Statistical Precipitation Downscaling over the Northwestern United States Using Numerically Simulated Precipitation as a Predictor*

Transcription

1 1MARCH 2003 WIDMANN ET AL. 799 Statistical Precipitation Downscaling over the Northwestern United States Using Numerically Simulated Precipitation as a Predictor* MARTIN WIDMANN AND CHRISTOPHER S. BRETHERTON Department of Atmospheric Sciences, University of Washington, Seattle, Washington ERIC P. SALATHÉ JR. Climate Impacts Group, Joint Institute for the Study of the Atmosphere and Ocean/School of Marine Affairs, University of Washington, Seattle, Washington (Manuscript received 8 October 2001, in final form 13 May 2002) ABSTRACT This study investigates whether GCM-simulated precipitation is a good predictor for regional precipitation over Washington and Oregon. In order to allow for a detailed comparison of the estimated precipitation with observations, the simulated precipitation is taken from the NCEP NCAR reanalysis, which nearly perfectly represents the historic pressure, temperature, and humidity, but calculates precipitation according to the model physics and parameterizations. Three statistical downscaling methods are investigated: (i) local rescaling of the simulated precipitation, and two newly developed methods, namely, (ii) downscaling using singular value decomposition (SVD) with simulated precipitation as the predictor, and (iii) local rescaling with a dynamical correction. Both local scaling methods are straightforward to apply to GCMs that are used for climate change experiments and seasonal forecasts, since they only need control runs for model fitting. The SVD method requires for model fitting special reanalysis-type GCM runs nudged toward observations from a historical period (selection of analogs from the GCM chosen to optimally match the historical weather states might achieve similar results). The precipitationbased methods are compared with conventional statistical downscaling using SVD with various large-scale predictors such as geopotential height, temperature, and humidity. The skill of the different methods for reconstructing historical wintertime precipitation ( ) over Oregon and Washington is tested on various spatial scales as small as 50 km and on temporal scales from months to decades. All methods using precipitation as a predictor perform considerably better than the conventional downscaling. The best results using conventional methods are obtained with geopotential height at 1000 hpa or humidity at 850 hpa as predictors. In these cases correlations of monthly observed and reconstructed precipitation on the 50-km scale range from 0.43 to The inclusion of several predictor fields does not improve the reconstructions, since they are all highly correlated. Local rescaling of simulated precipitation yields much higher correlations between 0.7 and 0.9, with the exception of the rain shadow of the Cascade Mountains in the Columbia Basin (eastern Washington). When the simulated precipitation is used as a predictor in SVD-based downscaling correlations also reach 0.7 in eastern Washington. Dynamical correction improves the local scaling considerably in the rain shadow and yields correlations almost as high as with the SVD method. Its combination of high skill and ease to use make it particularly attractive for GCM precipitation downscaling. 1. Introduction Atmospheric general circulation models (GCMs) as used in climate change experiments or seasonal forecasts have a typical resolution of a few hundred kilometers and thus regional aspects of the climate are rep- * Joint Institute for the Study of the Atmosphere and Ocean Contribution Number 888. Corresponding author address: Dr. Martin Widmann, Institute for Coastal Research, GKSS Research Centre, D Geesthacht, Germany. widmann@gkss.de resented by at most a few grid cells. Since GCMs are usually considered to yield unrealistic results on spatial scales smaller than several grid cells, there is in general little confidence in the simulated, regional-scale climate variability. In particular, the regional precipitation in the models is discounted, because precipitation can be strongly influenced by subgrid-scale processes, some of which are parameterized, while others, such as details of the land water distribution or topography, are not represented at all. Downscaling techniques are often used to derive precipitation variability from GCM simulations on or below the grid-cell scale. They are based on the assumption that atmospheric variability on small spatial scales is 2003 American Meteorological Society

2 800 JOURNAL OF CLIMATE VOLUME 16 conditioned, though not determined, by larger scales (Starr 1942; von Storch 1995, 1999). In recent years, a number of numerical dynamical and statistical downscaling methods have been developed, which are reviewed in Giorgi et al. (2001). Our paper focuses on statistical methods. They usually employ empirically derived, statistical formulations of the links between large and small scales, and select predictor variables that are considered to be well resolved and realistically simulated by the GCM. Common predictors for precipitation are, for instance, geopotential height (e.g., Kidson and Thompson 1998; Zorita and von Storch 1999), sea level pressure (Cavazos 1999), geostrophic vorticity (Osborn et al. 1999; Wilby et al. 1998), or wind speed (Murphy 1999). In this paper we present and evaluate three methods that statistically derive small-scale precipitation from GCM-simulated precipitation. The conceptual basis of our methods is different from the standard downscaling approach. When following the standard approach in cases where large-scale precipitation is used as the predictor, one would derive the statistical relationship between observed large-scale and small-scale precipitation, socalled disaggregation schemes, and apply these to simulated large-scale precipitation. This would only make sense if the simulated precipitation was realistic. Since GCM-simulated precipitation can deviate substantially from observations, especially around complex terrain, it is not useful to apply the standard downscaling approach to large-scale precipitation as the predictor variable. Instead we statistically link small-scale precipitation in the real world to the precipitation in the model world, which differs from the real world, for instance, due to a smoother topography or due to