1. Introduction
A long-standing problem in meteorology is to estimate, or retrieve, the state of the atmosphere in some domain given observations of reflectivity and radial velocity from one or more Doppler radars. (Alternatively, one may speak of assimilating these observations.) Doing this requires an algorithm that infers from the observations those variables not directly observed, such as vertical velocity and temperature, and this in turn requires additional information beyond the observations themselves. Such information is available, in principle, from our knowledge of the governing equations, and various techniques have been devised to utilize the governing equations in assimilation or retrieval.
This paper explores for the first time the use of an ensemble Kalman filter (EnKF) to assimilate single-Doppler radar observations in a cloud-scale model. The EnKF is a novel and flexible technique for data assimilation, first proposed in the geophysical literature by Evensen [1994; but see also Leith (1983, 375–377)]. The EnKF uses forecast covariances between observed and unobserved variables to spread information from the observations both spatially and between variables. These covariances are estimated from an ensemble of prior forecasts initialized when observations were last available. Section 2 provides further background on the EnKF.
Other approaches for analyzing the atmospheric state given radar observations fall into two categories: retrieval algorithms and four-dimensional variational assimilation (4DVAR). Retrieval algorithms typically begin by estimating the wind field given observations of radial velocity from one or more radars. Since only certain components of the velocity are observed, the entire velocity field is estimated using additional information or constraints (such as continuity, the prognostic equation for radial velocity, or the evolution of the reflectivity field). Given knowledge of the wind field, the thermodynamic variables are then estimated from the vertical momentum equation after first calculating pressure perturbations at each level from the horizontal momentum equations. Recent results from different algorithms, as well as earlier references, can be found in Xu et al. (2001), Montmerle et al. (2001), and Weygandt et al. (2002).
Four-dimensional variational schemes seek to fit a numerical simulation to observations spread over a time interval by adjusting the state at the beginning of the interval. Like retrieval algorithms, 4DVAR has shown practical value in estimating unobserved variables given radar observations (Sun and Crook 1997, 1998). Unlike the more empirical retrieval algorithms, however, 4DVAR analyzes all variables in a unified fashion. 4DVAR also allows the systematic treatment of observation error and information from recent forecasts, but covariance matrices for both the observation and forecast error must be specified. Further background on 4DVAR appears in Sun and Crook (1997), while Rabier et al. (2000) discuss the implementation of 4DVAR for an advanced numerical weather prediction model.
The EnKF has several appealing properties that motivate its consideration as an alternative to retrieval algorithms or 4DVAR. Like 4DVAR, it is a statistical scheme that handles uncertainty in the observations and the prior forecast gracefully and approximates, subject to certain assumptions, the Bayesian update for the forecast state given new observations. The EnKF also provides direct estimates of the forecast covariances from the forecast ensemble and then explicitly updates that ensemble to be consistent with the uncertainty of the analysis. Thus, because it produces not only an analysis (the ensemble mean) but also an ensemble of initial conditions, the EnKF is a natural foundation for ensemble forecasting schemes. This is particularly important at convective scales, where probabilistic forecasts may be warranted even at lead times of an hour. Finally, the EnKF is both relatively simple to implement, as tangent linear and adjoint versions of the forecast model are not required, and relatively straightforward to parallelize.
While applications of the EnKF to large-scale flows have progressed nearly to the point of operational testing (Mitchell et al. 2002; further references appear in section 2), similar successes are not guaranteed at convective scales, where the motions are fully three dimensional, are driven by distinctly nonlinear microphysical processes, and lack the approximate balances between the mass and wind fields, such as geostrophy, that pertain at larger scales. In order to test the EnKF at convective scales and with radar observations, we extract simulated radial-velocity observations from a numerical simulation of a splitting supercell and then use the EnKF to assimilate those observations. A detailed description of the numerical experiments appears in section 3.
The results of these experiments show that the EnKF has strong potential for convective-scale assimilation; specifically, the state of the simulated supercell can be accurately estimated given an initial sounding for the domain and several volume scans of radial velocity. Section 4 presents these results, together with some discussion of the characteristics of the forecast covariances. Since supercells are robust solutions, in the sense that a variety of initial conditions can and will produce qualitatively similar supercells given an appropriate environmental sounding, one might worry that prediction and data assimilation for a system of supercells are anomalously easy. We show in section 5 that forecast errors in fact grow significantly even in our idealized system. Section 6 addresses tuning the filter, the importance of the cross covariances between radial velocity and temperature or moisture or rainwater, and the role of the initialization of the ensemble. We summarize in the final section and outline a number of interesting issues that are not addressed in this initial study.
2. The ensemble Kalman filter
This section provides further background on the ensemble Kalman filter. We begin with a heuristic look at the central ideas in section 2a, then informally review the Kalman filter (section 2b), and finally detail the specific ensemble technique employed here (section 2c). Readers familiar with the EnKF may wish to skip directly to section 3, while those interested in a more complete and rigorous introduction to estimation theory and the Kalman filter are referred to Cohn (1997). Further discussion of aspects of the EnKF, together with results for large scales in the atmosphere, can be found in Houtekamer and Mitchell (1998), Hamill and Snyder (2000), Anderson (2001), Whitaker and Hamill (2002), and Mitchell et al. (2002). There is also a substantial oceanic literature; see Brusdal et al. (2003), Keppenne and Rienecker (2002), and references therein.
Except for the dimensions of the state and observation vectors, our notation will follow that of Ide et al. (1997).
a. Basic notions
Suppose that at some time, t = tk, and over some region of interest, we possess an ensemble of forecasts that are representative, in a sense to be made more precise below, of the forecast uncertainty. Typically, the ensemble-mean forecast is the best prediction of the atmospheric state at tk. Now suppose we receive an observation valid at the same time tk. Our problem is then to assimilate the observation or equivalently to update our prediction of the state and its uncertainty given this new observation. While the observation obviously carries information about the measured quantity, we wish also to extract information about the unobserved portion of the state. How can this be done?
