The characteristics of Hessian singular vectors using an advanced data assimilation scheme

Transcription

1 QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. 135: (2009) Published online 7 July 2009 in Wiley InterScience ( The characteristics of Hessian singular vectors using an advanced data assimilation scheme A. R. Lawrence,* M. Leutbecher and T. N. Palmer European Centre for Medium-Range Weather Forecasts, Reading, UK ABSTRACT: Initial condition uncertainty is a significant source of forecast error in numerical weather prediction. Singular vectors of the tangent linear propagator can identify directions in phase-space where initial errors are likely to make the largest contribution to forecast-error variance. The physical characteristics of these singular vectors depend on the choice of initial-time metric used to represent analysis-error covariances: the total-energy norm serves as a proxy to the analysiserror covariance matrix, whereas the Hessian of the cost function of a 4D-Var assimilation scheme represents a more sophisticated estimate of the analysis-error covariances, consistent with observation and background-error covariances used in the 4D-Var scheme. This study examines and compares the structure of singular vectors computed with the European Centre for Medium- Range Weather Forecasts (ECMWF) Integrated Forecasting System using these two types of initial metrics. Unlike earlier studies that use background errors derived from lagged forecast differences (the NMC method), the background-error covariance matrix in the Hessian metric is based on statistics from an ensemble of 4D-Vars using perturbed observations, which produces tighter correlations of background-error statistics than in previous formulations. In light of these new background-error statistics, this article re-examines the properties of Hessian singular vectors (and their relationship to total-energy singular vectors) using cases from different periods between 2003 and Energy profiles and wavenumber spectra reveal that the total-energy singular vectors are similar to Hessian singular vectors that use all observation types in the operational 4D-Var assimilation. This is in contrast to the structure of Hessian singular vectors without observations. Increasing the observation density tends to reduce the spatial scale of the Hessian singular vectors. Copyright c 2009 Royal Meteorological Society KEY WORDS singular vectors; observation targeting; ensemble prediction Received 22 March 2007; Revised 24 April 2009; Accepted 29 April Introduction In numerical weather prediction (NWP), singular vectors (SVs) are optimally amplifying perturbations that are defined from the singular-vector decomposition of a linearized dynamical model. The properties of SVs depend on the initial conditions, the forecast model and the intrinsic limitations of predictability (i.e. the chaotic nature of atmospheric flows: Palmer, 1999). These properties make them an attractive basis for representing the initial uncertainty in ensemble prediction systems (Molteni et al., 1996; Leutbecher and Palmer, 2008) and in observationtargeting strategies (Palmer et al., 1998; Bergot, 1999; Leutbecher, 2003; Langland, 2006). Depending on the type of problem being addressed, the size of the SV perturbations is defined using a metric or norm. In predictability and observation-targeting studies an ideal metric would be one that is based on the analysiserror covariances, as this would correctly describe the forecast-error covariances at the final time assuming no model error and linear model dynamics. The leading SVs are quantified by the maximum ratio between a final (forecast) time norm and initial (analysis) time norm Correspondence to: A. R. Lawrence, ECMWF, Shinfield Park, Reading, UK. a.lawrence@ecmwf.int and represent the most dynamically important part of the initial uncertainty, which evolves into the largest forecast uncertainty. In other words, SVs of the tangentlinear propagator of an NWP model evolve into the leading eigenvectors of an estimate of the forecast-error covariance matrix (Ehrendorfer and Tribbia, 1997). The analysis-error covariance matrix itself is dependent on the assimilation system, the characteristics of the model error, the observing network and the atmospheric flow. Unfortunately, in realistic NWP the absence of accurate analysis-error covariance information, the lack of knowledge about the individual sources contributing to the initial uncertainty and the large size of this operator present severe constraints, which call for approximations that mimic the expected distribution of analysis errors (Leutbecher and Palmer, 2008). In a previous study that focussed on predictability applications, Palmer et al. (1998) described SV structures using simple metrics based on total energy, kinetic energy, streamfunction variance and enstrophy. Of these, the total-energy norm provided white spectra of the dominant SVs that best agree with the spectra of analysiserror variance estimates. The total-energy norm is a simple geographically uniform metric with no spatial correlation of wind and temperature, but can be directly Copyright c 2009 Royal Meteorological Society