imperfect precipitation parameterizations. This is analogous to the model output statistics approach used in numerical weather forecasting. Our methods could be approximated by a first step, where the model errors on the model grid-cell scale are corrected, and a subsequent application of a disaggregation scheme. In practice both steps are performed together. The potential of GCM-simulated precipitation as a predictor for small-scale precipitation appears to have not been addressed earlier partly because of model fitting issues and partly because of the the poor reputation of GCM precipitation, some of which presumably stems from the validation of Atmospheric Model Intercomparison Project type (AMIP) GCM runs (Lau et al. 1996), where the models are forced by observed sea surface temperature and sea ice conditions. The unrealistic precipitation in these runs (Lau et al. 1996; Risbey and Stone 1996; Osborn and Hulme 1998; Leung and Ghan 1999) can either be due to unrealistic largescale circulation states in the GCMs or to errors in relating the large-scale circulation states to simulated precipitation on the model grid. Only the latter error is relevant for assessing whether downscaled precipitation is more realistic than the precipitation simulated in the GCM; if the simulated large-scale circulation is unrealistic, even a perfect downscaling method will yield meaningless results. In fact, the studies of Risbey and Stone (1996) and Leung and Ghan (1999) suggest that large-scale circulation model errors contribute substantially to model observation precipitation differences in AMIP-type GCM runs. Since even a perfect GCM will reproduce only the observed statistics of the observed atmospheric states when run in an AMIP-type experiment, rather than the historic sequence of the daily weather, all of the abovementioned studies were restricted to comparing statistical properties of simulated and observed precipitation, such as mean fields, variances, or probability distributions, and therefore were not suited to explore whether there are relationships between simulated and true precipitation time series, such as linear correlations, that could be exploited for downscaling purposes. The extent to which simulated precipitation is consistent with the simulated large-scale atmospheric state has been examined in several studies, most of which analyze whether GCMs are able to reproduce observed, statistically formulated links between large-scale variables and regional precipitation. Their results are not uniform. Busuioc et al. (1999) found that the ECHAM3 model reproduces well the observed link between European sea level pressure (SLP) and Romanian precipitation (in terms of canonical correlation patterns) in winter and to a moderate degree in fall and spring. Osborn et al. (1999) showed that the observed relations between precipitation over the United Kingdom and the local geostrophic flow strength, direction, and vorticity are well simulated by the HadCM2 model. However, Wilby and Wigley (2000) found the HadCM2 less satisfactory in reproducing the observed correlations between daily precipitation over six regions in the United States and a variety of different atmospheric predictor variables. A more direct approach to analyzing how well a GCM simulates downscale links can be taken if the simulated large-scale circulation represents a historic period. In this case the simulated precipitation time series can be compared with observations. Following this idea, Murphy (1999) relaxed the circulation in a GCM from the U.K. Meteorological Office toward operational analyses for the period May 1983 to February Grid-cell precipitation over Europe was found to be well correlated with station records within a given grid cell. The average correlation of monthly anomalies varies between 0.39 and 0.68, with the highest values occurring in winter. For most month of the year and in most regions, the GCM precipitation was found to be too high and to have too little variance. The latter discrepancy can be at least partly attributed to the fact that area means have less variance than point observations (cf. Osborn and Hulme 1997). As an alternative to GCM simulations nudged to operational analyses, Widmann and Bretherton (2000) pro-

3 1MARCH 2003 WIDMANN ET AL. 801 posed the examination of precipitation from atmospheric reanalyses to validate GCMs under historic flow conditions. By virtue of the data assimilation, the simulated large-scale atmospheric states of variables such as pressure, temperature, and humidity in a reanalysis are in excellent agreement with the real-world situation over data-rich regions. Reanalyzed precipitation, however, is derived purely from the model, without assimilation of precipitation observations. Roughly speaking, we think of the reanalysis as an ideal GCM, in which the timevarying, synoptic-scale circulation is accurately represented. Thus, a comparison of reanalyzed and observed precipitation quantifies the error in simulated precipitation that is purely caused by incomplete simulation of the links from the large-scale circulation state to local precipitation. Reanalyzed precipitation was investigated in several prior studies (Mo and Higgins 1996; Higgins et al. 1996; Gutowski et al. 1997; Janowiak et al. 1998). These studies did not address downscaling questions, but they did show that the correlation of reanalyzed and observed precipitation is quite high in most locations, though it varies strongly over the globe (see especially Janowiak et al. 1998). This is in accordance with the differing results of the statistical model analyses of Busuioc et al. (1999) and Osborn et al. (1999) on the one hand and those of Wilby and Wigley (2000) on the other. The reanalysis is directly comparable only to observed area-mean precipitation over a reanalysis grid box, rather than raw station data. This is of particular importance around mountainous terrain, in which elevation and slope aspect strongly influence local precipitation. Widmann and Bretherton (2000) constructed a 50-yr daily precipitation dataset over Oregon and Washington on a (50 km) 2 grid, interpolated from rain gauge observations and corrected for elevation and slope aspect. After averaging down to the coarser reanalysis grid resolution, they compared their observational estimates with the NCEP NCAR reanalysis (Kalnay et al. 1996) precipitation. Long-term means and month-to-month