporal and spatial scales that resolve most atmospheric variability on the scale of a typical GCM grid box.
Being able to spread the influence of the observations downstream with time.
The primary disadvantages of FDDA system discussed herein relative to the other techniques are the following.
The assimilated fields are directly dependent upon model characteristics and parameterizations. For example, although we used a fine grid in many of these tests, the boundary conditions are still influenced by convective parameterizations on the coarse outer grids. In addition, parameterizations are used for the surface fluxes, radiation, boundary layer, and microphysics. For this reason, the data assimilation fields should always be treated with caution.
Objective analysis is likely to outperform FDDA even for higher-order fields for dense observational arrays. The number of sites needed will depend upon the degree of small-scale variability (16–25 sites for this case).
Objective analysis is computationally very easy to implement.
For very poor initial or boundary conditions, this type of FDDA may not be able to sufficiently correct the accuracy of the derived fields.
The assimilated fields are not, strictly speaking, in balance due to the nudging terms. However, to overcome this limitation, one could use a nested approach where nudging is only implemented on the outer grids and the GCM-scale grid is left to satisfy the model equations without the additional nudging terms.
In order to provide boundary conditions to drive a coarse grid model such as an SCM, the assimilation field must be filtered in space to be consistent with GCM scales of motion. However, one would suspect that this spatial filtering should exceed the accuracy of filtering objective analysis in time, assuming it is equivalent to a spatial filter.
Some philosophical, but admittedly speculative, points can be made concerning the ARM measurement strategy based on these tests and previous work such as the recent study by Mace and Ackerman (1996). First, we feel that an ARM IOP strategy of using only a few rawinsonde sounding sites in the vicinity of the CART site to predict large-scale tendencies will likely fail when significant small-scale variability is present. The location of the CART site in a region where the warm-season convection is dominated by mesoscale convective systems, and where there is a climatological tendency for stationary fronts and drylines to be present, will guarantee the frequent presence of small-scale variability. The IOP strategy at the CART site may be improved by nearby wind profilers and the presence of some profilers with RASS systems. Hence for winds and, to a lesser degree, temperature (particularly above the boundary layer where the RASS virtual temperatures can be translated into sufficiently accurate temperatures), the situation is certainly improved over the ARM IOP sounding array. However, the errors may still be large for the CART site, since the spacing of the profiler sites lies between that found in our 2 × 2 and 3 × 3 station arrays. As mentioned earlier, the lack of measurements of humidity and microphysical parameters over what is provided by the rawinsondes is also a shortcoming. In view of these limitations, if an objective analysis approach is used, ARM may wish to strongly consider expanding the number of rawinsonde sounding sites and correspondingly decreasing the number of IOPs. Time filtering of the data or some other strategies aimed at reducing these errors will likely need to be implemented for SCMs and other models driven by these boundary conditions. On the other hand, FDDA has been shown to work well with sparse data, producing realistic 4D variability. Further exploration is warranted in making full use of available indirect data sources (radar, satellite, radiometer) with advanced 4D variational data assimilation techniques.
Acknowledgments
This work is supported by DOE-ARM Grant DE-A105-90 ER61070. We would also like to thank the ARM personnel and management team responsible for the central site, R. Cederwall and M. Dickerson for their help in designing this experiment, and the NCAR field staff for making this field deployment a success. W. Dabberdt, B. Kuo, Y.-R. Guo, and C. Davis (all of NCAR) are also to be acknowledged for their involvement in the design of these experiments and for numerous discussions. Discussions with J. Hack (NCAR), J. Petch (NCAR), D. Randall (Colorado State University), S. Krueger (University of Utah), J. F. Louis (Atmospheric and Environmental Research, Inc.), and S. Ghan (Pacific Northwest Laboratory) were useful in order to understand the needs of climate modelers and SCM strategies. M. A. Pykkonen and L. Lorusso patiently typed and edited the numerous versions of this manuscript.