2 1118 A. R. LAWRENCE ET AL. related to weather prediction as it provides a relative weighting between the mass and the wind fields (Buehner and Zadra, 2006). However, although computationally cheap, total-energy singular vectors (TESVs) are purely dynamical and do not contain any information about the statistics of the observing system. The properties of TESVs and Hessian SVs (HSVs) were previously compared by Barkmeijer et al. (1998, 1999). The Hessian (or second derivative) of the cost function provides an estimate of the inverse of the analysis-error covariance matrix that is consistent with the statistical assumptions made in the assimilation scheme and the observing network (Fisher and Courtier, 1995). In Barkmeijer et al. (1998, 1999), the HSVs were derived from the full Hessian of the 3D-Var cost function with a background-error covariance representation based on lagged forecast differences (also known as the NMC method). Since implementing a full background-error covariance matrix is much too costly, this matrix was obtained via a spectral approach, where the horizontalcorrelation matrix was reduced to a diagonal matrix. This produced homogeneous and isotropic correlations, thus negating the spatial variation of the correlations (Parrish and Derber, 1992). When compared at the initial time, they found that the horizontal structure of the HSVs was more large-scale than that of the TESVs. Also, the majority of the HSV energy at the initial time was at the jet level, whereas the TESV energy peaked at the baroclinic-steering level. The structure of the evolved TESVs and HSVs was similar. However, the growth rate of TESVs was twice as large as that of the HSVs. The present study continues this analysis of the structural characteristics of TESVs and HSVs, where emphasis is placed on an updated formulation of the Hessian norm, which includes 4D-Var and a background-error covariance derived from an ensemble of data assimilation (Fisher and Andersson, 2001; Fisher, 2003). In 4D-Var assimilation (Rabier et al., 2000), an initial condition is sought so that the forecast best fits the observations within an assimilation interval (i.e. every six hours in our study). The variational method to determine the analysis involves direct minimization of a cost function J and this can be efficiently minimized using the incremental approach (Courtier et al., 1994). This incremental approach is exploited in the operational 4D- Var at ECMWF. Some recent evaluation on the accuracy of this linearization is presented in Tremolet (2004). When the linearization of the incremental approach is used and when background errors and observation errors are assumed to be uncorrelated, the cost function involves the sum of two terms that are quadratic in the increment, the background term J b and the observation term J o.as a result, the Hessian (or second derivative) of the cost function is given by J = B 1 + H T R 1 H = A 1, (1) which is an estimate of the inverse of the analysis-error covariance matrix consistent with the assumed estimates of the background-error covariance matrix B and the observational-error covariance matrix R. The background-error covariance matrix B plays an important role by determining the distribution of the information in the data in space and between variables (Daley, 1991). This is essential for supplying information to data-void or data-sparse regions. However, it also acts as an additional obstacle when trying to correct errors in the background field with shallow structures. Even if observations with high vertical resolution were available, the information will always be filtered through the background-error covariance matrix. This will significantly damp any observational information with short vertical scales. One consequence of using the background-error covariance matrix derived from an ensemble of data assimilations (as in Equation (1)) over the NMC method is that it reduces the background-error correlation length-scale (i.e. the background-error correlation exhibits a more rapid decrease with distance). Therefore, correlations produced using these new background statistics are generally sharper in both the horizontal and the vertical than those generated using the NMC method (Fisher, 2003). Hence, the results shown here are not consistent with the results presented in Barkmeijer et al. (1999) and Hamill et al. (2003). The main effect of the observation term in Equation (1) is to decrease the error variances and to sharpen the correlations in well-observed areas compared with the variances and correlations implied by B (Barkmeijer et al., 1998). In certain situations where accurate observations are lacking, SVs can identify where additional observations can be placed in order to yield the optimum benefit to a forecast (Montani et al., 1999). The value of additional measurements and whether they contribute an improvement in weather forecasts forms one of the central scientific questions of THe Observing system Research and Predictability EXperiment (THORPEX) programme (Shapiro and Thorpe, 2004). This methodology is now commonly referred to as observation targeting (or adaptive observations). TESVs have previously been used to identify optimal regions in which to place additional observations for the Fronts and Atlantic Storm Track Experiment (FASTEX) (Bergot, 1999), numerous cases of cyclone development (Buizza and Montani, 1999) and to address the sensitivity of forecasts to the use of observations in TESV target locations (Buizza et al., 2007). HSVs, based on a sixhour 4D-Var assimilation, were used by Leutbecher et al. (2002) to compare forecast improvements of severe extratropical storm Lothar using TESVs. In this case study, observations in the region defined by HSVs produced a greater forecast error reduction than those in the region defined by the TESVs, leading to the conclusion that HSV targeting was superior to TESV targeting. Both TESVs and HSVs were also previously used in a targeting framework to designate sensitive areas in the Atlantic THORPEX Regional Campaign

