1. Introduction

A primary goal of climate change research is to understand variations in the frequency and intensity of severe weather events on decadal and longer time scales. An obvious prerequisite for achieving this goal is an accurate baseline estimate of the frequency and intensity of severe weather over the last century. Analyses of long-term changes in extratropical cyclone frequency and intensity have been hampered by the inadequacy of current datasets (Houghton et al. 2001, p. 163). Since synoptic-scale weather systems have time scales of less than a week, a century-long dataset of tropospheric circulation fields at daily resolution is required. The National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) 50- Year Reanalysis (Kistler et al. 2001) provides four- times-daily gridded circulation fields beginning in 1948, when digital upper-air observations became widely available. The only daily tropospheric circulation dataset available that extends back before 1948 is derived from charts of sea level pressure hand-drawn by U.S. Air Force meteorologists in the 1940s and 1950s (United States Weather Bureau 1944). Although a remarkable achievement for its time, this original reanalysis suffers from serious problems associated with incorrect assumptions made by the analysts in data-sparse regions (Jones 1987; Trenberth and Paolino 1980) and does not provide estimates of the full three-dimensional tropospheric structure. Clearly, a better dataset is needed— but is it possible to create a more accurate daily tropospheric circulation dataset for the first half of the twentieth century given the paucity of available observations?

In this study we examine whether advanced data assimilation systems have significant advantages over currently available systems for sparse networks of surface pressure observations representative of the early part of the twentieth century. Specifically, we examine the performance of the ensemble-based data assimilation system described by Whitaker and Hamill (2002) applied to a simulated 1915 observing network, created by subsampling the surface pressure observations for 2001. We focus on surface pressure observations since they are the most widely available and reliable observations in the early twentieth century and provide more information about the state of the free troposphere than surface wind and temperature observations. In a companion report we will carefully examine the available data record for the last 150 yr and assess the performance of several analysis schemes with surface-only observation networks representative of 1890–1940.

Previous studies using idealized ensemble data assimilation systems (e.g., Hamill and Snyder 2000) have shown that their flow-dependent background-error covariances are most beneficial when the observing network is sparse. When observations are very dense, the background-error covariances do not change as much from time to time, so static background-error covariance models [such as used in three-dimensional variational data assimilation (3DVAR)] can be nearly as effective. In addition, the computational cost of computing analysis increments in recently proposed ensemble data assimilation algorithms (Houtekamer and Mitchell 2001; Whitaker and Hamill 2002) is directly proportional to the number of observations being assimilated. Therefore, ensemble-based data assimilation should be more computationally feasible and provide the greatest benefit over current operational schemes in situations when observations are sparse. Reanalysis before the radiosonde era is just such a situation.

The paper is organized as follows: section 2 contains a description of the experimental design, including the simulation of the 1915 observing network, the ensemble data assimilation system, and forecast model. Section 3 presents the results of the assimilation experiments, which show that the ensemble square root filter (EnSRF) can produce midtropospheric analyses given surface observations at 1915 densities, which are as accurate as 2.5-day forecasts are today. Section 4 contains a summary of the results and a discussion of the unresolved issues that need to be addressed before these techniques can be applied to a real reanalysis of the first half of the twentieth century.

2. Experimental design

b. The ensemble data assimilation system

Ensemble data assimilation systems transform a forecast ensemble into an analysis ensemble with appropriate statistics. This can be done stochastically, treating the observations as random variables, (e.g., Houtekamer and Mitchell 1998; Burgers et al. 1998), or deterministically, requiring that the covariance of the updated ensemble satisfy the Kalman filter analysis error covariance equation. Deterministic analysis ensemble updates are Monte Carlo implementations of Kalman square root filters, hence we call them ensemble square root filters. Ensemble square root filters are not unique (Tippett et al. 2003), since different ensembles can have the same covariance. This nonuniqueness has led to the development of several different algorithms for updating the analysis ensemble (Bishop et al. 2001; Anderson 2001; Whitaker and Hamill 2002). Here we implement the latter variant, which reduces to a particularly simple form when observations are assimilated serially (one after another).