Figure 1a portrays the situation schematically. For definiteness, suppose that the observation is of radial velocity υr at some location and the unobserved variable is vertical velocity w at another location. The forecasts from each ensemble member for υr and w are shown as a scatterplot; in this case, the sample of ensemble members suggests that υr and w are positively correlated in the forecast. Our best forecasts of each variable, the ensemble means, are indicated on each axis by the thin arrows. The observed value of υr is assumed to be 14 m s−1 and is marked on the vertical axis by a thick arrow, along with the specified observational error distribution.
Intuitively, we expect that the best estimate for υr should lie between the observed value and the mean forecast, and that the uncertainty in this updated, or posterior, estimate should be smaller than that of either the observation or the forecast. In addition, it is clear that refining our estimate of υr provides information on w; the updated estimate of w should increase, since υr and w were positively correlated in the forecast and the observation indicated that υr was larger than was forecasted.
The updated situation given the observation is shown schematically in Fig. 1b. The updated ensemble is again displayed as a scatterplot and the updated means indicated by arrows on each axis, while the forecast and observed quantities from Fig. 1a are shown in gray. As intuition suggested, the updated ensemble has less spread in both variables, reflecting the additional information provided by the observation; the updated estimate of υr lies between the forecast mean and the observation; and the mean of w has increased from the forecast value. The EnKF, which provided the updated ensemble in Fig. 1b, will be discussed below after a brief review of the Kalman filter.
b. The Kalman filter
Let x of dimension Nx be the state of the system in some discrete representation, such as values on a regular grid. Thus, x consists of all gridpoint values for all variables concatenated into a single vector of length Nx. For the purposes of this paper, we will ignore the often prickly issues that surround projecting the continuous atmosphere onto a discrete representation and accounting for the uncertainties associated with that projection (see Cohn 1997).
Since we have a limited set of imperfect observations, the true state of the system, denoted by xt, cannot be determined precisely. It is therefore convenient to consider xt to be a random variable; the most that can be known about the system is then p(xt), the probability distribution function (pdf) of xt. Our goal becomes to estimate and forecast p(xt) given the available observations.
We will use one further property of the Kalman filter to simplify the implementation of the EnKF below. If the observations (1) have independent errors, then the update (2), which treats all observations (i.e., each element of y) simultaneously, is equivalent to serial assimilation of individual observations through repeated application of (4). To be more precise, one can assimilate the first element
Finally, it is important to emphasize that the Kalman filter update is optimal only in the case of linear observation operators and Gaussian errors [see (1) and the associated discussion]. Convective-scale dynamics, however, are often nonlinear, in part because of the importance of latent heat release and other microphysical processes. Thus, forecast pdfs will potentially be non-Gaussian and the Kalman filter update will at best approximate the mean and covariance of p(xt | yo).
c. The algorithm
As discussed above, the Kalman filter is a scheme for updating p(xt) given observations yo. The idea underlying ensemble filtering techniques is to work with a sample, or ensemble, drawn from p(xt) rather than with p(xt) itself. This notion was first proposed for data assimilation in the geophysical literature by Evensen (1994) but is of course at the heart of ensemble forecasting as well.
As an aside, we note that this algorithm unifies ensemble forecasting and data assimilation. The ensemble of forecasts provides the statistical information, required for data assimilation, concerning the uncertainty at a given time and location and the relations between observations and state variables. The analysis step, in turn, explicitly constructs an appropriate analysis ensemble to serve as initial conditions for subsequent forecasts.
1) The analysis step
In the analysis step, observations are processed serially, as described above. For each observation, the algorithm first calculates the mean of the prior ensemble from (5) and the difference of each member from the mean. The ensemble mean is then updated according to (4a), but replacing ci and d with sample covariances calculated from the ensemble.
In practice, the update is restricted to those state variables that are within a certain radius from the observation location. We do this because state variables far from the observation location typically have small covariances ci with the observation variable; at large distances, the sampling error (i.e., the error incurred by estimating covariances from a finite sample) then becomes comparable to or larger than ci unless the ensemble is very large (Houtekamer and Mitchell 1998; Hamill et al. 2001). Most previous applications of the EnKF have restricted the influence of observations in the horizontal but not in the vertical; here, for convective-scale motions that are not quasi–two dimensional, we allow observations to influence only state variables within a sphere of given radius. Besides the substantial computational savings, the resulting algorithm also performs better than if each observation influenced all state variables (Houtekamer and Mitchell 1998; Anderson 2001; Hamill et al. 2001). Houtekamer and Mitchell (2001) present a more sophisticated approach to restricting the influence of an observation on distant state variables; this will be discussed further in section 6b.
Two additional points deserve comment. First, this ensemble-based algorithm asymptotically approaches the Kalman filter update in the limit of a large ensemble and Gaussian distributions; differences arise only from the approximation of covariances by their sample values. Second, since (7) is deterministic, the ensemble produced by this scheme is not strictly a random sample from p(xt | yo), and the ensemble is perhaps better thought of a set of states whose sample covariance approximates 𝗣a.
2) The forecast step
Given the analysis ensemble, the forecast step simply involves forecasting each member forward to the time of the next available observations. This procedure is a Monte Carlo approximation to the computationally overwhelming propagation of the full pdf p(xt) forward in time, at least to the extent that the analysis ensemble is a random sample from p(xt). Moreover, if the forecast dynamics is nonlinear, this procedure generalizes the covariance propagation step of the extended Kalman filter, in which the analysis error covariance matrix is propagated by the tangent linear dynamics based on a linearization about the nonlinear forecast trajectory from the analysis mean. If the forecast dynamics is linear, the forecast step, like the analysis step, approaches that of the Kalman filter in the limit of a large ensemble.
In practice, forecast models are typically imperfect and the forecast step should account for such imperfections, either through the addition of noise to each member after the forecast (see, e.g., Mitchell and Houtekamer 2000) or through the incorporation of stochastic noise terms in the model itself. The simplified experiments presented here, however, all assume a perfect forecast model.
