## 1. Introduction

Reanalysis data are an important source of information for atmospheric and hydrometeorological studies. Reanalysis estimates are uncertain because they are a result of a combination of an imperfect model and uncertain observations using a data assimilation method. Unfortunately, uncertainty estimates are not normally provided with reanalysis data.

Global reanalyses currently use variational data assimilation methods, with three-dimensional variational data assimilation (3DVAR; Parrish and Derber 1992) used in the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) Global Reanalysis 1 (NCEP-R1) and NCEP/Department of Energy Global Reanalysis 2 (NCEP-R2). Sensitivity information associated with uncertainty can be obtained by the Hessian, or the second-order derivative information, which is associated with the cost function in variational methods (Wunsch 2006). However, operational centers do not compute this information because it is computationally expensive.

Some information on quality in the form of classes of variables (class A, class B, and class C) is given in current global reanalyses such as NCEP-R1 (Kalnay et al. 1996). These categories are subjective and depend on the relative influence of observations on the variable in the data assimilation. Winds and upper air temperature are denoted as class A—most reliable—because they strongly depend on observations. Specific humidity and surface temperature are denoted as class B, with direct influence of both observations and the model. Clouds and precipitation are not assimilated in NCEP-R1 and are denoted as class C, with no direct reliance on observations.

The reanalysis is a retrospective procedure, which uses a database of archived observations. These observations include surface, radiosonde, and satellite data that are available at varying times and locations. For example, radiosondes are land observations that provide vertical information, usually every 12 h and are reasonably dense over populous regions. In forecasting systems (i.e., NCEP-R1 and NCEP-R2), a model update is usually made every 6 h using the data assimilation method, which takes into account the available observations. The data assimilation method weights observations and model forecasts (from the last observation or analysis point), but it also updates locations not coincident with the observations using correlation (covariance) information. There is always model error involved in this step.

The data assimilation method attempts to yield an estimate with less uncertainty than either the model prediction or observations. Information associated with the data assimilation step of the reanalysis can be quite valuable in determining its quality. A key piece of information, the model “first guess” (6-h forecast) is sometimes saved with the reanalysis estimates in data archives. The analysis increment (AI), defined here as the analysis minus the model first guess (6-h forecast), can then be computed. The analysis increment is relevant where there are sufficient observations. In regions of few observations, the analysis increment will be small because unless the observations are very accurate, there will not be much difference between the model first guess and the analysis.

The data assimilation (DA) system itself has been used to monitor observations and data quality control (Hollingsworth et al. 1986) by computing statistics involving observations, such as observation increments. Observation increments are differences between the 6-h forecast and the observations and are considered an excellent measure of the quality of the 6-h forecast (Kistler et al. 2001). However, observation increments are not easily computed a posteriori for a large database like the reanalysis (Kistler et al. 2001). Kistler et al. (2001) state that the analysis increment can be used as a proxy for observation increments over data-rich regions. Also, because the AI is a measure of the 6-h forecast error started from the previous analysis, it provides a quantitative assessment of the quality of the analysis. Kistler et al. (2001) use AI root-mean-square (rms) statistics to study changes in the NCEP–NCAR 50-yr reanalysis. The effect of changes in the observation network from the 1950s to the 1980s on the quality of the reanalysis was analyzed by comparing the AI rms 500-mb geopotential height during the years 1950 and 1980.

Analysis increment information has also been used in a few studies involving global reanalysis data. Alpert et al. (1998) use the National Aeronautics and Space Administration (NASA) Goddard Earth Observing System (GEOS-1) reanalysis temperature AI to infer dust heating rates in the lower atmosphere. The patterns of monthly mean AI of temperature are found to be similar to the observed patterns of dust. Using a similar strategy, Jones et al. (2004) use the NCEP-R1 analysis increments of 700-mb geopotential height to infer the effect of African desert dust on easterly waves. In a hydrology-related study, Betts et al. (2003) investigate the 40-yr European Centre for Medium-Range Weather Forecasts Re-Analysis (ERA-40) over the period 1968–2001 in the Mackenzie River basin to assess systematic biases in water and energy budgets. The precipitable water analysis increment is found to be correlated with precipitation bias and spin up. The negative precipitable water AI in early years is associated with a dry bias in a particular assimilated humidity dataset. Betts et al. (2003) also analyze soil water AI and snow water equivalent (SWE) AI. The SWE analysis increment is found to be large and associated with problems in the snow model.