Following the notation of Ide et al. (1997), let x^b be an m-dimensional background model forecast; let y^o be a p-dimensional set of observations; let 𝗛 be the operator that converts the model state to the observation space; let 𝗣^b be the m × m-dimensional background- error covariance matrix; and let 𝗥 be the p × p-dimensional observation-error covariance matrix. The minimum error-variance estimate of the analyzed state x^a is then given by the traditional Kalman filter update equation (Lorenc 1986),

x^ax^by^ox^b(1)

where

^bTbT−1(2)

The analysis-error covariance 𝗣^a is reduced by the introduction of observations by an amount

View Expanded

In ensemble data assimilation, 𝗣^b𝗛^T is approximated using the sample covariance estimated from an ensemble of model forecasts. For the rest of the paper, the symbol 𝗣 is used to denote the sample covariance from an ensemble, and 𝗞 is understood to be computed using sample covariances. Expressing the variables as an ensemble mean (denoted by an overbar) and a deviation from the mean (denoted by a prime), the update equations for the ensemble Kalman filter (EnKF) may be written as

View Expanded

where 𝗣^b𝗛^T = (𝗛x′^b)x′^bT ≡ (n − 1)⁻¹Σⁿ_i=1 𝗛x′^b_ix′^bT_i, n is the ensemble size (=100 unless otherwise noted), 𝗞 is the traditional Kalman gain given by (2), and K̃ is the gain used to update deviations from the ensemble mean. Note that an overbar used in a covariance estimate implies a factor of n − 1 instead of n in the denominator, so that the estimate is unbiased. In the EnKF, K̃ = K, and y′^o is randomly drawn from the probability distribution of observation errors (Burgers et al. 1998). This choice of y′^o ensures that for an infinitely large ensemble, (3) will be satisfied exactly (Burgers et al. 1998). However, as pointed out by Whitaker and Hamill (2002), for a finite ensemble (3) will not be satisfied exactly, and the noise added to the observations acts as an extra source of sampling error, degrading the performance of the filter. In the EnSRF, y′^o = 0, and K̃ is given by

View Expanded

(Andrews 1968). This choice guarantees that (3) is satisfied exactly. If 𝗥 is diagonal, observations may be assimilated serially, the analysis after assimilation of the Nth observation becomes the background estimate for assimilating the (N + 1)th observation (Gelb et al. 1974), and the above expression simplifies to

View Expanded

where R and 𝗛𝗣^b𝗛^T are scalars, while 𝗞 and K̃ are vectors of the same dimension as the model state vector. This was first derived by J. Potter in 1964 (Maybeck 1979). Although (6) requires the computation of two matrix square roots, the serial processing version (7) requires the computation of only a scalar factor to weight the traditional Kalman gain, and therefore is no more computationally expensive than the EnKF.

As discussed in Whitaker and Hamill (2002), sampling error can cause filter divergence in any ensemble data assimilation system, so some extra processing of the ensemble covariances is usually necessary. The two techniques used here are distance-dependent covariance localization (Houtekamer and Mitchell 2001; Hamill et al. 2001) and covariance inflation (Anderson and Anderson 1999).

Covariance localization is a filter that forces the ensemble covariances to go to zero at some horizontal distance L from the observation being assimilated. It is intended to counter the tendency for ensemble variance to be excessively reduced by spurious long-range correlations between analysis and observations points. For all the experiments shown here, L is set to 5000 km and the horizontal structure of the filter is the same as used in Whitaker and Hamill (2002). We also use a covariance filter in the vertical, which forces ensemble covariances to go to zero at σ = 0.05 (roughly 50 mb). The vertical covariance localization function has a value of 1 below σ = 0.2, zero above σ = 0.05, and decreases linearly in between these σ levels. The same covariance filter is also applied to the SI analysis scheme, which uses a 100-member ensemble of randomly chosen December reanalysis states, instead of an ensemble of 6- h model forecasts.