variability on spatial scales of about 500 km or three grid cells were captured well by the NCEP NCAR model. At individual grid cells the NCEP NCAR reanalysis had systematic biases, which can be mainly attributed to the poor representation of the topography, but the temporal correlations with the observations are high (regionally dependent between 0.67 and 0.87 for monthly means). Widmann and Bretherton suggested that the good agreement was due to the reanalysis accurately simulating the overall structure of the midlatitude cyclones that bring this region most of its precipitation, even though their interaction with topography was underresolved. Their study suggests that in such a region, model-predicted precipitation should be considered as a central ingredient in a statistical precipitation downscaling approach. This paper extends their study by testing some statistical downscaling approaches that use GCM-simulated precipitation to predict local precipitation over Washington and Oregon using the NCEP NCAR reanalysis. As mentioned before, the reanalysis is regarded here as a GCM whose large-scale circulation follows the historical record over some period. The statistical models that are discussed in this paper are meant to be eventually applied to output from other GCMs, which are used, for instance, for climate change experiments or seasonal forecasts. The simplest method that we analyze is rescaling of the reanalyzed precipitation by a spatially varying, but time-invariant factor. To handle some deficiencies of this method in local rain shadow regions, we consider a second, nonlocal method based on singular value decomposition (SVD) of local precipitation and reanalyzed precipitation. Since model fitting for the SVD method is not straightforward if no reanalysis run is available, a third method, which calculates a nonlocal dynamical correction to the local scaling, is also proposed and tested. The results are compared with our (50 km) 2 areamean precipitation analyses on monthly to interannual timescales. The paper is organized as follows. In section 2 the precipitation dataset is presented and the SVD method is briefly described. Results of the conventional downscaling are shown in section 3, local rescaling and nonlocal corrections of simulated precipitation are discussed in section 4, and the two methods are compared in section 5. The dynamical correction for locally rescaled precipitation is the subject of section 6. Section 7 contains conclusions and a final discussion. 2. Data and method The grid-cell precipitation dataset used in this study is described in Widmann and Bretherton (2000). It provides daily data for the period and covers Oregon and Washington with a spatial resolution of about 0.48 (lat) 0.62 (lon) or 50 km (lat) 50 km (lon) at 45 N (Fig. 1). The grid-cell estimates are derived from 522 station records and account for biases due to the subgrid topography so as to be consistent with the Parameter-Elevation Regressions on Independent Slopes Model precipitation climatologies (PRISM; Daly et al. 1994, 1997). For most of the downscaling approaches that are investigated in this paper we use SVD of the cross-covariance matrix (Bretherton et al. 1992) to statistically describe the link between regional precipitation and predictors for it, such as simulated precipitation or synoptic-scale sea level pressure, temperature, or humidity. SVD is one among several statistical methods that could be used to estimate regional precipitation. An approach based on weather types was tested in Widmann and Schär (1997) and yielded poor results with respect to precipitation in Switzerland. Therefore we choose a linear method that objectively finds coupled anomaly patterns in the predictor and predictand fields. Among lin-

4 802 JOURNAL OF CLIMATE VOLUME 16 FIG. 1. Topography of the northwestern United States along with the grid for the precipitation dataset [dashed lines, resolution 52.5 km (lat) 50 km (lon)] and the Gaussian grid used in the NCEP NCAR reanalysis (solid lines). Coastlines, major rivers, and borders of the states Oregon and Washington are shown as well. ear methods SVD and canonical correlation analysis (CCA; e.g., von Storch and Zwiers 1999) are the ones most frequently used in climate research. CCA gives robust results only if both time-dependent fields are prefiltered by projecting them onto their leading empirical orthogonal functions. Whether the optimization criterion used in SVD or the one used in CCA gives better results if used for downscaling depends on details of the data. Prefiltered CCA and SVD performed similarly well in comparative studies (Bretherton et al. 1992). We choose SVD rather than CCA because its use is technically easier due to the orthogonality of the patterns, and because its application requires fewer subjective decisions. To introduce the nomenclature that will be used in subsequent sections, we summarize some aspects of the SVD method. Consider two time-dependent fields with zero temporal mean. The first, or left field is known at N s points, and the second, or right field is measured at N z points. Singular value decomposition of the crosscovariance matrix between these fields yields N min(n s, N z ) singular values i, N mutually orthogonal left singular vectors (LSVs) p i, and N mutually orthogonal right singular vectors (RSVs) q i, which are both represented by sets of row vectors. Orthogonal projection of the left (right) field at each time step onto p i (q i ) gives the ith left (right) time expansion coefficients LEC (REC) a i (b i ), which are represented by column vectors, and which satisfy cov(a i, b i ) i. The defining property of SVD is that the time expansion coefficients have the maximum possible covariance subject to the constraint of the patterns being mutually orthogonal. The right field can be reconstructed from knowledge of the left field by a linear regression of the ith REC onto the ith LEC, with the reconstructed REC given by bˆ i r i [var(b i)/var(a i)] a i [ i /var(a i )]a i, where r i denotes the correlation between the ith LEC and REC. The reconstructed right field based on n modes is Ẑ n n i 1 bˆ iq i, where we have used matrix notation for the fields. When we use cross validation, the singular vectors, as well as the variances, covariances, and correlations, are obtained from the fitting period, whereas the time series a i and bˆ i refer to the validation period. Measures for the strength of the linear coupling for a given mode include the squared covariance fraction SCF i 2 i/ rank(c) j 1 2 j (Bretherton et al. 1992) and the correlation r i. It is also useful to have a measure of how much of the overall variance var tot of the right field can be explained by a given mode. The ith RSV explains a fraction RVF i var(b i )/var tot of the total variance. Since it cannot be perfectly predicted from the left field, the reconstructed right variance fraction (RRVF) is smaller: 2 RRVFi var(bˆ i)/vartot rirvf i. (1) The variation in spatial data density in the left field is accounted for by weighting it with the square root of the inverse data density [i.e., with 1/ cos(latitude) ] prior to performing SVD. The weighted field is orthogonally projected onto the LSVs to obtain the LECs. Before displaying the singular vectors they are divided by the weights. 