3. Description of the assimilation experiments
We will test the EnKF using simulated observations of radial velocity from an isolated supercell thunderstorm. In the experiments, a numerical model first produces a reference solution of the supercell. Simulated observations are then constructed by adding random observational error to the radial velocity from the reference solution, and those observations are assimilated using the EnKF and the same numerical model. These experiments suffice to demonstrate the feasibility and potential of the EnKF for convective-scale data assimilation, but it is important to note that, in any practical implementation, we will have neither a perfect forecast model nor complete knowledge of the observation errors.
a. The reference solution
The reference solution begins with a warm, moist bubble in a horizontally uniform environment; that is, u, υ, θl, and qr vary only with height outside the bubble, and w is zero. The environmental sounding (Fig. 2) is based on the Oklahoma City sounding from 0000 UTC 25 July 1997, where 7 m s−1 has been substracted from the zonal wind in order to minimize the movement of the right-moving supercell through the domain. The warm bubble initiates a convective cell, which first forms rain after 20 min of simulation. As is common for soundings as in Fig. 2, the initial cell splits into a strong primary supercell that moves to the right of the environmental shear and a weaker, secondary supercell moving to the left of the shear. Splitting of the initial cell occurs at about 55 min, and the left-moving cell passes from the computational domain after 100 min. Snapshots of the reference solution will be shown in section 4.
The numerical model used in the reference simulation (and in the assimilation experiments) is that of Sun and Crook (1997) and is documented in detail there.2 Briefly, the model solves the nonhydrostatic equations using θl, the liquid-water potential temperature, as the thermodynamic variable and including only warm-rain microphysics. The equations are discretized spatially using second-order centered differences and a second-order Adams-Bashforth time step is used. The lateral boundary conditions depend on the sign of the normal component of velocity at the boundary. Where there is flow into the domain, gradients normal to the boundary are computed by assuming that, outside the domain, each variable is given by the environmental sounding, while at out-flow boundaries normal gradients are computed using one-sided differences.
For all simulations, the computational domain is a 70-km × 70-km square in the horizontal with 2-km grid spacing and extends from surface to 17 km in the vertical with 500-m grid spacing. The origin of the Cartesian coordinates (x, y, z) is taken to lie at the lower left (southwest) corner of the domain.
b. The experiments
Simulated Doppler radar wind observations are extracted from the reference solution as follows. We assume that 1) the radar is located at the southwest corner of the computational domain, at (x, y, z) = (0, 0, 0); 2) it measures υr, the radial velocity in a spherical coordinate system centered on the radar; 3) the observations have independent, Gaussian random errors of zero mean and variance R = 1 m2 s−2; and 4) observations are available at 5-min intervals and at each grid point where the rainwater qr > 0.13 g kg−1. While observations of radar reflectivity undoubtedly will provide useful information in practice, they are not assimilated in the present experiments for simplicity.
The assimilation experiments begin at t = 20 min, when rain first begins to form in the reference simulation. Observation sets, consisting of υr at all points with qr exceeding the threshold given above, are then assimilated every 5 min thereafter. The analysis variables are the same as the forecast model's prognostic variables (u, υ, w, water vapor, and qr). A typical observation set includes O(103) inividual observations.
The initialization of the ensemble begins with the environmental sounding shown in Fig. 2, which is assumed known. (Estimation of the environmental sounding is an outstanding problem for convective-scale forecasting and will be addressed in a subsequent study.) Each ensemble member is initialized at t = 0 by adding realizations of Gaussian noise to the environmental sounding. This noise is independent at each grid point and for each variable, has zero mean, and has standard deviation 3 m s−1 for each velocity component and 3 K for θl. Water vapor and cloud water are initialized using the environmental sounding at each level.
Our choice of these statistics for the initial ensemble is motivated entirely by simplicity. As will be discussed in sections 6c and 7, more sophisticated initializations are possible and, because of the relatively short duration of the experiments, will influence the performance of the EnKF. Except where noted, the EnKF uses 50 members in all experiments, and each observation is allowed to influence state variables within a sphere of radius 4 km.
4. Results
This section presents results from the basic assimilation experiment outlined above. We first show that the EnKF analyses closely approximate the reference solution after a few assimilation cycles and then discuss the characteristics and quality of the ensemble covariances.
a. Ensemble-mean analyses
Figure 3 compares the ensemble-mean analysis of vertical velocity (
The analysis also faithfully approximates the thermodynamic variables, which unlike w have no direct influence on the observations. Figure 3 also shows the −0.75-K contour of the temperature perturbation at a height of 1 km, which broadly outlines the low-level cold pool. As for the vertical velocity, the analysis contains some information after a few assimilation cycles and then beyond t = 60 min becomes increasingly detailed and accurate.
The overall quality of the ensemble-mean forecasts and analyses can be obtained from Fig. 4, where the rms errors for the horizontal wind, θl, w, and rainwater are shown as a function of time. Errors are averaged only over the portion of the domain where qr > 0.1 g kg−1 in order to provide a more accurate and sensitive measure of the analysis quality near the cells. (Over the rest of the domain, errors tend to be small simply because the variability of the reference simulation about the initial sounding is small.) As was evident in Fig. 3, the analyses improve rapidly over the first 20–30 min of assimilation. The errors in all fields then level off at a magnitude that is small compared to the O(10 m s−1, 10 K) variations typically found near the convective cells in the reference solution.
The results shown to this point are all based on a single set of initial ensemble members. Since the initial ensemble is drawn randomly with a specified pdf, as are the observation errors, one may expect some random variation of the results for different realizations of these random quantities. To quantify this variation, the rms error of the ensemble-mean analysis of w is shown in Fig. 5 for 12 different realizations. Variations are most significant over the first four cycles (up to t = 40 min), after which all realizations reach similar error levels.
b. Individual members
Before turning to the covariance information in the ensemble, it is useful also to examine how individual members behave. The vertical velocity from the first and second members of the ensemble is shown in Fig. 6 at t = 80 min. As would be expected given the excellent agreement between the ensemble mean and the reference solution at this time, each member is similar to the reference solution in the region near the actual convective cells where observations are available. They differ elsewhere. In particular, a line of spurious convective cells is evident in the second member along the northern boundary of the domain (y = 70 km).