In this study, we focus on the monthly standard deviation of the AI, which is associated with quality, and compute the analysis increment statistics for two different global reanalyses, NCEP-R1 and NCEP-R2, at the same location and for the same variable. The objective of this study is to use AI statistics to rank or compare the analysis accuracy of the same variable at the same location.

Note that the NCEP-R1 and NCEP-R2 reanalyses have the same assimilated observation database (Kanamitsu et al. 2002). Therefore, the same fixed observation network is used in both reanalyses, and the AI rmse monthly statistics can be interpreted as a rank in regions of sufficient observations.

Currently, uncertainty information is not provided with reanalysis estimates. A future goal would be to provide reanalysis users with a quantitative uncertainty value—for example, an estimate of the standard deviation of analysis variables. This study can be considered a step toward this goal, achieved by using information that is sometimes available, the model first guess.

We compute the AI statistics for zonal and meridional wind (*u*, *υ*) and specific humidity (*q*) variables at three atmospheric levels (850, 500, and 300 mb), which are relevant to atmospheric moisture transport. Both a North American and a South American location are investigated.

## 2. NCEP-R1 and NCEP-R2 reanalyses

The NCEP-R1 and NCEP-R2 datasets have the same model resolution (T62L28; horizontal resolution ∼200 km) and share a similar observation database; therefore, differences can be attributed to different model parameterizations. There are several important model differences between NCEP-R1 and NCEP-R2 that are relevant to hydrometeorology variables, particularly the boundary layer, radiation, and soil hydrology models. The NCEP-R1 boundary layer model uses a local-K vertical diffusion scheme based on local gradients of wind and temperature.

Betts et al. (1996) compared the First International Satellite Land Surface Climatology Project (ISLSCP) Field Experiment (FIFE) field observations to NCEP-R1 reanalysis and found that the scheme produces a realistic well-mixed boundary layer but underestimates boundary layer deepening by entrainment. Subsequently, Hong and Pan (1996) tested and implemented in NCEP-R2 a nonlocal vertical diffusion scheme, which considers a countergradient term in the diffusion equation. The height of the boundary layer is solved iteratively based on the critical bulk Richardson number. Above the mixed layer, the free atmosphere is modeled as in the NCEP-R1 local-K scheme.

Radiation is computed on the full resolution grid in NCEP-R2 and on an hourly basis rather than three every hour as in NCEP-R1 (Kanamitsu et al. 2002). A new shortwave radiation scheme is also implemented in the NCEP-R2 model (Chou 1992; Chou and Lee 1996).

In NCEP-R1, model precipitation is used to force a two-layer soil hydrology model [thickness of 10 and 190 cm; Betts et al. (1996)], and a nudging term is used to prevent soil wetness from drifting too far from climatology. For NCEP-R2, a soil moisture correction procedure is used for incorporating observed 5-day mean precipitation data from a retrospective global precipitation analysis (refer to Kanamitsu et al. 2002 for details).

## 3. Data and processing

For NCEP-R1 reanalysis, the 6-h forecast (first guess) files are available in World Meteorological Organization (WMO) gridded binary (GRIB) format from the data support section (DSS) archive at NCAR in Boulder, Colorado. For NCEP-R2 data, first guess data were obtained directly from the National Oceanic and Atmospheric Administration (NOAA; W. Ebisuzaki 2005, personal communication) also in GRIB format, which was regenerated for the period of the original NCEP-R2 reanalysis, 1979–2001. (The archive at NCAR DSS does not contain NCEP-R2 first guess information.) The NCEP-R1 and NCEP-R2 reanalysis values, four-times-daily pressure level data on a 2.5° grid, were obtained in netCDF format from the NOAA Climate Diagnostics Center (CDC) archives.

Specific humidity is available in NCEP-R1 archives. For NCEP-R2, specific humidity is calculated from relative humidity and temperature. Because the forecast is valid 6 h ahead of the reference time, and the analysis is valid at the reference time, the forecast time series is shifted 6 h back to compute the analysis increment. The monthly standard deviation of the analysis increment is computed based on the 6-hourly data; therefore, there are approximately *N* = 120 samples for each month.

## 4. Results

Monthly analysis increment statistics are presented for a North American location (40°N, 80°W) and a South American location (10°S, 62.5°W) for a 4-yr period (1998–2001) using the NCEP-R1 and NCEP-R2 datasets. These two locations were chosen on the basis of different observation density and climate. The North America (40°N, 80°W) location is downwind of a high observation density region (i.e., radiosondes), whereas the South American location is a low observation density region.