Covariance inflation simply inflates the deviations from the ensemble mean first guess by a factor r > 1.0 for each member of the ensemble, before the computation of the background-error covariances and before any observations are assimilated. We have found that different inflation factors were required in the Northern and Southern Hemispheres because of the large differences in the density of the observing networks. In the limit that there are no observations influencing the analysis in a given region, it is easy to see how inflating the ensemble every analysis time can lead to unrealistically large ensemble variances, exceeding the climatological variance. The simulated 1915 network has very few observations in the Southern Hemisphere extratropics, generally less than 20 per analysis time. Therefore, only a very small inflation factor is needed there. In the Northern Hemisphere, however, we found it necessary to use a significantly larger inflation factor to avoid filter divergence. For all the results presented here, we use an inflation factor that varies smoothly across the equator at σ = 1 from a value r = 1.07 in the Northern Hemisphere to a value of r = 1.007 in the Southern Hemisphere. Our use of vertical covariance localization means that no observations ever affect the analysis above σ = 0.05. Therefore, in order to avoid the development of excessive ensemble variance in the stratosphere we use a vertically varying r that is constant from σ = 1 to σ = 0.2, then decreases linearly to unity at σ = 0.05.

Observations of surface pressure are processed serially. The ensemble mean and ensemble deviations are updated using Eqs. (4) and (5), with the Kalman gains given by Eqs. (2) and (7). Covariance inflation is applied to the background ensemble deviations before assimilating any observations. Only those grid points within L km of the observation are updated, where L = 5000 km is the covariance localization length scale. The analysis is performed on a 128 × 64 Gaussian grid, and the forward operator 𝗛 represents bilinear interpolation to the observation location (and, if necessary, a reduction to the model orography as described in section 2a). The analysis updates for each observation are parallelized by partitioning the model state vector by vertical level. For example, with our vertical covariance localization, only the lowest 23 (out of 28) model levels are updated. To run the assimilation on four processors, 23 levels of zonal wind are updated on processor 1, 23 levels of meridional wind on processor 2, 23 levels of temperature on processor 3, and 23 levels of specific humidity plus surface pressure on processor 4. The only communication between processors needed during the analysis update involves the propagation of 𝗛x^b from the processor updating surface pressure (processor 4 in this example) to the other processors. The results shown here were run on thirty-one 2.2-GHz Intel processors. Only 6 s of wall-clock time were needed to process 417 surface pressure observations for a 100-member ensemble with L = 5000 km, or about 0.015 s per observation. Running the forecast ensemble to generate the background estimates for the next assimilation time is trivially parallel, since every ensemble member runs independently on a separate processor.

Although we do not employ an explicit treatment of model error, our implementation of covariance localization and inflation can be considered a crude parameterization of model error. Covariance inflation increases the magnitude of the background-error covariance estimate. This is necessary to deal with biases caused by sampling error (as discussed by Whitaker and Hamill 2002), but it can also be thought of as accounting for unrepresented model-error covariance. This can only account for model errors that are in the same subspace as the background ensemble. Covariance localization forces the covariance between background-error estimates at two locations to go to zero as the distance between the two locations increases. This is necessary to deal with sampling error, even in the absence of model error. However, the effect of covariance localization is to increase the rank of the background-error covariance estimate (Hamill et al. 2001). The extra degrees of freedom introduced into the background-error estimate can be thought of as representing model errors that do not project onto the subspace spanned by the background ensemble.

3. Assimilation results

Figure 3 summarizes the assimilation results for the simulated December 1915 network. Shown are time series of the root-mean-square (rms) analysis error for (a) mean sea level pressure (MSLP) and (b) 500-hPa geopotential height (Z500). The analysis error (defined relative to the NCEP–NCAR reanalysis and averaged over the Northern Hemisphere poleward of 20°N) is shown for the EnSRF (black curve), SI (red curve), and CDAS- SFC (blue curve) analyses, along with the spread of the EnSRF analysis (green curve). The CDAS-SFC analysis is significantly better than the SI analysis, indicating that using a 6-h model forecast as a first guess in a 3DVAR analysis is an improvement over a climatological background, even for this small number of surface pressure observations. However, the EnSRF analysis is about 50% more accurate than the CDAS-SFC analysis. In fact, there is not a single case in the 1-month period where the CDAS-SFC is as accurate as the EnSRF. We note that since we have made no attempt to tune the structure of the background-error covariances used in CDAS for this very sparse observation network, it is possible that a better specification of static background- error covariances in the CDAS-SFC analysis system would improve the CDAS-SFC analysis. The SI analysis, which uses climatology as a background, has a large diurnal cycle associated with the differences in the number of observations available at 0000 (1200) and 0600 (1800) UTC. This occurs because there is no mechanism in the SI analysis to propagate information forward in time.