Somewhat less than one-half of the members possess such spurious cells at t = 80 min. These cells can be traced to the initialization of the members, which typically excites a few weak cells in each member spread throughout the domain. The spurious cells that survive do so because they are located away from υr observations, so their evolution is not altered during analysis updates. Even the surviving cells exhibit slow decay, however, since convective cells prefer to be widely separated, as their narrow plume of ascent and broad region of subsidence suppress nearby cells. Since the observed cells are continually reinforced during the assimilation, their subsidence gradually weakens spurious cells.
Finally, we note that although many members have spurious cells, their positions and strength are largely a random consequence of the ensemble initialization. Thus, the spurious cells average to (nearly) zero in the ensemble mean.
c. Ensemble covariances
Within the EnKF, the forecast ensemble provides the covariance information required in the assimilation of observations. This section discusses two issues related to the ensemble covariances: their consistency with the statistics of the error of the ensemble-mean forecasts and analyses, and their form and characteristics.
The issue of consistency arises because we are using the variations of the ensemble about its mean to estimate the statistics of the error of the ensemble mean. To see this, recall from section 2 that the KF assumes p(xt), the forecast pdf for the true state, to have mean
Figure 7 shows this ratio in the present experiments, where we have replaced the expected values in both the numerator and denominator with averages over the 12 realizations shown in Fig. 5. In addition, both the squared error and the ensemble variance are again (as in Figs. 4, 5) summed over only those grid points where qr > 0.1 g kg−1. The ensemble variance at the first analysis time is a factor of 2–4 smaller than the error of the ensemble mean. Their ratio then increases steadily with time, although it remains generally less than Ne/(Ne + 1).
The results of Fig. 7 suggest that there is scope to improve the performance of the EnKF. For example, the ratio of variance to squared error at early times is determined (and at later times, influenced) by our choice of the ensemble's initial variance. Although we have not tested this possibility, the ratio shown in Fig. 7 could likely be improved by increasing the initial variance. Further discussion of tuning the EnKF to optimize its performance appears in section 6.
The increase of the ratio with time differs from the behavior found in other implementations of the EnKF. In those implementations, the ratio typically decreases through successive assimilation cycles (e.g., Houtekamer and Mitchell 1998) owing to a systematic underestimation of the analysis variance by the EnKF.3
To understand why the ensemble variance steadily grows relative to the error of the ensemble mean, recall from Fig. 6 that, throughout the experiment, many members retain spurious convective cells in unobserved portions of the domain. During the forecast for a given member, these spurious cells (if present) interact with the “observed” cells and increase the rate at which the forecast of the observed cells diverges from the reference simulation. The ensemble-mean forecast, however, diverges more slowly from the reference solution, since the spurious cells are spread randomly through the unobserved areas and their averaged effect on the mean is small (except perhaps for some spatial smoothing of the observed cells). Thus, consistent with the shape of the curves in Fig. 7, the ensemble variance increases more rapidly than the squared error during forecasts and overcomes the tendency for the EnKF update to underestimate the analysis variance.
Consideration of the ratio of variance to squared error over the entire domain (not shown) also supports this view. The ensemble variance is initially uniform throughout the domain, yet away from the observed cells the errors of the ensemble mean are small since the motions themselves are weak. The ratio of variance to squared error is then larger than Ne/(Ne + 1) (typically between 2 and 4); this “extra” uncertainty outside the observed regions contaminates forecasts in the observed regions and leads to more rapid growth of variance than squared error there.
We emphasize that the spurious cells in some members are a direct consequence of the initialization of the ensemble with spatially white noise throughout the domain. Thus, the choice of the initial ensemble strongly influences the results of diagnostics, such as shown in Fig. 7, even at t = 100 min. Indeed, as will be shown in section 6c, the initial ensemble also affects the performance of the EnKF throughout the assimilation experiments.
We now turn to the ensemble estimates of the forecast covariances. These are of interest both because little is known about them for convective-scale prediction and because they provide some justification for the radius of influence assumed in our implementation of the EnKF, although that justification is by no means complete.
The left-hand panels of Fig. 8 display variances (i.e., diagonal elements of
As might be expected given that the EnKF analyses estimate unobserved variables, the ensemble reveals significant correlations between υr and the state variables. Three examples appear in the right-hand panels of Fig. 8, which show the correlation of wf, uf, or
Figure 8 provides justification for our choice of a 4-km radius of influence in the assimilation, at least in its order of magnitude: that radius is comparable to the scale of variation of the covariances. Moreover, the largest covariances are found within that radius of the observation point. It is clear, however, that the 4-km radius does not include all locations with significant covariance with υr at the observation point, nor does it exclude all locations with small covariances that are likely strongly contaminated by sampling error.
Figure 8 also demonstrates that covariances have complex structure that is highly inhomogeneous and anistropic, and are flow dependent in that their structure is related to position and form of the convective cell in reference state. It would likely be difficult to model the covariance structure and its relation to reference state with just a few parameters.
5. Forecast-error growth
In many geophysical systems, the accuracy of state estimates is limited in part by forecast-error growth. It is not obvious, however, that the present system of two isolated supercells will behave similarly. In particular, each supercell is long lived and quasi-steady, which indicates that they are at least structurally stable to small perturbations. This in turn raises the possibility that the decrease of analysis error with time shown in Figs. 4 and 5 arises mainly because forecast errors grow slowly or decay. This section examines the forecast-error growth in our experiments.
Perhaps the simplest diagnostic is to compare the error of the ensemble-mean 5-min forecast to that of the preceeding analysis. As can be seen in Fig. 4, errors do typically grow over the course of the forecast, although there are instances in the initial few cycles in which decay occurs.
Errors also grow during longer forecasts. Figure 9a shows rms errors for w for forecasts beginning from the ensemble-mean analysis at various times. (Although they are not shown, forecast errors for other variables behave similarly. Also note that in Fig. 9 the errors are averaged over the entire domain.) The time required for the error to double is roughly 10–20 min in all cases, consistent with the dynamical timescale of the supercells. Moreover, it is clear that the assimilation of observations with the EnKF provides a significantly better state estimate than would be available given only a forecast from an earlier time.