3 CHARACTERISTICS OF HESSIAN SINGULAR VECTORS 1119 (ATReC) in 2003 (Leutbecher et al., 2004; Richardson and Truscott, 2004). To explore the consequences of using the new formulation of the HSVs for adaptive-observation studies, this study will contrast this methodology with previous adaptive-observation studies by Leutbecher et al. (2002). From this we can evaluate qualitatively the suitability of different observation platforms (e.g. wind and temperature measurements from aircraft, dropsondes and radiosondes) used for targeted observations. However, satellite data not only provide more comprehensive global coverage for routine observations, but are hugely important for analyzing baroclinic developments over remote geographical regions or the oceans (E. Andersson, personal communication). Recent improvements of the assimilation of satellite data into NWP models has increased forecast skill significantly (Kelly et al., 2004). Despite this, data-retrieval methods often fail to constrain initial conditions in the directions of shallow structures, and the presence of cloud cover can attenuate infrared (IR) radiances of channels sensitive to information in the lower troposphere (McNally, 2002). These limitations may constitute a satellite Achilles heel that restricts the ability of satellites to sample dynamically important structures (and regions) accurately in order to adequately constrain forecast uncertainty. On the other hand, assimilation methods such as 4D-Var allow satellite information in nearby clear-sky regions to be propagated into neighbouring cloudy regions. This article is structured as follows. Section 2 provides a description of the experiments, before section 3 documents the structural characteristics of TESVs and HSVs and examines the influence of observation and background terms in the Hessian metric on SV horizontal wavenumber spectra, vertical energy profiles and vertical correlations. In this section, we also address the potential implications for adaptive sampling strategies and the notion of the satellite Achilles heel. In section 4, TESVs and HSVs are used as initial perturbations for ensemble forecasts to investigate further the properties of these SVs for operational numerical weather prediction. Concluding remarks are provided in section Experimental framework The following experiments are designed to identify the structuralcharacteristicsof SVs computed with both totalenergy and several Hessian norms as initial-time metrics. To estimate the influence of observations within the Hessian norm, experiments are performed with both the background and observation terms in the Hessian (as in Equation (1)) and with just the background-error term (thus omitting H T R 1 H from Equation (1)). Herein, these configurations are referred to as the full Hessian and the partial Hessian respectively. The sensitivity of the full Hessian with respect to different configurations of observation operator is also assessed. The singular vectors are computed with an optimization time of 48 hours with total energy, full Hessian and partial Hessian used as initial-time norm. A final-time local projection was employed to optimize final-time perturbation energy within a local verification region. Thus, the initial SVs (measured with TE, full Hessian and partial Hessian initial-time norms) indicate regions where the forecast in the verification region is most sensitive to changes in the analyses. The SVs are computed with a time step of 900 s and are valid at 1200 UTC (note the HSVs are influenced by observations during a sixhour assimilation window from 0900 to 1500 UTC). The horizontal resolution of the SVs is T 42 (roughly equivalent to ) with 60 vertical levels. This differs from the spatial resolution of the SVs in the studies of Barkmeijer et al. (1998) (T21 with five levels) and Barkmeijer et al. (1999) (T42 with 31 levels). The singular vectors in the following experiments are optimized for the layer from the bottom of the atmosphere to 100 hpa, consistent with the configuration for the extratropical SVs in the ECMWF Ensemble Prediction System (EPS). Comparisons between the different types of singular vectors are made over two specific periods. The first coincides with the Atlantic THORPEX regional campaign (ATReC) and the Northern Hemisphere winter of 2003/4 and is hereafter referred to as the ATReC winter period (or AW03 for short). The SVs are obtained for 15 dates between 8 November 2003 and 28 January 2004 (on the 8th, 13th, 18th, 23rd and 28th of each month). The individual dates do not necessarily coincide with specific cases from ATReC. The experiment uses observations from a large portion of the Northern Hemisphere (120 W 90 E and N) and verified over Northern Europe (15 W 30 E and N). The SVs are computed using a version of the ECMWF operational model designated cycle 26R3 (operational from October 2003 to March 2004). The second period extends from 1 July 2005 to 24 December 2005 and is referred to as the summer-towinter period (or SW05). The SVs, obtained for 12 cases spaced at regular 16 day intervals, are computed with ECMWF model cycle 29R2 (operational from June 2005 to February 2006). Observations are limited to a region incorporating continental USA and the North Atlantic (150 W 0 and N), which is sufficiently large to give very similar results to those for global observation coverage but at significantly reduced computational cost. The area includes a wealth of routine in situ measurements. The corresponding verification region is located over North-west Europe (15 W 15 E and N) and simulates a realistic domain for a targeted observation experiment. The experimental scenarios used for the SW05 period include (a) all observation types, as in Table I (the full Hessian ), (b) just the radiosonde observations (referred to as TEMP only ), (c) just the aircraft observations (referred to as AIREP only ) and (d) no observations (the partial Hessian ). In order to investigate whether results are sensitive to the choice of a localized verification region over Europe, further sets of TESVs and full HSVs are computed with the Northern Hemisphere extratropics as optimization