We present results at the 850-, 500-, and 300-mb atmospheric levels. The three atmospheric levels chosen are relevant to atmospheric moisture transport and are a subset of the available atmospheric levels (there is negligible moisture above 300 mb). The full set of 17 pressure levels for wind variables is 1000, 925, 850, 700, 600, 500, 400, 300, 250, 200, 150, 100, 70, 50, 30, 20, and 10 mb. The specific humidity is only available for the lowest eight pressure levels, up to 300 mb.

### a. Specific humidity

The specific humidity results for the North American location (40°N, 80°W) and for two (37.5°N, 80°W and 42.5°N, 82.5°W) of the surrounding eight points on the 2.5° pressure level grid are shown in Fig. 1. These results are qualitatively representative of the eight surrounding points of 40°N, 80°W for which we have computed AI statistics. For the North American location, the mean monthly standard deviation AI for specific humidity is smaller for NCEP-R2 than for NCEP-R1 at the 300-mb atmospheric level. The statistics are about the same for the other two levels. The mean of the monthly standard deviation for the four years is shown to allow a more direct comparison of the standard deviation AI magnitude of the two reanalyses.

Figure 2 gives the specific humidity AI statistics for the South American location (10°S, 62.5°W) with two (12.5°S, 62.5°W and 7.5°S, 65°W) of the surrounding eight points that were analyzed. For NCEP-R1, there is not much interannual variation in the period 1998–2001 (data not shown). However, for NCEP-R2 there is a large increase in monthly standard deviation AI for 1998 at the 500- and 850-mb levels starting in September (Fig. 3), when monthly statistics are shown for each year individually (instead of a mean of four years of monthly statistics in Fig. 2). This increase in standard deviation AI peaks in either September or October for the 850- and 500-mb levels and in November for the 300-mb level.

### b. Wind fields

For the North America location (40°N, 80°W) and the two surrounding points (37.5°N, 80°W and 42.5°N, 82.5°W), the monthly AI statistics are very similar for NCEP-R1 and NCEP-R2 at each atmospheric level (850, 500, and 300 mb) for both meridional (*υ*) and zonal wind (*u*). The mean monthly AI standard deviation of the four years for zonal wind is shown in Fig. 4 and corresponding meridional wind statistics are shown in Fig. 5.

The monthly standard deviation AI could be considered a crude estimate of the standard deviation of the analysis estimate. The standard deviation of the analysis estimate is expected to be smaller than the standard deviation of the observation error because data assimilation should lead to estimates with smaller uncertainty than either observations or first guesses. To compare with observation error, a typical radiosonde error standard deviation at the 500-mb level used in data assimilation is *σ _{u}* =

*σ*= 2.8 m s

_{υ}^{−1}(Cohn et al. 1994). The 500-mb monthly standard deviation AI for zonal (Figs. 4, middle row) and meridional wind (Figs. 5, middle row) is about 1.5 m s

^{−1}, up to 2 m s

^{−1}, which is smaller than typical observation error and thus consistent with expectations.

For the South American location and the two surrounding points, the mean monthly AI standard deviation of the four years for zonal wind is shown in Fig. 6, and corresponding meridional wind statistics is shown in Fig. 7. For NCEP-R1 monthly AI statistics, there is not much interannual variation in the period 1998–2001 (data not shown). However, for NCEP-R2 there is a large increase in monthly standard deviation AI around October 1998, most clearly seen at the 500-mb level, Fig. 8 (zonal wind) and Fig. 9 (meridional wind).

## 5. Discussion

The reason for the large increase in NCEP-R2 monthly AI standard deviation for specific humidity during September–December 1998 compared to the other years (Fig. 3) may be related to the strong transition from an El Niño event in the first few months of 1998 to the La Niña event in December 1998. This increase may also be related to the fact that the NCEP-R2 outgoing longwave radiation (OLR) over the tropical warm pool and upper level tropical moisture are known to be less accurate in NCEP-R2 than in NCEP-R1 (Kanamitsu et al. 2002).

The monthly standard deviation (*σ*) statistics presented in this paper include the monthly mean. The standard deviation, *σ*(*x*) = {*E*(*x*^{2}) − [*E*(*x*)]^{2}}^{1/2}, contains a term involving the mean *E*(*x*). A large persistent AI mean would indicate potential model error (Dee 2005). We decompose the variance into its two parts for the months of 1998, and the largest component by far is the *E*(*x*^{2}) part of the variance in Fig. 3. This suggests that the large AI standard deviation anomaly in September–December 1998 is not a result of bias caused by model error.