Although it does not prevent error growth, the structural stability of the supercells manifests itself in certain characteristics of the forecast errors. Specifically, initial errors can have two effects on the forecast: they may alter the position and the instantaneous intensity of the supercells, or they can extinguish one or both cells. At t = 100 min, for example, the forecasts shown in Fig. 9a each contain a right-moving supercell, displaced by a few grid points from that in the reference solution and typically weaker, while none of the forecasts, except that from t = 80 min when splitting has already begun, capture the left-moving cell.
In addition, the structural stability of the supercells means that smaller initial errors should produce smaller forecast errors, even beyond the time at which error growth begins to saturate. To test this possibility, we have performed additional forecasts whose initial conditions (at each of the times shown in Fig. 9a) were created by scaling the initial error by a factor of 2, 0.4, or 0.1 and then adding that rescaled perturbation to the reference solution. Figure 9b illustrates the results of these experiments in terms of the rms error of w. As expected, smaller errors at any initial time give rise to smaller forecast errors through the entire length of the forecast. In a chaotic system, error growth would proceed from even very small initial errors until the forecast was completely decorrelated from the reference solution, and the curves in Fig. 9b would, in contrast, all approach a limiting value with time, regardless of the size of the initial error. Presumbably, this would also occur in a more complex convective situation with multiple cells and outflows.
6. Other issues
a. Importance of covariance information
We have suggested that the EnKF is particularly appealing for use at convective scales because of its ability to estimate forecast covariances with a minimum of prior assumptions, and thereby infer state variables that are unobserved. Here, we test this assertion with an experiment in which the EnKF updates only the three components of the wind given observations of υr, and does not use the information available in cross covariances between υr and θl, qr and qt.
The rms error of the ensemble-mean analysis of w is shown as a function of time in Fig. 10 for 10 realizations of the experiments (i.e., for 10 realizations of the initial ensemble and 10 realizations of the observation errors). The error is typically more than three times what it would be if all variables were updated during the analysis (see Fig. 5), and even in the best realization the error is still doubled. The error in variables other than w behaves similarly.
Except in the single, anomalously accurate realization evident in Fig. 10, the analyses (not shown) fail to capture either the updraft or surface cold pool of the observed cell at least through t = 45 min. Instead, the analyses (and the individual members) contain a noisy mix of updrafts and downdrafts spread over the region of observations, with a variety of spurious cells outside that region in individual members. Beyond an hour, the analyses begin to exhibit narrow updrafts in the locations of the observed cells and small cold pools beneath these updrafts. The analyzed updrafts then decay markedly during the course of the 5-min forecasts to the next analysis time.
We have also performed experiments in which the observed variable is u and only u is updated by the EnKF, so that cross covariances between components of velocity are also ignored in the assimilation. Analysis errors are even larger and the analyses never capture the updrafts of the observed cells (not shown). Together, these two sets of experiments show that covariances estimated by the EnKF contain significant information about relations among the state variables, despite the sampling errors associated with an ensemble of 50 members.
b. Tuning of algorithm
To implement EnKF as described here, one must choose the ensemble size Ne and the radius of influence for the observations. Although we have not systematically tuned the algorithm (by determining optimal radius for a given Ne), we have performed a number of experiments with values other than Ne = 50 and r = 4 km.
It is clear for these additional experiments that the results in section 4 depend quantitatively on the choice of Ne and r. In particular, decreasing Ne to 25 increases the rms error of the ensemble-mean analyses and also makes the results less robust, in the sense that there will be realizations of the initial ensemble for which the analyses are always a poor approximation to the reference simulation, as was the case for the experiments described in section 6a above that did not update θl or the moisture variables. (Increasing Ne has smaller, but opposite, effects.) For sufficiently large r (roughly 20 km or larger, using Ne = 50), the ensemble-mean analysis noticeably deterioriates and “outlying” realizations again become frequent, although more modest variations of r, from 2 to 6 km, have little effect on the analysis quality. Such dependence on Ne and r is broadly consistent with other examples of the EnKF (e.g., Houtekamer and Mitchell 1998; Anderson 2001; Whitaker and Hamill 2002).
The results of our experiments also depend on how the analysis update is performed within the radius r. A more sophisticated approach is to decrease the influence of an observation on the state with increasing distance from the observation location by multiplying ci in (4a) by a correlation function with local support, as in Houtekamer and Mitchell (2001). In this way, the influence of an observation on the analysis decreases smoothly to zero at finite radius, rather than jumping discontinuously to zero. Using such an approach improves the results presented here (A. Caya 2002, personal communication).
Another aspect of the algorithm that can likely be improved is our choice of the probability distribution from which the initial ensemble is drawn. In the results just presented, the initial ensemble is centered on the environmental sounding and each member is initialized with independent Gaussian noise at each grid point and in each variable. Although the initial variance in this simple scheme could presumably be tuned, we will show in section 6c below that a careful choice of that initial distribution is likely more important.
A final aspect of ensemble filters that is often subject to tuning is the covariance inflation (Anderson 2001; Hamill et al. 2001), in which the ensemble covariances are multiplied by a scalar factor slightly greater than 1 to compensate for the usual bias of the EnKF to underestimate the analysis uncertainty. We have performed experiments with several inflation factors and always find that inflation degrades the results. There appears to be two related reasons for this: first, the tendency for the ensemble forecasts to overestimate, because of the presence of spurious cells, the growth of variance in the region of the observed cells; and second, the fact that the inflation enhances the spurious cells by increasing deviations from the ensemble mean. Since the spurious cells are an artifact of the ensemble initialization, it seems likely that covariance inflation may yet prove beneficial given a more sophisticated initialization. We leave a more systematic exploration of tuning the EnKF for subsequent work. The following section provides further discussion of the role of the initial ensemble.
c. Dependence on the statistics of the initial ensemble
Sections 4b,c discuss how the initialization of each member with noise throughout the domain leads to spurious convective cells in many members, which in turn degrade the forecasts from those members and affect the properties and performance of the EnKF. A natural question is then the extent to which the result might be improved with a different initialization of the ensemble.