4 1120 A. R. LAWRENCE ET AL. Table I. Contributions of different data types (for all variables) within the North America North Atlantic target region and in the Northern Hemisphere extratropics for 1 July Data count n Observation cost function J o J o /n Observation type Target NH ET Target NH ET Target NH ET Land stations and ships (SYNOP) Aircraft (AIREP) Atmospheric motion winds (SATOB) Drift buoys (DRIBU) Radiosondes (TEMP) Balloons and profilers (PILOT) Satellite-sounding data (SATEM) Scatterometer (SCATT) TOTAL region. For the hemispheric HSVs, all observations in the Northern Hemisphere are included in the observation term. The hemispheric SVs are used later in this study for the initial perturbations of the ensemble forecasts described in section Results: Singular vector structure 3.1. Comparison between TESV and HSV structures This section presents the results of structural differences between the TESVs and the two types of HSVs for the AW03 sample period described in the previous section. For a quantitative comparison of the SV characteristics, average vertical profiles of energy and horizontal wavenumber spectra are shown in Figure 1 for TESVs (top row), full HSVs (middle row) and partial HSVs (bottom row). Each curve represents an average of all 15 cases and the leading 12 SVs. At the initial time (shown by the thin curves), values have been multiplied by a factor of 100 so that the curves could be plotted on the same scale as the corresponding evolved-time values (represented by the thicker curves). Structures at the final time (using the total-energy norm) are similar for all three initial-time norms. This latter result is similar to the findings of Barkmeijer et al. (1998) and Barkmeijer et al. (1999). In Figure 1(a), the average vertical profiles for the TESVs are plotted. The kinetic-energy and potentialenergy components are displayed, as well as the combined total energy. At the initial time, potential energy dominates and exhibits a single peak at about model level 40 (corresponding to 540 hpa). The kinetic energy peaks at the same altitude, but has a relatively smaller magnitude. However, at the final time the dominant energy is kinetic and peaks near the tropopause at about level 34 ( 315 hpa). As referred to in Buizza and Palmer (1995), the kinetic energy from the basic flow is transferred to the perturbations over the 48 hour optimization time, and this transfer of energy supports the mechanism of wave propagation and amplification characteristic of baroclinic systems (Badger and Hoskins, 2001). The horizontal wavenumber spectra are shown in Figure 1(b) and aid the interpretation of upscale energy propagation. At the initial time, the average TESVs are characterized by a spectrum with a broad maximum at about wavenumber 21 (corresponding to a wavelength of 2000 km), although the majority of the total energy is at larger wavenumbers (smaller wavelengths). At the final time, the spectrum peaks at wavenumber 13 ( 3000 km) and the energy of the evolved TESVs is more concentrated close to the peak. Figure 1(c) and (d) show the corresponding plots for the full HSVs. The initial and evolved total-energy profiles peak at approximately model levels 40 and 34 respectively, indicating similar upward energy propagation to the TESVs. However, in contrast to the TESVs, the majority of the energy at the initial time is kinetic energy. The initial full HSV horizontal spectrum peaks at wavenumber 18 (equivalent to 2200 km) and, similarly to the TESVs, a significant portion of the energy is present at larger wavenumbers (smaller wavelengths). This resemblance illustrates that both initial norms result in leading SVs that imply small-scale baroclinic-type features at the initial time (i.e. a westward slope with height, see also Figure 4 discussed later). In contrast, Barkmeijer et al. (1999) found that the initial energy spectrum of the full HSVs was more large-scale-dominated than the spectrum of TESVs. However, they also implied that both sets of SVs explain similar parts of the forecast error at 48 h. Now, we compare full HSVs with the partial HSVs to examine how the SV structure depends on the observation term in the Hessian. The peak of the vertical profile of the partial HSVs at the initial time is much broader than the peaks of both the TESVs and the full HSVs (Figure 1(e)). The wavenumber spectra in Figure 1(f) indicate that most of the energy is concentrated at smaller wavenumbers (larger horizontal wavelengths) with a dominant peak at wavenumber 12. There is less energy in the larger wavenumbers (smaller horizontal wavelengths) relative to the TESVs and full HSVs (i.e. the total energy at wavenumber 40 is about a fifth of the equivalent values for TESVs and full HSVs (Figure 1(b) and (d)).

5 CHARACTERISTICS OF HESSIAN SINGULAR VECTORS 1121 (a) (b) (c) (d) (e) (f) Figure 1. Average vertical energy profiles and average horizontal energy spectra (over the leading 12 SVs) for (a), (b) TESVs; (c), (d) full HSVs and (e), (f) partial HSVs for 15 cases in winter Values at the initial time (thin lines) have been scaled by a factor of 100. Relating the initial-time curves to the final-time curves in Figure 1(e) and (f), it is apparent that there is a less distinct upscale propagation of energy using this partial HSV formulation. The less-distinct upscale energy propagation indicates that the Hessian metric without the observation term strongly penalizes the small-scale structures (evident in the full HSVs). To provide some quantitative comparison to those results from Barkmeijer et al. (1999), Table II displays diagnostics for TESVs and both formulations of HSVs. (Values are derived from Figure 1 in Barkmeijer et al. (1999) and Figure 1 of the present study.) The comparisons involve (i) the wavenumber k of the peak total-energy spectra TE max, (ii) the ratio between total energy at k = 15 and k = 30 and (iii) the approximate altitude of the vertical energy-profile maxima. For the latter, equivalent hpa have been denoted given the different vertical resolutions of the two studies. It is clear that the TESV values for the diagnostic (i) are similar in both studies. The partial HSV values from this current study are much closer to the HSV values from Barkmeijer et al. (1999) for all three diagnostics.