For both zonal and meridional wind, the magnitude of monthly AI standard deviation is similar for NCEP-R1 and NCEP-R2 at all atmospheric levels for the North American location. This is expected because of the high observation density upwind of the location (40°N, 80°W). For the South American location, the AI standard deviation statistics are similar for NCEP-R1 and NCEP-R2 winds, except for the September–November 1998 period, where the monthly standard deviation AI is much larger for NCEP-R2. The wind statistics have a similar pattern to the specific humidity NCEP-R2 AI statistics in the late 1998 period and could be related to the transition to La Niña. The wind field AI results are consistent with the Kanamitsu et al. (2002) statement that the NCEP-R2 reanalysis may not necessarily provide better analyses (estimates) than the NCEP-R1 analysis.

Analysis increment statistics take some care to interpret. First, analysis increment statistics alone are not an absolute measure of the quality of the analysis. Kistler et al. (2001) state that the analysis increment is a proxy for the observation increment. In optimal linear filtering, the observation increment is also known as the innovation. Theoretically, we could have an example in which the mean AI is approximately zero and the monthly standard deviation of AI is small, but it is not an optimal filter (i.e., the mean innovation sequence is not zero). However, we are using the reanalysis output from the fixed forecast data assimilation system of NCEP (NCEP-R1 and NCEP-R2), and it is reasonable to assume that the assimilation is trying to give the best forecast and was not tuned to arbitrarily lower analysis increments.

There are several issues associated with this work. There may be inaccuracies as a result of the use of pressure level data for the analysis and first guess, which is interpolated from the original model sigma-level data. Over the course of a month (*N* = 120 samples), this is probably not a significant effect. Pressure level data is used to compute monthly AI statistics at various fixed atmospheric levels. The specific humidity was computed for NCEP-R2 from air temperature and relative humidity, which are uncertain variables. This may not be significant over a month period. There is also a peculiarity in the wind data, which is archived in tenths of a meter per second significance. There is probably not much effect on the AI statistics.

Despite the above concerns, the results show that AI statistics can provide useful information when comparing reanalysis models. In this case study, we learn that the quality of NCEP-R1 and NCEP-R2 specific humidity and wind products are generally comparable over a well-sampled North American location, with the possible slight advantage of NCEP-R2 in predicting specific humidity at upper levels of the atmosphere (where it is generally less important.) Over the South American location—with less observations—it is clear that NCEP-R2 performs worse in some years, and this is traceable to reported accuracy problems of NCEP-R2 with outgoing longwave radiation over tropical warm pools. The year in question here involved a strong El Niño–La Niña transition in 1998. We posit that this type of diagnostic information is very useful.

The NCEP-R1 and NCEP-R2 datasets were chosen for a first study because these datasets have the same model resolution (T62) and share a similar observation database; therefore, differences can be attributed to different model parameterizations. A future study could be to compare ERA-40 (Uppala et al. 2005) with resolution T159 (approximately 125-km horizontal resolution) analysis increment statistics with NCEP-R1 and NCEP-R2 statistics.

Analysis increment information can be obtained for any data assimilation method. The ensemble Kalman filter (EnKF) is currently being tested at operational centers and has been tested in a reanalysis application (Whitaker et al. 2004). The AI is relevant for the EnKF because although the filter provides an error estimate in the form of the spread of the ensemble filter, the accuracy of the second moment information is problematic because of sampling effects (Furrer and Bengtsson 2007).

## 6. Summary

We have examined monthly standard deviation analysis increment statistics for two reanalyses datasets, NCEP-R1 and NCEP-R2, associated with a North American location and a South American location. The analysis increment, the analysis minus the model first guess (6-h forecast), can be computed by the user if the model first guess is available in the reanalysis archives. The standard deviation of the analysis increment could provide a quantitative measure of quality in which there are sufficient observations.

The monthly AI statistics of specific humidity, zonal, and meridional wind were investigated for the period 1998–2001. For specific humidity, NCEP-R2 was found to have smaller monthly standard deviation AI than NCEP-R1 for the 4-yr period at the North American location at the 300-mb level. The South American location was found to have much larger NCEP-R2 standard deviation AI for specific humidity than NCEP-R1 in September–November 1998, which may be related to the transition to La Niña during that period.

For both the meridional and zonal wind, the monthly standard deviation AI are of similar magnitude for NCEP-R1 and NCEP-R2 North America at the atmospheric levels 850, 500, and 300 mb. This was expected partly because of the high observation density. The NCEP-R2 South American location wind statistics were found to have a similar pattern as the specific humidity, which showed to have much larger monthly standard deviation AI in the September–November 1998 period than NCEP-R1.