One easy modification, which should reduce the spurious cells, is to restrict the initial noise in each member to the vicinity of the radar echoes. To be more specific, we have performed experiments in which the members are initialized at t = 0 with noise confined to a 20 km × 20 km box centered on the location of the first echoes (i.e., the first nonzero values of qr) at t = 20 min. Except for this restriction of the initial noise to a portion of the domain, these experiments are identical to those discussed in section 4. Our use of information from t = 20 min to initialize the ensemble at t = 0 is akin that of 4DVAR or the Kalman smoother (Cohn 1997), in which observations influence the state estimate at earlier times as well as the present time. Note also that this initialization is feasible in practice, as one would simply wait until observations were available and then initialize the ensemble at a somewhat earlier time.
The rms error for w is shown in Fig. 11. Analyses are significantly improved over those based on the original initialization of the ensemble. Averaging over 12 realizations of the experiments, error at t = 50 min is reduced by a factor of 2, while that at t = 100 min is reduced by a factor of 3.
In the individual members (not shown), some spurious cells are still excited by the initial noise, but these are of course located much closer to the observed cell on average. The subsidence surrounding the observed cell then suppresses the spurious cells more strongly than in the previous experiments, so that they are weaker throughout the simulation and almost all have disappeared by t = 70 min. The fact that the forecasts from individual members are not degraded by the presence of spurious cells is at least one factor contributing to the improved performance of the EnKF.
In section 4c, we also argued that the presence of spurious cells in some members led to a continual increase in the ensemble variance relative to the squared error of the ensemble mean. Figure 12 shows the ratio of these quantities for the present “local perturbation” experiment and should be compared to Fig. 5. The ratio of variance to error increases through the first half of both experiments. By t = 70 min, the ratio stops its systematic increase in the local perturbation experiment; this coincides with the time at which most spurious cells have been suppressed.
7. Summary and discussion
In the experiments presented here, we have used the EnKF to assimilate simulated Doppler radar observations of radial velocity in a nonhydrostatic, cloud-scale model. The observations are taken (together with random observational noise) from a reference simulation of a splitting supercell storm, produced with the same numerical model. We assume that observations are available every 5 min, but only in the small fraction of the domain where the rainwater in the reference simulation exceeds a threshold.
These experiments demonstrate the potential of the EnKF for assimilation of radar data at the convective scale. Using an ensemble of 50 members, the EnKF produces analyses that accurately approximate the true state (i.e., that from the reference simulation) after about seven assimilation cycles and a half hour of observations. Variables not directly observed, including the vertical velocity and temperature, are accurately estimated in the analyses. Examination of forecasts also shows that the simple supercell simulation considered here supports significant growth of forecast error and, thus, that the ability of the EnKF to track the growth and splitting of the cell is not simply a consequence of stable system dynamics.
In principle, the crucial element of the EnKF is its direct estimate of forecast covariances between radial velocity (or other observed quantities) and the state variables. To test the importance of such covariances, we also performed experiments in which the covariances of υr with the thermodynamic and moisture variables were set to zero in the assimilation, so that the observations did not influence the analyses of temperature, moisture, and cloud water. The rms analysis error more than tripled in these experiments.
It is worth emphasizing that the dynamics of moist convection represents a significant test for the EnKF. Unlike the large-scale flows in the atmosphere or basin scales in the ocean, which have been the setting of all previous implementations of the EnKF, convective-scale motions generally lack approximate, static balances, such as geostrophy, that link the velocity to thermodynamic fields. Our results indicate that the lack of such balances is not a fundamental obstacle to the use of the EnKF at convective scales, as the dynamically produced relations among state variables (i.e., those that arise through the evolution of the flow) are sufficiently strong and develop rapidly. In addition, moist convection is driven by distinctly nonlinear microphysical processes, such as condensation, and tends to form discrete, coherent structures, such as supercells, whose dynamics are nonlinear; both of these facts call into question the Gaussian assumptions that underlie the EnKF analysis step. The success of the EnKF in the present problem suggests that the nonlinearity inherent in moist convection is also not a fundamental obstacle to the EnKF. Nevertheless, continued development of the EnKF for convective scales will undoubtedly require further consideration of nonlinear and non-Gaussian effects, particularly in relation to reflectivity observations.
Our experiments also differ from those of previous studies with the EnKF in that they cover a limited time interval, spanning only a few dynamical timescales. The quality of the analyses, and even diagnostics of the performance of the scheme such as the ratio of variance to error, are therefore strongly influenced throughout our experiments by the choice of the initial ensemble. A particular example appears in section 6c. (In contrast, other studies have collected results over long simulations of systems that possess a statistically steady state, so that the initial ensemble is not important.) Although the initial conditions for the ensemble should clearly reflect our best knowledge of p(xt) prior to any observations, considerable latitude remains for choosing them reasonably. We expect that improving the initial ensemble and diagnosing its role are likely to be persistent issues for the EnKF at convective scales.
The performance of the EnKF relative to retrieval techniques or 4DVAR is an important question that we have not addressed in this paper. In fact, our use the of the model of Sun and Crook (1997) was motivated by the possibility of comparing against the existing 4DVAR scheme for that model. Comparisons are under way and will be reported elsewhere (A. Caya and J. Sun 2003, personal communication).
There are a number of other important issues that were beyond the scope of this initial paper. These include the use of reflectivity observations; estimation of the environmental sounding and its uncertainty; accounting for imperfections in the forecast model, particularly the microphysical parameterizations; the treatment of lateral boundary conditions and their uncertainty; and quantification of errors in real radar observations and in the forward operators for both radial velocity and reflectivity, as well as quality control for radar observations. Progress on all of these issues is likely important if the EnKF, or indeed another technique, is to be applied routinely and successfully to the assimilation of radar observations.
Acknowledgments
We are particularly indebted to Juanzhen Sun for the use of her cloud model in this study. Both Alain Caya and David Dowell have shared with us their results related to various refinements of the algorithm used here. It is a pleasure to acknowledge helpful discussions with them and with William Skamarock and Jeff Anderson. This research was supported at NCAR by the U.S. Weather Research Program and by NSF Grant 0205655.