6 1122 A. R. LAWRENCE ET AL. Table II. Quantitative comparison of total-energy and Hessian singular vector structures from Barkmeijer et al. (1999) and the present study, where k represents wavenumber and TE max is the peak value of total energy (TE). Barkmeijer et al. (1999) Present study Comparison diagnostic TESV HSV TESV Full HSV Partial HSV k at TE max TE k=15 TE k= Level TE max ( hpa) 22 (700) 14 (360) 40 (560) 41 (600) 36 (400) However, the full HSV values for diagnostics (ii) and (iii) are much more similar to the TESV values from Barkmeijer et al. (1999). Although Barkmeijer et al. (1999) and this study consider a different set of start dates, we expect that results can be compared as the characteristics of the two sets of TESVs are very similar. As a further measure of the vertical structure of the different sets of SVs, the average vertical correlations have been estimated. For this purpose, the leading N SVs are interpreted as a sample from a hypothetical distribution of initial errors. (One can assume a Gaussian distribution of initial errors in the space spanned by the leading N SVs with unit variance in the direction of each SV and zero covariance between pairs of SVs.) From the sample, the vertical correlations are determined via the horizontally averaged vertical covariance matrix, as described in Franke and Baker (2000). Figure 2 shows the vertical correlation for temperature mapped onto a linear pressure grid for the AW03 period, calculated over an average of the 15 cases and leading 12 SVs with respect to two different levels in the mid-troposphere: (a) model level 39 (corresponding to 500 hpa) and (b) model level 42 (corresponding to 615 hpa). Both the TESVs and the full HSVs have relatively tight vertical correlations, an indication of shallow, baroclinic-type structures. Furthermore, note that the value of the vertical correlation of the TESVs is almost zero at high and low altitudes, whereas the HSVs both oscillate with negative correlation values. The broader correlations of the partial HSVs indicate that the partial HSV structures are deeper and more characteristic of barotropictype features. This emphasizes again the importance of the observation term in determining the shallower structure of the SVs computed with the full Hessian metric. On a case-by-case basis, by estimating the half-height width of each curve centred around the reference level (i.e. finding the layer depth in hpa across the peak where correlation has a value of 0.5), the variability of the vertical correlation over the individual cases is now examined. Note that at altitudes below 615 hpa, the peak was too broad to provide a measurable width at a correlation of 0.5. From the time-series of AW03 cases in Figure 3, there is a clear separation of vertical correlation thicknesses between the three norms, which appear to be modulated by variations of the atmospheric flow that produce in-phase peaks and troughs in the vertical correlations. The TESV has the smallest values and exhibits vertical correlations that are generally hpa tighter than the correlations of the full HSV. The partial HSVs exhibit the broadest correlations of about 300 hpa. This is particularly evident for two of the AW03 cases in the 615 hpa plot (8 December 2003 and 18 January 2004), where wider structures are noticeable in the partial HSVs. These intermittent peaks are not as apparent in the full HSV and TESV curves. This result implies that the broader correlations, solely contributed by the background in the partial HSVs, are damped by the presence of the observations in the full HSV. To examine SV structures in more detail, Figure 4 shows temperature contours for the leading SV (at the initial time) for the case of 23 December 2003, which are optimized for the northern Europe verification area. Each plot depicts the plan views (at 750 hpa) and vertical cross-sections (along a 50 N parallel) of the leading total energy, full Hessian and partial Hessian SV respectively. The temperature structure of the TESV, shown in Figure 4(a), are mostly confined to the north-west Atlantic and between 900 and 400 hpa. The horizontal location correlates with the position of the mid-latitude jet stream during this particular case. It is also apparent that there is a westward slope in the vertical which is characteristic of the tilt against vertical shear often seen in examples of growing baroclinic structures located in the lower and middle troposphere (Farrell, 1990). The upshear tilt at these altitudes is also evident (to a lesser extent) in the initial-time u-wind and v-wind components of the total energy (not shown here). Figure 4(b) shows similar plots for the full HSV. From visual inspection, it is clear that the structures display similarity to the dominant TESV structure in both geographical location and scale. These tilted patterns are consistent with those shown in Leutbecher et al. (2002). The partial HSV in Figure 4(c) displays a much larger-scale structure with lower amplitudes than both the TESV and full HSV. These structures extend vertically through the depth of the troposphere and over a larger horizontal range, representing more barotropic characteristics. The multiple small-scale features that are abundant in the TESV and full HSV are suppressed and the westward tilt is not as prevalent. The structural characteristics of this single case in Figure 4 are consistent with the average structures documented in Figures 1 3 for the AW03 period.

7 CHARACTERISTICS OF HESSIAN SINGULAR VECTORS 1123 (a) (b) Figure 2. Average vertical correlations of temperature for TESVs, full HSVs and partial HSVs with respect to (a) 500 hpa level and (b) 615 hpa level over the 15 cases of the ATReC winter period. These comparisons confirm that the observations in the Hessian metric rotate the SV structure from the larger spatial scale of the partial HSVs to smaller scales. This notion has previously been used as a motivation for the use of TESVs as a proxy for the analysis-error covariance metric (Palmer et al., 1998). With these results in mind, the following analysis of the SW05 cases will explore the effects of different observing system configurations on the HSV structure as described in section Dependences of HSV structure on observations Now we focus on the differences in the manner in which the different types of observation rotate the HSVs to smaller scales, and how this relates to their spatial distribution and relative weighting in the Hessian matrix. Figure 5 displays the initial-time horizontal wavenumber spectra and vertical profiles of the SW05 period where all curves are averaged over the 12 cases and the leading 12 SVs. Besides the full Hessian and the partial Hessian, there are two HSV scenarios that include selected types of observations, as introduced in section 2. In general, the profiles show a more complex average structure than depicted in the similar plot in Figure 1, possibly due to the differences in atmospheric flow over the sixmonth sample period. The peaks of the initial horizontal spectra for all HSV scenarios in Figure 5(a) display similar ranges between wavenumbers 7 17, corresponding to wavelengths in the region of km. The spectrum for the partial HSV case has less energy at larger wavenumbers, consistent with the comparable plot for the AW03 cases in Figure 1(f). Note that the initial-time spectra have not been multiplied by a factor of 100 as in Figure 1, as the final-time curves are not included here. However, it is worth mentioning that for a given type of initial norm there is a higher proportion of energy in the smaller wavelengths in the AW03 cases than in the SW05 cases. Again, this is likely to be due to changes in the atmospheric flow between the sample periods as the SV computation method is identical in each instance. Generally, the average vertical-energy profiles in Figure 5(b) display relatively broad maxima at the initial time, either near the tropopause (model level hpa) or in the mid-troposphere (model level hpa). For the HSVs, the kinetic energy dominates; however, there is a small secondary peak in the lower troposphere (model level hpa) influenced by the potential energy contributions. Average vertical correlations for temperature are displayed in Figure 6. The curves correspond to an average over the 12 cases and leading 12 SVs at the 615 hpa reference level. The results from this period are consistent with those from the AW03 period. Using AIREP only generally tends to provide more similarity with the full HSVs than just using TEMP only measurements. It is likely that this is due to the geographical distribution of AIREP and TEMP measurements over the region upstream of the North-west European optimization region. The time series of the 615 hpa vertical correlation thickness variability is displayed in Figure 7. All curves display similar variability, regardless of observation type. Overall there is a progressive trend from partial HSVs (with the broadest correlations) to TESVs (with the narrowest correlations). Throughout the 12 cases, it is clear that using in situ observations in the Hessian norm results in SV structures that are shallower than the structures obtained when excluding observational information. These results are consistent with the AW03 results in Figure 3.