3. SVD with predictors other than precipitation Before analyzing the link between precipitation over Oregon and Washington and the large-scale flow, we briefly discuss the spatial organization of the precipitation field. We focus on winter, which is the season of heaviest precipitation in most of this region. The mean precipitation and the first two EOFs of monthly precipitation for DJF are shown in Fig. 2. The Coast and Cascade Mountains cause very large east west gradients in the mean field, from a rain forest climate on the west side of the Cascades to a near-desert climate on the east side. Strong across-mountain gradients are also visible in the EOFs, but neither of the EOFs changes its sign from one side of the mountains to the other. In that respect precipitation variability in Oregon and Washington is fundamentally different from that in the European Alps (Widmann et al. 1995; Widmann and Schär 1997; Frei and Schär 1998; Schär et al. 1998). In the Alps the mean precipitation is approximately the same on both sides, and EOF2 of Alpine precipitation is an across-mountain dipole, whereas the EOF2 dipole in the Cascades divides the northern and southern parts of the domain. The reason for this difference is pre-

5 1MARCH 2003 WIDMANN ET AL. 803 FIG. 2. (left) Mean precipitation and (middle),(right) the first two EOFs of monthly means for DJF The EOFs represent values associated with principal components of one std dev. The explained variance fraction is 69% for EOF1 and 16% for EOF2. sumably mostly the fact that moisture sources exist on both sides of the Alps (the Atlantic and the Mediterranean), whereas in the northwestern United States the Pacific is the primary moisture source. This allows for strong foehn effects in the Alps for both across-mountain wind directions, whereas in Oregon and Washington it is very rare during winter that it rains on the east side but not on the west side. The main effect of the Cascades is only a strong scaling of the monthly (and daily) precipitation rather than dividing two regions with a very different temporal variability. This can not only be seen in the EOFs but also in the RSVs discussed below and in one-point correlation maps, which have a fairly isotropic decay of the correlation with increasing distance (Widmann and Bretherton 2000, their Fig. 7). Thus, while the complex orography of this region greatly complicates the mean patterns of precipitation, it has much less impact on its temporal variability. Several left fields were used to predict precipitation over Oregon and Washington by means of SVD, namely, Z 1000 (1000-hPa geopotential height), q 850 (850-hPa specific humidity), and the combined fields Z 1000 /Z 500, Z 1000 / T 850 (850-hPa temperature), and Z 1000 /q 850. The model skills for the various predictor fields and for daily and monthly temporal resolution for model fitting are compared in Table 1. Monthly q 850 is the best one-field predictor for monthly precipitation, followed by monthly Z The dynamical origin of the couplings can be best understood by analyzing the Z 1000 case, for which detailed results will be presented later on. Of the most importance for downscaling is the RRVF, since this measures the fraction of the precipitation variance that is explained by the predictor field. For the first two modes combined, and using the entire period for both fitting and validation, the RRVF is 45% for q 850 and 35% for Z 1000, indicating respectable but not outstanding skill. Combined monthly left fields yield only small increases in the correlations between LECs and RECs, and therefore only slightly higher RRVFs. If the fitting is based on daily fields, the singular values (SVs) are very similar to their monthly counterparts, but the correlations of monthly means of the leading ECs and the RRVFs are slightly lower than if monthly fields are used. This similarity of daily and monthly SVD modes indicates that the coupling on the monthly timescale is basically governed by daily processes rather than by mechanisms TABLE 1. Characterization of the leading two SVD modes. Rows refer to different left fields and temporal resolutions. The four columns per mode refer to correlation coefficients, right variance fraction, reconstructed right variance fraction, and squared covariance fraction. All values given in percent and are based on DJF For SVDs based on daily fields, monthly and daily values (in parentheses) are given. Left field Monthly Z 1000 Monthly q 850 Monthly Z 1000 and q 850 Monthly Z 1000 and T 850 Monthly Z 1000 and Z 500 Daily Z 1000 Daily q 850 Daily Z 1000 and q 850 Daily Z 1000 and T 850 Daily Z 1000 and Z 500 Mode 1 r 2 RVF RRVF SCF (32) 41(40) 37(41) 48(37) 44(38) (53) 66(54) 66(54) 66(54) 66(54) (17) 27(22) 24(22) 31(20) 29(20) (90) (86) (88) (90) (90) Mode 2 r 2 RVF RRVF SCF (13) 26(17) 26(17) 22(13) 28(16) (15) 15(15) 15(15) 15(15) 15(15) (2) 4(3) 4(3) 3(2) 4(2) (7) (11) (8) (7) (7)