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Schematic depiction of the EnKF update (or analysis). (a) Forecast values of w and υr for an ensemble of 50 members (black dots). Thin arrows on each axis show the ensemble means, each roughly 9 m s−1. A hypothetical observation of υr equal to 14 m s−1 is indicated by the bold arrow on the υr axis, while the accompanying curve is meant to suggest the observational uncertainty. (b) As in Fig. 1a but for the updated ensemble given the observation of υr. The forecast ensemble and observation are reproduced in gray for reference
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Schematic depiction of the EnKF update (or analysis). (a) Forecast values of w and υr for an ensemble of 50 members (black dots). Thin arrows on each axis show the ensemble means, each roughly 9 m s−1. A hypothetical observation of υr equal to 14 m s−1 is indicated by the bold arrow on the υr axis, while the accompanying curve is meant to suggest the observational uncertainty. (b) As in Fig. 1a but for the updated ensemble given the observation of υr. The forecast ensemble and observation are reproduced in gray for reference
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Schematic depiction of the EnKF update (or analysis). (a) Forecast values of w and υr for an ensemble of 50 members (black dots). Thin arrows on each axis show the ensemble means, each roughly 9 m s−1. A hypothetical observation of υr equal to 14 m s−1 is indicated by the bold arrow on the υr axis, while the accompanying curve is meant to suggest the observational uncertainty. (b) As in Fig. 1a but for the updated ensemble given the observation of υr. The forecast ensemble and observation are reproduced in gray for reference
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Skew T diagram for the environmental sounding. Temperature and dewpoint (°C) profiles are indicated by thick solid and thick dashed lines, respectively. Wind vectors are shown at the right (half barbs, 2.5 m s−1; full barbs, 5 m s−1; flags, 25 m s−1)
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Skew T diagram for the environmental sounding. Temperature and dewpoint (°C) profiles are indicated by thick solid and thick dashed lines, respectively. Wind vectors are shown at the right (half barbs, 2.5 m s−1; full barbs, 5 m s−1; flags, 25 m s−1)
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Skew T diagram for the environmental sounding. Temperature and dewpoint (°C) profiles are indicated by thick solid and thick dashed lines, respectively. Wind vectors are shown at the right (half barbs, 2.5 m s−1; full barbs, 5 m s−1; flags, 25 m s−1)
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Vertical velocity at z = 6 km in (a)–(e) the reference simulation (wt) and (f–j) the EnKF analysis (the ensemble mean). Shades of red and blue indicate upward and downward motion, respectively, with gradations of color every 2.5 m s−1 beginning at ±1.25 m s−1 and up to a maximum of 26.25 m s−1. Contours of the −0.75-K temperature perturbation at z = 1 km are also displayed (black lines). Fields are shown at t = 30, 35, 45, 60, and 80 min, as marked on each panel
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Vertical velocity at z = 6 km in (a)–(e) the reference simulation (wt) and (f–j) the EnKF analysis (the ensemble mean). Shades of red and blue indicate upward and downward motion, respectively, with gradations of color every 2.5 m s−1 beginning at ±1.25 m s−1 and up to a maximum of 26.25 m s−1. Contours of the −0.75-K temperature perturbation at z = 1 km are also displayed (black lines). Fields are shown at t = 30, 35, 45, 60, and 80 min, as marked on each panel
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Vertical velocity at z = 6 km in (a)–(e) the reference simulation (wt) and (f–j) the EnKF analysis (the ensemble mean). Shades of red and blue indicate upward and downward motion, respectively, with gradations of color every 2.5 m s−1 beginning at ±1.25 m s−1 and up to a maximum of 26.25 m s−1. Contours of the −0.75-K temperature perturbation at z = 1 km are also displayed (black lines). Fields are shown at t = 30, 35, 45, 60, and 80 min, as marked on each panel
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
The rmse of the ensemble mean from the EnKF, averaged over all points at which qr > 0.1 g kg−1, for four quantities: horizontal wind (vector magnitude in m s−1; thick solid lines), w (m s−1; gray), θl (K; dotted), and rainwater (g kg−1; thin solid). Errors for both the forecast and analysis means are shown at each analysis time, producing the “sawtooth” appearance of the curves. Results for the first analysis (at t = 20) are omitted because only 12 observations are available at that time
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
The rmse of the ensemble mean from the EnKF, averaged over all points at which qr > 0.1 g kg−1, for four quantities: horizontal wind (vector magnitude in m s−1; thick solid lines), w (m s−1; gray), θl (K; dotted), and rainwater (g kg−1; thin solid). Errors for both the forecast and analysis means are shown at each analysis time, producing the “sawtooth” appearance of the curves. Results for the first analysis (at t = 20) are omitted because only 12 observations are available at that time
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
The rmse of the ensemble mean from the EnKF, averaged over all points at which qr > 0.1 g kg−1, for four quantities: horizontal wind (vector magnitude in m s−1; thick solid lines), w (m s−1; gray), θl (K; dotted), and rainwater (g kg−1; thin solid). Errors for both the forecast and analysis means are shown at each analysis time, producing the “sawtooth” appearance of the curves. Results for the first analysis (at t = 20) are omitted because only 12 observations are available at that time
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
The rmse of the ensemble-mean analysis of w at various times and for 12 realizations of the initial (random) ensemble perturbations and the observation errors. For clarity, four of the realizations are indicated by thin black lines, four others by gray lines, three others by dotted lines; the realization shown in Figs. 2–4 is indicated by a thick black line
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
The rmse of the ensemble-mean analysis of w at various times and for 12 realizations of the initial (random) ensemble perturbations and the observation errors. For clarity, four of the realizations are indicated by thin black lines, four others by gray lines, three others by dotted lines; the realization shown in Figs. 2–4 is indicated by a thick black line
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
The rmse of the ensemble-mean analysis of w at various times and for 12 realizations of the initial (random) ensemble perturbations and the observation errors. For clarity, four of the realizations are indicated by thin black lines, four others by gray lines, three others by dotted lines; the realization shown in Figs. 2–4 is indicated by a thick black line
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
As in Fig. 3, but showing the vertical velocity for the first and second members of the ensemble at t = 80 min.
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
As in Fig. 3, but showing the vertical velocity for the first and second members of the ensemble at t = 80 min.