Averaged over the Northern Hemisphere extratropics, the spread in the EnSRF has nearly the same mean value as the rms error, indicating that our implementation of covariance localization and inflation has successfully countered the loss of ensemble variance expected from sampling error and the lack of an explicit treatment of model error. The pattern of ensemble spread, averaged over the monthlong assimilation period, is also quite similar to the pattern of the ensemble mean rms error (Fig. 4). However, close inspection of Fig. 4 reveals that in data-dense (data-sparse) regions, the ensemble mean spread is somewhat smaller (larger) than analysis error. This is in part due to our implementation of covariance inflation. Over data-sparse areas that are only weakly influenced by observations, covariance inflation increases ensemble variance too much. We have tuned the inflation factor so that the Northern Hemisphere average value of spread is similar to analysis error, consequently ensemble variance over data-dense areas must be deficient in order to balance the tendency for ensemble spread to be too large over data-sparse areas. This behavior may also be in part a manifestation of spatially correlated observation error, perhaps associated with “representativeness error” (e.g., Liu and Rabier 2002). Our analysis algorithm assumes that the errors for each observation are independent. If observation errors are actually spatially correlated, the filter may reduce the variance of the ensemble too much. Further study is needed to assess the impact of misspecification of observation-error covariances on the performance of ensemble data assimilation systems, particularly with dense observation networks.

The inhomogeneity of the ensemble variance shown in Fig. 4 helps explain how the EnSRF is able to outperform the CDAS-SFC system, which assumes that the background-error variance is a function of latitude only (Parrish and Derber 1992). A 3DVAR system could probably be tuned to the 1915 surface pressure network to produce a better estimate of the spatial varying nature of the error in the first guess, and hence a better analysis. However, the EnSRF is able to do this with little tuning (only the specification of the covariance localization and inflation parameters). We consider the ability of the EnSRF to adapt to large changes in the structure of the observing network to be a very desirable property for an analysis system to be used for a reanalysis of the entire twentieth century.

Also included in Fig. 4 is a map showing the rms error of 60-h forecasts run with the same version of the MRF used in the assimilation experiments, but initialized from the 0000 UTC reanalyses for all Decembers between 1979 and 2002.¹ The mean rms error averaged over the Northern Hemisphere extratropics is 39 m, very close to the mean December EnSRF analysis error for the simulated 1915 network (Fig. 3). Therefore, we expect that a reanalysis of the early twentieth century using only surface pressure observations should be about as accurate at 500 hPa (in a Northern Hemisphere average sense) as 60-h forecasts are today. In fact, Fig. 4c shows that the EnSRF is actually more accurate than modern 60-h forecasts over most of the Northern Hemisphere, except over the polar region and Asia, where there are voids in the simulated 1915 observation network.

A sample 500-hPa EnSRF analysis is shown in Fig. 5 for 0000 UTC 14 December, along with the corresponding map from the NCEP–NCAR reanalysis (using all available observations). The EnSRF analysis, using only a few hundred surface pressure observations, is clearly able to reproduce most of the significant midtropospheric flow features present in the reanalysis, including the synoptic-scale short waves.

One source of error in the EnSRF scheme is associated with sampling error in the estimate of the background-error covariances. Assuming this source of error is significant compared to other possible sources of error (e.g., misrepresentation of model-error or observation- error covariance), then increasing the ensemble size should improve the accuracy of the EnSRF analysis. We have performed additional assimilation experiments with 50- and 200-member ensembles to assess the impact of ensemble size on analysis quality. The 50- (200) member ensemble was run with a covariance localization length scale L of 4000 (6000) km (the 100-member ensemble experiment was run with L = 5000 km), since covariance localization is intended primarily to remove spurious long-range correlations between analysis and observation points that arise from sampling error less localization should be needed for larger ensembles. The covariance inflation parameters for the 50- and 200- member experiments were fixed to the values used in the 100-member experiment. Figure 6 shows time series of 500-hPa analysis error for these experiments. For the 16-day assimilation period, the error of the 50-member ensemble is about 8% more than the 100-member ensemble. Increasing the ensemble size from 100 to 200 members results in a very slight reduction in analysis error. From these results we conclude that increasing the ensemble size beyond 100 provides little benefit. This could either be because all sources of analysis error for this sparse observation network are well sampled with an ensemble of 100, or more likely, other sources of error, such as model error, become relatively more important than sampling error when ensemble sizes exceed 100.