6 804 JOURNAL OF CLIMATE VOLUME 16 FIG. 3. (upper) First and (lower) second singular vectors (divided by the area weights used for SVD) obtained from an SVD between monthly 1000-hPa (left) geopotential height and (right) precipitation in Oregon and Washington based on DJF The left-hand patterns are associated with expansion coefficients of one std dev (isoline spacing is 10 m). The right-hand patterns depict the precipitation anomalies that can be predicted from the left-hand patterns. Note the different scales for the two precipitation patterns. specific to the monthly timescale. The variance explained by REC2 is in all cases only about 15%, which leads to RRVFs of not more than 5%. Therefore, although the second mode appears to be physically based (see below), it is of little practical relevance for predicting monthly precipitation. The leading RSVs obtained from the different SVDs are practically indistinguishable. The first RSVs are also practically identical to EOF1 of the precipitation field (Fig. 2); the second RSVs are very similar to EOF2 in most of the domain. This means that one of the two factors entering the RRVF [Eq. (1)] is already maximized. Note that this does not follow from the definition of SVD and is advantageous since our goal is to predict the right field from the left field. Physically interpreted it means that the two dominant modes of precipitation variability (i.e., the EOFs) are the imprints of the two most important large-scale forcings of precipitation rather than being the result of a superposition of the effects of multiple forcings. The LSVs obtained in the two-left-field SVDs are, after they are projected on the one-field spaces, almost identical to those found in the SVDs using one left field. The leading LSVs in the different fields are synoptically consistent and capture basically the same processes. The q 850 and T 850 SVs (not shown for brevity) are, for instance, similar to the anomaly patterns one would expect from advection by the Z 1000 SVs. This indicates that geopotential height variability is what primarily forces Oregon and Washington precipitation. However, these interfield connections are not very strict. This is indicated by only moderate correlations between the various LECs [e.g., r 0.7 (0.73) for LEC1s of Z 1000 and q 850 (T 850 )]. The strong coupling between precipitation and q 850 indicates that an important part of the Z 1000 precipitation link is established via the moisture field. The leading two modes of an SVD with Z 1000 as the left field are depicted in Fig. 3 using physical units. According to the asymmetry of the problem, these units are defined differently for the left and the right field. For the former, anomalies associated with LECs of one standard deviation are shown. For the latter we show the predictions associated with LECs of one standard deviation. These are the patterns for one standard deviation of the right coefficients times the correlations between the left and the right coefficients, or, equiva-

7 1MARCH 2003 WIDMANN ET AL. 805 FIG. 4. (left) Mean 1000-hPa geopotential height for DJF and (middle),(right) its sum with the singular vectors times / one std dev of the respective monthly expansion coefficients. The upper (lower) row shows the first (second) mode. The middle (right) column shows the mean with positive (negative) anomalies added. lently, the regression coefficients obtained from regressing the right data on the normalized (to standard deviation one) LECs. For a meteorological interpretation of these two modes, it is useful to look also at the sum of the mean field and realistically scaled anomalies (Fig. 4). We choose LECs of plus or minus one standard deviation, that is, the patterns shown in the left-hand panels of Fig. 3 were added or subtracted to the mean field. In the mean Z 1000 field (Fig. 4, left-hand panel) the southwesterly flow over the northern Pacific only weakly penetrates Oregon and Washington. For positive (negative) states of mode 1 this southwesterly flow is strengthened (weakened) and the Z 1000 field is lower (higher) over Oregon and Washington, leading to positive (negative) precipitation anomalies which are approximately proportional to the mean precipitation (Fig. 2, left-hand panel). The changes in Z 1000 as well as the changes in precipitation uniformly affect the entire analysis domain. The second mode is associated with a strong flow anomaly over Washington and only a small flow anomaly but strong Z 1000 anomaly over Oregon. In the positive state the strong southwesterly flow over Washington and the weak flow and high pressure over Oregon lead to drier (wetter) than normal conditions in Oregon (Washington). In the negative state there is a moderate southwesterly flow toward the Coast and Cascade Mountains in Oregon and along-mountain flow in Washington, leading to wetter (drier) than normal conditions in Oregon (Washington). Since the variance of precipitation in the western parts of the domain is much higher than that in the eastern parts, the SVD will be dominated by statistical connections between the large-scale circulation and precipitation in the west. We examined this effect by performing also an SVD with precipitation that was normalized to unit variance at each grid cell. The resulting SVs (not shown) have the same overall structure as those depicted in Fig. 3 so we did no further analysis in this direction. This means that the large-scale circulation patterns that have a dominant influence on precipitation in the west are very similar to those that have a dominant influence on precipitation in the east. All statistical downscaling methods rely on the stationarity of the statistical link between predictor and predictand fields and we investigated the effect of overfitting by applying cross validation. The results for Z 1000 as the predictor field are shown in Table 2. A TABLE 2. Cross-validation statistics of the leading two SVD modes with Z 1000 as predictor. Rows refer to different validation and fitting periods (P1 is , P2 is ). Columns are defined as in Table 1. All vales are given in percent. Mode 1 Mode 2 Val Fit r 2 RVF RRVF SCF r 2 RVF RRVF SCF P1 P2 P1 P1 P2 P2 P1 P2 P1 P2 P2 P