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APPENDIX
Formulation of the Data Assimilation Technique
Shaded contours of 3-h rainfall accumulation for the 10–11 April 1979 storms for the truth simulation and the series C experiments: (a) truth, (b) control, (c) 5 × 5, and (d) 10 × 10, as described in the text. The rainfall fields are shown for 0900–1200 UTC 11 April. Levels of shading indicate 2-, 10-, and 25-mm rainfalls. (e) The station locations for the 3 × 3 network used in experiment series D.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
The differences between the truth simulations and the predicted values for the control and FDDA simulations expressed in term of rms errors for experiment series C. The time series are shown for three heights (850, 500, and 200 hPa). The solid line is the control simulation, and the short- and long-dashed lines the values for assimilating data from 5 × 5 and 10 × 10 networks. (a) The rms errors for the u component of the wind and (b) for the temperature.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
As in Fig. 2 but for assimilating temperature only (short-dashed line), wind only (long-dashed line), and wind and temperature (dot–dash line). The control is once again the solid line and (a) and (b) once again indicate the u component of the wind and the temperature rms errors.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
A three-dimensional bar graph of the mean threat scores derived from taking the values over the 5-km grid shown in Table 2 and averaging over time for each of the experiments and threshold levels in series D. The threshold level labeled mean is derived from taking the average over all the time-averaged threshold levels.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
As in Fig. 2 but for experiment series D. The solid line is the control and the dashed lines indicate the station density ranging from a 2 × 2 (shortest-dashed line) to a 5 × 5 network (longest-dashed line). (a) and (b) The rms errors for the u component of wind and the temperature, respectively.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
The mean errors for series D for (a) the u component of the wind and (b) the temperature for the control simulation and the four data assimilation experiments as shown in Fig. 5.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
As in Fig. 2 but for experiment series E. The solid line is the control, the dotted line (labeled 33T) is for assimilating wind and temperature measurements for a 3 × 3 network, the short-dashed line (labeled 55) is for assimilating both measurements for a 5 × 5 network, the longer-dashed line (33V) for the error for assimilating wind data only with a 3 × 3 network, and the longest-dashed line (labeled 33T) is for assimilating temperature only for a 3 × 3 network. In this figure (a) is the u component of the wind and (b) is the temperature.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
As in Fig. 7 but with the dot–dash line (labeled CB) being the simulation with improved boundary conditions and degraded initial conditions and the dashed line (labeled 33NC) indicating the simulation with a larger nudging coefficient.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
Near-surface temperature (K) for a 20-km simulation at 1200 UTC 13 February 1990. The contour interval is 2 K. The 5-km domain is indicated by the heavy solid line and the state outlines are also shown. Negative values are dashed and positive values indicated by solid lines with contours labeled every 4 K. The latitude and longitude in degrees are indicated on the outside of the domain for the 20-km grid. From Warner et al. (1992). (b) The 5-km simulated temperature and 3-h rainfall with shading levels of 1, 2, and 5 mm, and (c) location of the stations in the 3 × 3 network.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
(Continued)
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
As in Fig. 2, the solid line is the control, the short-dashed line is the 3 × 3 network, the medium-dashed line is the 4 × 4 network, and the long-dashed line is the 5 × 5 network. The errors are plotted as a function of time for 12 h starting at 0000 UTC 13 February 1990. Three heights (800, 500, and 200 hPa) are shown for each field: (a) the u component of the wind, (b) the temperature, and (c) the normalized water vapor mixing ratio (see text). Only the two lower levels are shown for the water vapor plots.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
As in Fig. 10 but the errors are mean differences averaged over the 5-km domain.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
A three-dimensional bar graph of the threshold scores plotted as in Fig. 4 but for the WISP simulations.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
The model-predicted behavior of (a) the u component of wind, (b) temperature, (c) divergence, and (d) temperature advection as a function of time (solid lines). The simple hourly estimates provided by a 2 × 2 network are shown by the dashed lines. (e) The addition of instrumental error on the temperature advection (d) is indicated by the dashed lines.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
(Continued)
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
(a) The mean rms errors in the estimation of the temperature advection term plotted as a function of height. (b) The rms errors in the estimation of the vertical motions. While the largest contribution to these errors in the aliasing of unresolved scales, interpolation errors also contribute to this large error.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
A scatterplot of the differences between the estimation of the mean winds and the wind at the center of the domain versus the error in the estimation of the divergence.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
(a) FDDA and (b) objective analysis estimates of mean divergence. The thick solid line is the divergence estimated from the truth simulation. The thinner solid line is the model divergence for the independent simulation. The series of dashed lines are the model and objective analysis results for increasingly dense arrays of measurements as in the previous formats.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
A schematic illustrating the instrumentation associated with the ARM CART site during the IOP for data assimilation.
Citation: Monthly Weather Review 125, 10; 10.1175/1520-0493(1997)125<2353:OSSEAO>2.0.CO;2
Simulations performed on the SESAME 10–11 April 1979 case.
Threat scores for 3-h periods ending at 9–24 forecast hours, for various thresholds (cm) in the five simulations; no FDDA (NOF); 2 × 2, 3 × 3, 4 × 4, 5 × 5 array FDDA runs; and a 3 × 3 array, 3-h FDDA run.
Simulations performed on the WISP 1990 12–13 February case.
Threat scores for 3-h periods ending at 3, 6, 9, 12 forecast hours, for various thresholds (cm) in the four simulations: no FDDA (NOF), 3 × 3, 4 × 4, and 5 × 5 array FDDA runs.