We have also performed experiments using observations for June 2001 subsampled to simulate the 1915 network (not shown). The 500-hPa geopotential height rms Northern Hemisphere analysis errors for both the EnSRF and CDAS-SFC analyses are about the same for June as they are for December. However, since the climatological variance of 500-hPa geopotential height is smaller in June than December, the analysis errors expressed in terms of anomaly correlation degrade in June relative to December (0.85 versus 0.95 for the EnSRF). When compared to forecasts initialized from the NCEP– NCAR reanalysis for all Junes from 1979 to 2001, the EnSRF analyses generated from the simulated June 1915 network have an rms 500-hPa geopotential height error roughly equivalent to the 84-h forecast (as compared to 60-h forecasts for the simulated December 1915 network). The degradation in the June analysis error relative to December, which occurs for all candidate analysis schemes, is likely due to the fact that covariances between surface pressure and other variables at other levels in the troposphere is larger in winter, when coherent baroclinic systems are more prevalent.

4. Discussion and conclusions

A Northern Hemisphere reanalysis of the middle and lower troposphere for the first half of the twentieth century is feasible using only surface pressure observations. Ensemble data assimilation techniques are particularly well suited to the task and can be expected to produce 500-hPa analyses with errors similar to current 2–3-day forecasts.

Before such a reanalysis can be undertaken, methods for quality control of the historical surface pressure observations will need to be developed. Background-error covariance estimates from the ensemble data assimilation system could provide a basis for a simple “background check” (Dee et al. 2001) that marks as suspect all observations whose deviation from the ensemble mean first guess is greater than some factor times the background-error variance estimate at the observation location. Typically, those observations flagged as suspect by the background check are then subjected to a “buddy check” (Dee et al. 2001) that compares suspect observations to nearby observations that passed the background check. Unfortunately, when observations are sparse, there may not be very many “buddies.” Observation errors are likely to be larger than they are assumed to be in this study, particularly for ships. Currently we have no reliable estimates of surface pressure observation error for observations taken in the early twentieth century.

Data assimilation is typically used to generate initial conditions for numerical weather forecasts. Therefore, each analysis is based upon only current and past observations. However, when producing a retrospective reanalysis, one is free to use all available observations, including those data collected after the analysis time. A Kalman smoother is a direct generalization of the Kalman filter, which incorporates observations both before and after the analysis time. Whitaker and Compo (2002) introduced an ensemble square root smoother (EnSRS), which is a generalization of the EnSRF based upon the fixed-lag Kalman smoother proposed by Cohn et al. (1994). In future work, we will be investigating the use of the EnSRS to see if using observations taken past the analysis time can improve upon the accuracy of the EnSRF analyses presented here.

Ensemble data assimilation systems are only now moving from the realm of perfect model, “identical twin” experiments to real-world cases with actual observations. There is still research needed to fully realize the potential that ensemble data assimilation holds for improving analyses and forecasts. In particular, our results suggest that spatially correlated “errors of representation” (which are incorporated into the overall observation error in most data assimilation schemes) may adversely affect the performance of ensemble filters that assume spatially uncorrelated observation errors. In addition, parameterizations of model error that are more sophisticated than the combination of covariance localization and inflation employed here will almost certainly improve the performance of ensemble data assimilation systems.

Our results demonstrate that with some further development, advanced ensemble data assimilation systems and the available surface pressure observations could be used to create a reanalysis of the entire twentieth century. Such a dataset would be useful for determining decadal variations of synoptic-scale variability in the Northern Hemisphere extratropics.