8 1124 A. R. LAWRENCE ET AL. (a) (b) Figure 3. Time series of the vertical correlation length-scale (thickness in hpa of layer with correlation 0.5) for (a) 500 hpa and (b) 615 hpa for the 2003 winter period. One remaining difference from the SV configurations used by Barkmeijer et al. (1998) and Barkmeijer et al. (1999) is that they focused on SVs optimized for the Northern Hemisphere extratropics, while in this study the SVs were optimized for much smaller Northern European regions. Therefore, it is of interest to examine whether the small-scale baroclinic structure of the HSVs is generic or dependent on a particular optimization region. To this purpose, the SVs optimized for the North-west European region will be compared with SVs optimized for the entire Northern Hemisphere extratropics (30 90 N). In order to include all observations relevant for the enlarged optimization region, the observation region was also extended to the entire Northern Hemisphere. Figure 8(a) shows the average horizontal wavenumber spectra of cases from the SW05 period for the entire Northern Hemisphere extratropics optimization region. The results for the hemispheric experiment are consistent with those for the smaller North-west European optimization region. Both full HSV and TESV spectra exhibit small-scale structures and narrow vertical scales similar to the SVs optimized for the smaller North-west European region shown in Figure 5(a). The corresponding relationship of vertical correlation of temperature is displayed in Figure 8(b). For comparison, the correlations for the SVs optimized for Europe/North Atlantic are also shown. The width of the correlations (at model level 42) is almost identical for the targeted and non-targeted TESVs, whilst the non-targeted full HSV correlations are only slightly tighter than the targeted full HSV correlations. Overall, there is relatively little difference between the vertical correlations. These sets of results have revealed that HSV structures strongly depend on including observational information in the Hessian formulation. The observations produce smallscale baroclinic-type features that are similar to those structures derived from TESVs. Incorporating just one

9 CHARACTERISTICS OF HESSIAN SINGULAR VECTORS 1125 (a) (b) (c) Figure 4. Plan views (at 750 hpa) and vertical cross-sections (along 50 N) of initial temperature fields for the leading singular vector of (a) TESVs, (b) full HSVs and (c) partial HSVs on 23 December Contours are at intervals of K (positive values: solid; negative values: dashed) and all singular vectors have been normalized with respect to the total-energy norm. component of the in situ measurement network also had a substantial effect on the spectral response (i.e. the size, shape and magnitude of the spectra). The upscale and upward propagation of the full HSVs (from the mid-troposphere to the tropopause level over the optimization period) also resembles that of the TESVs. This is in direct contrast to the results noted by Barkmeijer et al. (1999), where the (full) HSV structure was of a more barotropic nature and no upscale energy propagation was obvious. In both AW03 and SW05 periods, the partial HSVs (without observations included in the Hessian formulation) more closely resembled the larger barotropic-type structures revealed in Barkmeijer et al. (1999). The discrepancy between their results and our results suggests that the structure of Hessian singular vectors with observations in the metric are sensitive to the details of the background error covariance formulation. Qualitatively, one would expect that the observations, and their density, change the correlations implied by the Hessian norms. It is well known that the process of assimilation has a whitening effect on the spectrum of error variance. Daley (1985) illustrates this in a simple case study. He sampled a range of observation spacings in a 2D observation network and showed that when the spacing between the observations is large, the spectral response favours large-scale waves (small wavenumbers) more than small-scale waves (large wavenumbers). As observation spacing decreases, the spectral response shifts in favour of smaller-scale waves. In order to gain a better understanding of the sensitivity of the Hessian norm to (a) the spatial scale of a perturbation and (b) the density of the observing network, we include an idealized 1D example. For this example, the correlations of background error are set to exp( 1 2 (x/l B) 2 ), where L B is the correlation length-scale. A periodic domain of size D = 20L B is considered. Furthermore, it is assumed that observation error and background error have the same variance. The domain is discretized with 128 points. Then perturbations x k = a k sin(xk + ϑ) are considered, where k denotes the wavenumber and ϑ a random wave phase. The amplitude a k is determined in order to obtain perturbations with unit L2 norm. Figure 9 shows the value of the analysis-error covariance metric x T k A 1 x k (2)

10 1126 A. R. LAWRENCE ET AL. (a) (b) only, can be viewed as a situation with intermediate observation density. Despite the total-energy norm not including any information about the distribution of the observations, this similarity appears to confirm that it can be an appropriate approximation to a norm based on the analysis-error covariance metric, under the specific circumstances of this study (i.e. in well-observed regions and within an assimilation system that relies on an accurate background, such as that at ECMWF). What causes our results to differ significantly from those of Barkmeijer et al. (1999), who computed HSVs that are larger-scale and more barotropic than the HSVs of this study? The root cause is believed to be associated with differences in the background-error correlations. While here the background-error statistics are based on three-hour forecasts from an ensemble of analyses, Barkmeijer et al. s background-error statistics are based on 24 hour/48 hour forecast differences (Derber and Bouttier, 1999). As these lagged differences use longer lead-time forecasts to define the sample of background errors than do the ensemble of analyses, the latter method inevitably leads to tighter correlations due to the upscale cascade of forecast uncertainty in nonlinear atmospheric dynamics (Tribbia and Baumhefner, 2004) Implications of SV structure for observing