8 806 JOURNAL OF CLIMATE VOLUME 16 FIG. 5. (left),(middle) Scaling factors (observed/reanalyzed precipitation) for DJF precipitation based on two different periods. (right) The relative difference. moderate overfitting problem occurs for the validation period P1. If the SVD is fitted using P1, the RRVF1 is clearly higher than if P2 is used for fitting. This effect is less pronounced for validation period P2. Averaged over the two periods, the cross-validated RRVF for the first two modes combined is 27%, compared to 35% when the entire period P1 P2 is used for both fitting and validation. Further analysis showed that the left SV1 but not the right SV1 is somewhat sensitive to the fitting period, but all key features of the SVs discussed above remain the same. 4. Statistical downscaling using simulated precipitation as the only predictor In this section we present two methods to statistically estimate local precipitation purely from simulated precipitation. For reasons discussed in the introduction the simulated precipitation is taken from the NCEP NCAR reanalysis. The first approach uses local scaling, while the second one employs SVD to allow for nonlocal relations. a. Local scaling Systematic over- and underestimations of precipitation over Oregon and Washington in the NCEP NCAR reanalysis and local correlations of the simulated precipitation with observations have already been discussed in Widmann and Bretherton (2000), where scaling factors and correlations are calculated on the 150 km 200 km NCEP NCAR reanalysis grid. Figure 5 depicts scaling factors on the 50-km grid. When applying the local scaling method the simulated precipitation is multiplied by these factors. Since the local scaling method will be compared later on with SVD-based methods that use only one joint statistical model for all three winter months, the scaling factors are also determined with respect to DJF averages, rather than individually for each month of the year as in Widmann and Bretherton (2000). In order to determine how well scaling factors obtained from one period are applicable in another period, we split the dataset into two halfs. Over most of the area the scaling factors in the second half of the analysis period were higher than in the first half. To the west of the Cascade Mountains the differences are very small; to the east they reach moderate values up to 30%. The scaling factors with respect to both periods range from 0.17 to 3.2. b. SVD A nonlocal correction of reanalyzed precipitation can be obtained if coupled anomaly patterns between reanalyzed and observed precipitation are found. We determine these patterns by means of SVD. The leading SVs based on monthly data are displayed in Fig. 6. The fact that the leading SVs have the same large-scale structure for the left and the right field indicates that the large-scale spatial structure of wintertime precipitation over Oregon and Washington is well captured by the NCEP NCAR model. Since the model topography (not shown) does not resolve the Coast and Cascade Mountains, we conclude that mainly synoptic-scale forcings, rather than mesoscale, topography related processes, determine whether precipitation occurs at a given location. The effect of the topography is mainly a scaling of the precipitation. The under- and overestimations by the reanalysis, which were examined in section 4a, are reflected in the different values of the loadings in the first LSV and first RSV. The second mode shows a very good simulation of the node of the north south precipitation dipole, which indicates that the large-scale geopotential height and humidity fields are almost perfectly represented in the reanalysis. Note that the simulated amplitude of this dipole is in very good agreement with the observations averaged over the 150 km 200 km grid of the reanalysis.

9 1MARCH 2003 WIDMANN ET AL. 807 FIG. 6. (upper) First and (lower) second singular vectors (divided by the area weights used for SVD) obtained from an SVD between (left) monthly reanalyzed precipitation and (right) observed precipitation in Oregon and Washington using DJF The left-hand patterns are associated with expansion coefficients of one std dev (isoline spacing is 0.5 mm day 1 ). The right-hand patterns show the precipitation anomalies that can be predicted from the left-hand patterns. The correlations and reconstructed right variance fractions (Table 3) are substantially higher than in the cases where Z 1000 or q 850 were used as predictors (Tables 1 and 2), which means that on large scales reanalyzed precipitation is a better predictor field than Z 1000 or q 850. In the physical units we use to display the RSVs, the advantage of using reanalyzed precipitation as a predictor is also apparent in the higher values of the RSVs compared to those in Fig. 3. A systematic comparison on the local scale will follow in the next section. Singular valves obtained from fitting the SVD model with daily data (not shown) are almost identical to their monthly counterparts and consequently the skill on the monthly timescale is identical for both models (Table 3, first two rows). When cross validation is used, the SVD with reanalyzed precipitation performs very well. The RRVFs for the first mode are about 60% for the cases where fitting and validation period are identical, and they drop to around 50% when they are independent. In particular, the low performance for one of the TABLE 3. As in Table 2 but with reanalyzed precipitation as the left field. The first row is based on fitting with daily data. The daily statistics are given in parentheses, all other values are based on monthly values. For rows 2 6 monthly data were used for fitting the model. Mode 1 Mode 2 Val Fit r 2 RVF RRVF SCF r 2 RVF RRVF SCF P1 P2 P1 P2 P1 P1 P2 P2 P1 P2 P1 P2 P1 P2 P2 P1 86(77) (54) (41) (89) (52) (15) (8) (9)

10 808 JOURNAL OF CLIMATE VOLUME 16 FIG. 7. Correlations of observed and predicted monthly precipitation for different downscaling methods. All values are based on DJF and cross validation. cross-correlation pairs that occurred when conventional predictors were used is not a problem anymore. If the second mode is taken into account as well, about 60% of the variance can be reconstructed using independent fitting and validation periods, which is about twice as good as with Z orq 850 -based downscaling. 