This study provides an example of a diagnostic analysis comparing analysis increments between two reanalyses, providing a quantitative assessment of reanalysis quality. The AI statistics require careful interpretation because they are a proxy to the observation increment, an indicator of quality. By computing the analysis increment, the user can determine a quantitative index of any variable of interest at any location where there are sufficient observations. Reanalysis data is an essential part of the observing system, and a clearinghouse of reanalyses is planned. It is envisioned that a database of reanalyses with different models, model resolutions, and physics parameterizations (e.g., boundary layer and convection models) would include these diagnostics, which would either be readily available or be able to be computed by the user.

## Acknowledgments

Wesley Ebisuzaki of NOAA kindly provided the NCEP-R2 first guess reanalysis data. This research was supported by NASA TRMM (Tropical Rainfall Measuring Mission) Grant NAG5–13638. We thank the reviewers for helping to improve this paper.

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*Discrete Inverse and State Estimation Problems: With Geophysical Fluid Applications*. Cambridge University Press, 372 pp.

Same as Fig. 1 but for (left column) the South American location 10°S, 62.5°W, and surrounding locations (middle column) 12.5°S, 62.5°W and (right column) 7.5°S, 65°W.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 1 but for (left column) the South American location 10°S, 62.5°W, and surrounding locations (middle column) 12.5°S, 62.5°W and (right column) 7.5°S, 65°W.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 1 but for (left column) the South American location 10°S, 62.5°W, and surrounding locations (middle column) 12.5°S, 62.5°W and (right column) 7.5°S, 65°W.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 2 but showing the individual years 1998 (points), 1999 (circles), 2000 (triangles), and 2001 (crosses); NCEP-R2 dataset.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 2 but showing the individual years 1998 (points), 1999 (circles), 2000 (triangles), and 2001 (crosses); NCEP-R2 dataset.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 2 but showing the individual years 1998 (points), 1999 (circles), 2000 (triangles), and 2001 (crosses); NCEP-R2 dataset.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 4 but for meridional wind *υ* (m s^{−1}).

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 4 but for meridional wind *υ* (m s^{−1}).

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 4 but for meridional wind *υ* (m s^{−1}).

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 4 but for (left column) the South American location 10°S, 62.5°W, and surrounding locations (middle column) 12.5°S, 62.5°W and (right column) 7.5°S, 65°W.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 4 but for (left column) the South American location 10°S, 62.5°W, and surrounding locations (middle column) 12.5°S, 62.5°W and (right column) 7.5°S, 65°W.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 4 but for (left column) the South American location 10°S, 62.5°W, and surrounding locations (middle column) 12.5°S, 62.5°W and (right column) 7.5°S, 65°W.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 6 but for meridional wind *υ* (m s^{−1}).

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 6 but for meridional wind *υ* (m s^{−1}).

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 6 but for meridional wind *υ* (m s^{−1}).

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 8 but for meridional wind *υ* (m s^{−1}); NCEP-R2 dataset.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 8 but for meridional wind *υ* (m s^{−1}); NCEP-R2 dataset.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 8 but for meridional wind *υ* (m s^{−1}); NCEP-R2 dataset.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Mean monthly standard deviation AI of zonal wind *u* (m s^{−1}) for 1998–2001 using NCEP-R1 (points), and NCEP-R2 (circles) at (left column) North American location 40°N, 80°W, and surrounding locatins (middle column) 37.5°N, 80°W and (right column) 42.5°N, 82.5°W.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Mean monthly standard deviation AI of zonal wind *u* (m s^{−1}) for 1998–2001 using NCEP-R1 (points), and NCEP-R2 (circles) at (left column) North American location 40°N, 80°W, and surrounding locatins (middle column) 37.5°N, 80°W and (right column) 42.5°N, 82.5°W.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Mean monthly standard deviation AI of zonal wind *u* (m s^{−1}) for 1998–2001 using NCEP-R1 (points), and NCEP-R2 (circles) at (left column) North American location 40°N, 80°W, and surrounding locatins (middle column) 37.5°N, 80°W and (right column) 42.5°N, 82.5°W.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 6 but showing the individual years 1998 (points), 1999 (circles), 2000 (triangles), and 2001 (crosses); NCEP-R2 dataset.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 6 but showing the individual years 1998 (points), 1999 (circles), 2000 (triangles), and 2001 (crosses); NCEP-R2 dataset.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1

Same as Fig. 6 but showing the individual years 1998 (points), 1999 (circles), 2000 (triangles), and 2001 (crosses); NCEP-R2 dataset.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM946.1