strategies Figure 5. (a) Average initial horizontal energy spectra and (b) average initial vertical energy profile (over the leading 12 SVs) for full HSVs, full HSVs with TEMP only, full HSVs with AIREP only, partial HSVs and TESVs for 12 cases during as a function of wavenumber k for various observation densities. For all observation densities ρ obs the value of the metric increases with wavenumber. Adding observations penalizes the large scales most, whereas the metric remains largely unaltered at small scales, kl B > 2.5. The value of the metric at large scales, kl B < 0.3, increases by a factor of 6 from the background-error covariance metric (ρ obs L B = 0) to the analysis-error covariance metric for the dense network (ρ obs L B = 2). The wavenumber at which the metric is twice the value for k = 0 changes from 1.2LB 1 to 2.0L 1 B as one moves from the no-observation case to the dense-observation case. Therefore, the dominant spatial scales of SVs computed with an analysis-error covariance metric will decrease with increasing observation density in those situations where the dynamics favours the growth of small spatial scales, say kl B > 1.5. This idealized example is consistent with the results for the Hessian SVs. The full Hessian norm (with a full observation network and high observation density) tends to penalize the large scales more than the partial Hessian norm (zero observation density). The SW05 experiments that use just individual components of the in situ observing network, AIREP only and TEMP This study has shown that HSV structures that include observational information generate small-scale baroclinic-type features that are similar to those structures derived from TESVs. Some observation platforms may be better suited to sample these types of structures than others. The hypothesis of a satellite Achilles heel Figure 6. Average vertical correlations of temperature at 615 hpa for the cases in 2005.

11 CHARACTERISTICS OF HESSIAN SINGULAR VECTORS 1127 Figure 7. Time series of the vertical correlation length-scale (thickness in hpa of layer with correlation 0.5) at 615 hpa for the cases in represents the limitations posed by mainly IR satellite sounders when sampling dynamically important structures with small-scale characteristics beneath or within cloud. Although tropospheric temperature profiles can be conveniently sampled by high-spectral-resolution IR radiometers in clear-sky conditions (McNally and Watts, 2003), the presence of cloud significantly increases the radiance signal such that it dwarfs the signal of the intended quantity. Whilst there are continual efforts to use cloud-screening or removal algorithms to correct radiances that have been affected by cloud (Joiner and Rokke, 2000) or to omit those sounding channels that are affected by cloud contamination (Rabier et al., 2002), these methods do not favour satellite targeting over in situ methods. McNally (2002) discussed the impact that cloud cover has on the observability of 3D error structures in the initial atmospheric conditions that develop into medium-range forecast errors. He noted that above a certain threshold of cloud cover, sampling these structures using advanced IR sounders would be impossible. To qualitatively assess the impact of observing SV structures, a realistic scenario of cloud cover from the sample case of 23 December 2003 (illustrated in Figure 4) is considered. Figure 10 displays the cross-section of cloud cover fraction as a function of model level, where a value of 1.0 represents total (opaque) cloud cover. The grey contour depicts cloud with a fraction above 0.01 and the black contour signifies a cloud fraction above 0.6, where it is assumed that this arbitrarily chosen cloud-fraction value represents a conservative threshold in which IR satellite radiances are significantly attenuated. In this specific case, two of these opaque areas are located at 60 W (approximately model levels 35 and 49), which corresponds directly to the location of the TESV temperature maxima in Figure 4, centred at 60 W between hpa (approximately model levels 42 47). The amount of energy above these cloudy areas that would be visible to a space-based satellite represents a small percentage of the total. These values are listed for reference in Table III. Each row lists the proportion of the total energy (under the verticalprofile curve, as in Figure 1), above a specific model level for 23 December 2003 for all three types of SV. The equivalent pressure level is listed for purpose of comparison with Figure 4. Above the cloud-top, shown in Figure 10 to be at approximately model level 32, less than 10% of the energy for the TESV and full HSV is visible (i.e. without any amount of cloud contamination). Comparatively, 20% of the energy of the partial HSV is above this same level. Furthermore, about 40% of the total energy for the TESV and full HSV is above 500 hpa (model level 39), with a corresponding value of almost 60% for the partial HSVs. A significant proportion of the energy for all three types of norm is located above a height of 850 hpa (model level 50). The vertical cross-section of cloud for all other cases exhibits similar coverage to the representative case presented here. This fact, together with the average vertical position of the SV temperature perturbations (see Figure 1), suggest that a large fraction of the structures could be obscured by opaque cloud. This would make it difficult to correct background errors in the direction of these structures through the assimilation of radiance data. The weighting functions, which describe how the radiance observed by a space-based instrument depends on the Planck function and thus temperature at a given level, are broad functions. Thus, shallow structures as those identified by HSVs will be not well observed by a radiance measurement from space, which represents an integral of the temperature information over a deeper layer. Even though IR sounders exhibit a relatively high vertical resolution compared with microwave soundings, the vertical resolution exhibited by AIRS temperature retrievals is considerably coarser than the narrow vertical correlations shown in Figures 2 and 3 (McNally et al., 2006). Under these circumstances, a satellite Achilles heel could be overcome by using high-resolution in situ observations, such as radiosondes, to gain accurate vertical temperature profiles.