5. Comparison of the methods Different methods to estimate regional precipitation have been presented in the previous sections. Now they will be compared using correlation maps between reconstructed and observed precipitation. The comparison is entirely based on independent fitting and validation periods. The whole period DJF is split into two halves, with the reconstructions of the first (second) half obtained from models fitted with the data from the second (first) half. These two reconstructions are concatenated so that they cover the whole period DJF and are then used to calculate the correlation maps. Correlations between reconstructed and observed monthly means on the 50-km grid are shown in Fig. 7. Precipitation downscaled from 1000-hPa geopotential height using the leading two SVD modes is moderately correlated with the observations in western Oregon (up to 0.67), but practically uncorrelated (down to 0.25) in large areas in eastern Oregon and parts of eastern Washington (upper left panel). A better reconstruction in the eastern parts of the analysis domain is obtained when 850-hPa specific humidity is used as a predictor (again using two modes, upper right panel). In this case the correlation coefficients are higher than 0.5 in most areas; the maximum is still Rescaled reanalyzed precipitation yields higher correlations (up to 0.92) than the two conventional downscaling techniques everywhere but in eastern Washington, where the correlations are as low as 0.34 (lower left panel). Precipitation estimates based on SVD with reanalyzed precipitation as the predictor field (lower right panel) are also highly correlated with observations in the western parts of the domain (up to 0.93) but compared to the rescaled reanalyzed precipitation correlations drop in some locations in the west. In eastern Washington the correlations are substantially higher than in the previous case (minimum 0.50). In order to assess where the differences between the

11 1MARCH 2003 WIDMANN ET AL. 809 FIG. 8. Differences of the correlation maps shown in Fig. 7. Only differences that are significant on the 5% level are shown. correlations in Fig. 7 are statistically significant and where they merely reflect sampling variability, we performed a standard significance test for the difference between correlations. By means of Fisher s z transform the correlations r were transformed into z 0.5 ln[(1 r)/(1 r)]. The transformed correlations z are realizations of a normal distribution with variance 1/ N 3, where N is the length of the time series used to calculate the correlation. The expected means of these normal distributions are a priori unknown, but they are not needed for the significance test, which tests the null hypothesis that two values z 1 and z 2 are likely to be two realizations of a normal distribution of the same (unknown) mean and standard deviation. The test compares z 12 (z 1 z 2 )/ 2/(N 3) with the percentiles of the standard normal distribution. Since the differences can be either negative or positive, a two-sided test was used. Figure 8 shows all difference maps between the correlation maps of Fig. 7, with only those grid cells shaded that passed the significance test on the 5% level. The influence of potential serial correlation on this test is small. A test based on only ( N 3)/2 temporal degrees of freedom yielded almost identical results. The difference between the SVDs with Z 1000 and q 850 as predictors is almost nowhere significant (upper left panel; note that there are about 180 grid cells and that we expect 5 out of 100 grid cells to pass the test if the null hypothesis is true), whereas the increase in correlations using the rescaled reanalysis precipitation compared to the Z 1000 and q 850 SVDs is significant in large areas (upper middle and upper right panel). The same is true for the SVD with reanalyzed precipitation used as the predictor (lower left and lower middle panel). If this method is compared with the rescaled reanalysis (lower right panel) differences of both signs are significant. Thus using reanalyzed precipitation as a predictor for local precipitation is clearly better than using the conventional predictors, while not much is gained by using q 850 instead of Z The ranking between the two precipitation-based methods depends on how one weights the advantages and disadvantages in different areas. In order to investigate whether the ranking of the methods depends on the spatial scale, we also calculated correlation maps with coarser spatial resolutions (not shown) along with significance tests, which confirm the results obtained on the finer spatial scale. Similar results were also obtained with a resolution of 300 km 400 km (the whole area), for which the correlations between precipitation reconstructions and observations are 0.65 for the Z 1000 SVD, 0.67 for the q 850 SVD, 0.94 for the rescaled reanalyzed precipitation, and 0.91 for the SVD with reanalyzed precipitation as the predictor. The difference between either one of the first two and either one of the last two correlations is significant at the 1% level, and the differences between the first two or between the last two correlations is not significant at the 5% level. The quality of downscaled precipitation can depend strongly on the timescale and the ranking between different methods may change. We investigated this dependence by calculating correlation maps on the NCEP NCAR grid also for seasonal means (DJF) and 3-yr and 5-yr running means thereof. For brevity only the results

12 810 JOURNAL OF CLIMATE VOLUME 16 FIG. 9. Correlations ( 100) of observed and predicted 3-yr running means of wintertime (DJF) precipitation. The values are based on cross validation. for the 3-yr running means are shown (Fig. 9). Note that the various models are still fitted using monthly data as described above. In general the SVD based on Z 1000 performs better on the 3-yr timescale than on the monthly timescale (which was not shown on the NCEP NCAR grid for brevity). With the exception of eastern Oregon and southeastern Washington, the correlations are above 0.75, while for monthly means most of the values are between 0.5 and 0.6, and are as low as 0.43 in the eastern part. If q 850 is used as a predictor the correlations on the two timescales are roughly the same, and the apparent (but nonsignificant) advantage of using q 850 rather than Z 1000 as predictor on the monthly timescale turns into an apparent disadvantage of q 850 on the longer timescale. The correlations between running means of scaled reanalyzed precipitation and observations are similar to those on the monthly timescale, around 0.9 (better than the Z based method) in the west, but drop to around 0.5 (worse than the Z based method) in northeastern Oregon and southeastern Washington. If reanalyzed precipitation is used as a predictor for SVD the results on the 3-yr timescale are everywhere similar or better than on the monthly timescale. The correlations in the west are slightly smaller than those of the rescaled precipitation, but they are clearly higher in the east (the difference reaches significance on the 5% level at one grid cell in southeastern Washington, based on 12 effective temporal degrees of freedom.) Overall, the SVD method gives the best results, as at shorter timescales. Running