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
As in Fig. 3, but showing the vertical velocity for the first and second members of the ensemble at t = 80 min.
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Ratio of the ensemble variance to the squared error of the ensemble mean for horizontal wind (thick solid lines), w (gray), and θl (dotted). Both the numerator and denominator of the ratio are averaged over 12 realizations of the experiment. The horizontal line indicates Ne/(Ne + 1), the ratio of the expected values of sample variance and squared error for an ensemble drawn from the same distribution as the reference solution.
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Ratio of the ensemble variance to the squared error of the ensemble mean for horizontal wind (thick solid lines), w (gray), and θl (dotted). Both the numerator and denominator of the ratio are averaged over 12 realizations of the experiment. The horizontal line indicates Ne/(Ne + 1), the ratio of the expected values of sample variance and squared error for an ensemble drawn from the same distribution as the reference solution.
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Ratio of the ensemble variance to the squared error of the ensemble mean for horizontal wind (thick solid lines), w (gray), and θl (dotted). Both the numerator and denominator of the ratio are averaged over 12 realizations of the experiment. The horizontal line indicates Ne/(Ne + 1), the ratio of the expected values of sample variance and squared error for an ensemble drawn from the same distribution as the reference solution.
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Variances and correlations estimated from the ensemble at t = 80 min in the x–z plane along y = 36 km, which passes through the maximum updraft. (left) Variances of (a) wf, (c) uf, and (e)
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Variances and correlations estimated from the ensemble at t = 80 min in the x–z plane along y = 36 km, which passes through the maximum updraft. (left) Variances of (a) wf, (c) uf, and (e)
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
Variances and correlations estimated from the ensemble at t = 80 min in the x–z plane along y = 36 km, which passes through the maximum updraft. (left) Variances of (a) wf, (c) uf, and (e)
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
(a) The rmse for forecasts of w (black lines), starting from the ensemble-mean analysis at t = 30, 45, 60, 80 min and averaged over the entire domain. Errors for the forecast and analysis means of w are indicated by the gray lines as in Fig. 4 (b) As in Fig. 9a, but for forecasts starting from initial conditions whose error has been rescaled by factors of 2 (thin lines), 0.4 (thick), and 0.2 (dotted). To be more precise, initial conditions for each forecast are created by calculating the error of the ensemble-mean analysis at the appropriate time, multiplying the error field by a constant scalar factor (2, 0.4, 0.1), and adding the rescaled error to the reference solution. The rmse for the original forecast (scale factor of 1) is shown in gray
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
(a) The rmse for forecasts of w (black lines), starting from the ensemble-mean analysis at t = 30, 45, 60, 80 min and averaged over the entire domain. Errors for the forecast and analysis means of w are indicated by the gray lines as in Fig. 4 (b) As in Fig. 9a, but for forecasts starting from initial conditions whose error has been rescaled by factors of 2 (thin lines), 0.4 (thick), and 0.2 (dotted). To be more precise, initial conditions for each forecast are created by calculating the error of the ensemble-mean analysis at the appropriate time, multiplying the error field by a constant scalar factor (2, 0.4, 0.1), and adding the rescaled error to the reference solution. The rmse for the original forecast (scale factor of 1) is shown in gray
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
(a) The rmse for forecasts of w (black lines), starting from the ensemble-mean analysis at t = 30, 45, 60, 80 min and averaged over the entire domain. Errors for the forecast and analysis means of w are indicated by the gray lines as in Fig. 4 (b) As in Fig. 9a, but for forecasts starting from initial conditions whose error has been rescaled by factors of 2 (thin lines), 0.4 (thick), and 0.2 (dotted). To be more precise, initial conditions for each forecast are created by calculating the error of the ensemble-mean analysis at the appropriate time, multiplying the error field by a constant scalar factor (2, 0.4, 0.1), and adding the rescaled error to the reference solution. The rmse for the original forecast (scale factor of 1) is shown in gray
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
As in Fig. 5, but for the experiment in which θl, qr, and qt are not updated in the analysis; i.e., observations of υr influence only the velocities
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
As in Fig. 5, but for the experiment in which θl, qr, and qt are not updated in the analysis; i.e., observations of υr influence only the velocities
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
As in Fig. 5, but for the experiment in which θl, qr, and qt are not updated in the analysis; i.e., observations of υr influence only the velocities
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
As in Fig. 5, but for the local perturbations experiment in which the initial ensemble perturbations are nonzero only in a 20 km × 20 km box centered on the location of the first echoes
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
As in Fig. 5, but for the local perturbations experiment in which the initial ensemble perturbations are nonzero only in a 20 km × 20 km box centered on the location of the first echoes
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
As in Fig. 5, but for the local perturbations experiment in which the initial ensemble perturbations are nonzero only in a 20 km × 20 km box centered on the location of the first echoes
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
As in Fig. 7, but for the local perturbations experiment
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
As in Fig. 7, but for the local perturbations experiment
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
As in Fig. 7, but for the local perturbations experiment
Citation: Monthly Weather Review 131, 8; 10.1175//2555.1
To be more precise, p(xt) is the pdf for xt at t = to conditioned on all observations prior to yo; this is why we refer to p(xt(t = to) as a forecast. Since all the equations in this section are valid at t = to and p(xt) is always conditioned on the prior observations, we will suppress explicit references to to or the observations prior to to.
In a more recent version of the model, developed after the bulk of the results reported here, the numerical smoothing algorithms have been modified and the effective dissipation in the model has been significantly reduced (J. Sun 2002, personal communication). We have repeated a limited number of experiments using the newer version of the model and find no qualitative change in our results, although the new version produces only a single, right-moving supercell when initialized as in our reference simulation.
One reason for this can be seen from (4b), the scalar-observation update for the KF covariances, which shows that the total analysis variance tr(𝗣a) is reduced by an amount proportional to tr(ccT) = c · c relative to the forecast variance. When c is estimated from a finite sample, sampling error biases the estimate of c · c to be too large, so that the estimate of 𝗣a is correspondingly too small. See van Leeuwen (1999) for a more rigorous analysis.
The National Center for Atmospheric Research is sponsored by the National Science Foundation.