12 1128 A. R. LAWRENCE ET AL. (a) expect that it is this cumulative reduction of uncertainty through adding observations in subsequent assimilation cycles which leads to the significant positive impact of satellite radiances seen in Observing System experiments (Kelly et al., 2007). An additional technique that can be used to quantify the impact of changes of the observing network on the uncertainty of analyses, background state and forecasts is ensembles of data assimilations (Tan et al., 2007). (b) Figure 8. (a) Average initial horizontal energy spectra (over the leading 12 SVs) for non-targeted (NH) TESVs and full HSVs and (b) comparison of average vertical correlations of temperature at 615 hpa for both targeted and non-targeted TESVs and full HSVs for 12 cases during The potential limitations of satellite radiances discussed here consider a single assimilation cycle only. This will be relevant for instance in an observation-targeting context where observations are added only at one time. When changes of the observing network persist over two or more assimilation cycles, the reduction of uncertainty in the background state needs to be taken into account. We 4. Results: Ensemble performance The ability of SVs to identify the most unstable structures of the initial uncertainty makes them an ideal method for generating initial perturbations for an ensemble prediction system (Palmer, 1999). So far, this study has examined how SV structure at the initial time depends on the choice of norm. This section examines how the structural characteristics of both total energy and the full 4D-Var Hessian norms translate to performance of the ECMWF ensemble prediction system (EPS) when the respective SVs are used for the initial perturbations. The ECMWF EPS was initiated with perturbations from (a) TESVs and (b) full HSVs, optimized for the Northern Hemisphere extratropics. Their structural characteristics were described in section 3. Initial perturbations for 50 perturbed forecasts are generated from the leading 25 initial SVs using a Gaussian sampling technique (Leutbecher and Palmer, 2008). Forecasts were run up to a range of 10 days at forecast horizontal resolution T L 255 (with 60 vertical levels, as opposed to forecast resolution T L 159 with 31 levels in Barkmeijer et al. (1999)). The comparison of ensemble performance is based on the 12 cases of the SW05 period. Three experiments will be discussed: (i) an ensemble with TESV initial perturbations; (ii) an ensemble with HSV initial perturbations using the same scaling factor for the initial perturbation amplitude as (i); (iii) an ensemble with HSV initial perturbations with a scaling factor for the initial perturbation amplitude inflated by 15% (denoted HSV inf ). The inflation in experiment (iii) was chosen to render the average ensemble spread of 500 hpa geopotential (Z500) at day 2 equal to the spread in experiment (i). Results were compared using standard verification methods to estimate the quality of the forecasts; the Table III. Percentage of total energy above selected model levels for 23 December % of total energy above level Model level Equivalent pressure (hpa) TESV Full HSV Partial HSV (surface)

13 CHARACTERISTICS OF HESSIAN SINGULAR VECTORS Figure 9. Dependence of analysis-error covariance metric on observation density ρ obs and wavenumber k for a 1D periodic domain where the correlation length-scale of the background error is L B. The arrows denote the value of wavenumber where the corresponding metric is twice its value at k =0 forρ obs =0and2L 1 B W 100 W 80 W 60 W 40 W 20 W 0 20 E 50.0N Figure 10. Cross-section (along 50 N) of cloud fraction for 23 December Grey areas represent a cloud fraction >0.01; black areas represent a cloud fraction >0.6. ensemble mean root-mean-square (RMS) error, the area under the relative operating characteristic (ROC) curve and the Brier score (Wilks, 2005). Figure 11(a) shows the average Z500 RMS error of the ensemble mean and spread (around the ensemble mean) for TESV, HSV and HSV inf ensembles over all 12 cases for the Northern Hemisphere extratropics. Beyond day 4 (forecast step 4), all three ensembles exhibit less spread than the ensemble mean RMS error, indicating that these ensembles have insufficient variability (i.e. they are underdispersive meaning that the spread is below that of the ensemble mean spread). The HSV inf ensemble has similar spread to the TESV ensemble at day 2, but still significantly less spread than the TESV ensemble in the later forecast ranges. The inflation of the initial perturbation amplitude in the HSV inf ensemble is not sufficient to correct the underdispersion at these later forecast ranges. A larger inflation of the initial