means of total precipitation over the whole area are correlated with 0.84 for the Z 1000 SVD, 0.70 for the q 850 SVD, 0.94 for the rescaled reanalyzed precipitation, and 0.94 for the SVD with reanalyzed precipitation as the predictor. Only the difference between 0.7 and 0.94 is significant on the 5% level. Some of the 3-yr running mean time series used in Fig. 9 are shown in Fig. 10. We selected two grid cells with very different climatic conditions. The upper panels refer to the grid cell located in southern Washington centered around N and W, which covers the western slopes and the crest region of the Cascade Mountains; the middle panels refer to the cell in southeastern Washington centered around N and W. The lower panels show the mean precipitation over the entire analysis domain. The SVDs with Z 1000 and q 850 as predictors are shown in the left-hand panels; the rescaled reanalysis precipitation and the SVD with reanalysis precipitation as predictor are shown in the right-hand panels. The relatively small reconstructed right variance fractions on the monthly timescale that are obtained with the Z 1000 and q 850 SVD (Tables 1 and 2) are associated with an underestimation of the variability at individual grid cells and on longer timescales. In addition, the reconstructions based on Z 1000 and q 850 deviate considerably from the observations during the last decade. The reconstructions using SVD with reanalyzed precipitation have almost the right variability and follow the observations closely throughout the period. Rescaling of the reanalyzed precipitation is a linear regression with the regression line forced through the origin, with a regression coefficient that is independent of the temporal resolution of the fitting data (because it is solely determined by the ratio of the long-term means). Therefore the squared correlation r 2 is not equal to the ratio of the reconstructed and the true variance (in contrast to the fitting period of a standard linear regression). Thus the reconstructions can have a larger variance than the observations, which actually is the case here. The observed variances from the top to bottom panel are 1.90, 0.11, 0.56, while the variances of the rescaled NCEP NCAR simulation are 4.33, 0.28, In the first half of the period rescaling yields relatively large errors, which, in agreement with the discussion in section 4a, show that the scaling factors in the second half period, which are used to reconstruct the first half, are in general larger than those in the first half period. 6. Dynamical correction to local scaling Over most of the geographical domain, the correlation of locally scaled precipitation to observations is similar to that for the SVD method using reanalyzed precipitation as a predictor (SVD-RP). The exception is the Columbia Basin of eastern Washington, where substantially lower correlations are obtained with the locally scaled precipitation. Although SVD-RP might therefore seem preferable, it may not be straightforward to implement for down-

13 1MARCH 2003 WIDMANN ET AL. 811 FIG. 10. Observed (solid lines) and reconstructed (dashed and dashed-dotted lines) 3-yr running means of wintertime (DJF) precipitation at two 150 km 200 km grid cells to the (upper) west and (middle) east of the Cascade Mountains and (lower) area mean for Oregon and Washington. Left- and right-hand panels refer to different downscaling techniques. scaling a GCM simulation in climate change experiments or seasonal forecasts, which are typical applications for downscaling methods. The downscaling methods that we propose must be calibrated by comparing a sequence of model simulations to a historical record of observations. The local scaling method can be fitted with a control climate run of a GCM long enough to capture the distribution of weather events occurring in the historical record. In contrast, fitting the SVD method requires a GCM run in reanalysis mode, in which the well-resolved fields are nudged to agree with observations over the historical record to simulate the observed sequence of weather events. Alternatively, the SVD method might be fitted using a historical analog method instead of a reanalysis. We have not yet tried this; one anticipates some degradation in the fit and predictions. In this section we instead extend the local scaling method to retain its straightforward fitting procedure but improve its skill in the eastern parts of the domain. The region poorly represented by the local scaling is in the rain shadow of the Cascades, which is very poorly resolved by the reanalysis model. In reality, we expect the intensity of the rain shadowing effect to depend strongly on the low-level flow, and especially its crossmountain component. Local scaling does not account for this effect, but it can be improved as follows. For predictors, we use not only the simulated precipitation P mod at the GCM grid cells, but also circulation-related fields that we consider to be well resolved by the GCM, specifically, 1000-hPa geopotential height. We do not assume that we have a run of the GCM in reanalysis mode. As with pure local scaling, we do assume that we have a run of the GCM whose statistics of wellresolved fields can be compared with observations over some extended period. Let x be a set of points at which we wish to estimate local precipitation P(x, t) at times t n. Let Z(t) be a vector whose components are the principal component time series of the leading EOFs of 1000-hPa geopotential height. In general, Z(t) could be taken to be any predictor field (or combination of fields) well resolved by the GCM, given as a vector of modal amplitudes or gridpoint values. For simplicity, we assume Z(t) has zero mean, which removes any possible errors associated with small global-scale shifts in the climatology of the well-resolved fields that might spuriously project onto the precipitation prediction. We define the components Z j (t)ofz(t) as the leading principal components of the 1000-hPa geopotential height field, evaluated from gridpoint data over a synoptic-scale region that includes the Pacific Northwest, because these optimally compress the geopotential height into a small set of modes. Let Pˆ (x, t) be the downscaled estimate of local precipitation P based on P mod and Z j. Let subscript i refer to prediction location x i and subscript j to the jth principal component (again, each such series could be taken