Singular Vector Calculations with an Analysis Error Variance Metric

Transcription

1 1166 MONTHLY WEATHER REVIEW Singular Vector Calculations with an Analysis Error Variance Metric RONALD GELARO,* THOMAS ROSMOND, AND ROGER DALEY Naval Research Laboratory, Monterey, California (Manuscript received 18 January 2001, in final form 21 September 2001) ABSTRACT Singular vectors of the navy s global forecast model are calculated using an initial norm consistent with an estimate of analysis error variance provided by the Naval Research Laboratory s (NRL) Atmospheric Variational Data Assimilation System (NAVDAS). The variance estimate is based on a decomposition of the block diagonal preconditioner for the conjugate-gradient descent algorithm used in NAVDAS. Because the inverse square root of the operator that defines the variance norm is readily computed, the leading singular vectors are obtained using a standard Lanczos algorithm, as with diagonal norms such as total energy. The resulting singular vectors are consistent with the expected distribution of analysis errors. Compared with singular vectors based on a total energy norm, the variance singular vectors at initial time have less amplitude over well-observed areas, as well as greater amplitude in the middle and upper troposphere. The variance singular vectors are in some ways similar to the full covariance (Hessian) singular vectors developed at the European Centre for Medium-Range Weather Forecasts (ECMWF). However, unlike the Hessian singular vectors, the variance singular vectors exhibit only minor difference in structure and growth rate compared with total energy singular vectors. This is because the variance singular vectors exclude covariance information used in NAVDAS that significantly penalizes smaller scales. The 20 leading analysis error variance singular vectors explain approximately the same fraction of forecast error variance as the total energy singular vectors in a linear context, but less in a nonlinear context. Deficiencies in the current experimental configuration are among the reasons suspected for this. Implications for targeted observing are also examined. The results show that the variance norm can have a significant impact on determining the locations for supplemental observations. 1. Introduction The singular vectors (SVs) of the forward tangent propagator of a nonlinear dynamical model provide a mathematically rigorous, but tractable, approach to quantifying perturbation growth over a finite time interval. The dominant SVs describe the most rapidly growing structures with respect to a given metric, or norm, over this interval in a tangent linear sense. As such, SVs have been applied to a variety of problems in meteorology and other geophysical sciences. In meteorology, for example, SVs have been used to study baroclinic instability (Farrell 1982, 1985, 1989), forecast error growth (Lorenz 1965; Molteni and Palmer 1993; Barkmeijer et al. 1993; Ehrendorfer and Errico 1995; Ehrendorfer and Tribbia 1997) and ensemble weather prediction (Mureau et al. 1993; Molteni et al. 1996). Most recently, SVs have been used as a means * Current affiliation: Data Assimilation Office, NASA Goddard Space Flight Center, Greenbelt, Maryland. Corresponding author address: Ronald Gelaro, Data Assimilation Office, NASA Goddard Space Flight Center, Greenbelt, MD gelaro@dao.gsfc.nasa.gov of identifying locations for collecting adaptive observations to improve numerical weather forecasts (Palmer et al. 1998; Bergot 1999; Gelaro et al. 1999, 2000; Montani et al. 1999; Reynolds et al. 2000). Of primary importance in applying SVs to a given problem is the choice of norm(s) used in the optimization, which depends on the problem being studied (e.g., Palmer et al. 1998; Errico 2000). With respect to the chosen norm, the SVs have unit amplitude at initial time in obeyance of the constraint, T x Cx 1, (1) where x is the perturbation state vector, C is a matrix that defines the norm and the superscript T denotes the transpose operation. Physically, the appropriate norm is the one for which each of these unit vectors occurs with equal probability. Palmer et al. (1998) distinguish between two general classes of problems in this regard: those in which SVs are used to study perturbation growth in a phenomenological context, such as in describing the life cycle of a cyclone (referred to as an instability problem), and those in which SVs are used to study the growth of errors in the initial conditions of a forecast (referred to as a predictability problem). In this paper, we are concerned with the latter.

2 MAY 2002 GELARO ET AL Inherent in the application of SVs to problems in atmospheric predictability is the principle that the leading SVs, if appropriately constructed, evolve into the leading eigenvectors of the forecast error covariance matrix (e.g., Ehrendorfer and Tribbia 1997). In this case, the appropriate metric at initial time is the inverse analysis error covariance matrix, which depends on the observational network and the properties of the data assimilation scheme used to construct the forecast initial conditions. Unfortunately, for realistic systems such as those used in numerical weather prediction, the lack of accurate error covariance information, combined with the enormous size and complexity of the corresponding operator, C, present a major practical problem. As a result, SVs for these systems are routinely calculated using highly simplified approximations to the analysis error covariance matrix based, for example, on perturbation total energy, enstrophy, or streamfunction variance. Of these and other simple norms, total energy has proven to be an effective and popular choice. Not only does the energy norm yield a simple form of the SV equation suitable for solution with a Lanczos algorithm (as do all diagonal norms), but empirical evidence suggests that the resulting SVs have properties that are appropriate for studying atmospheric predictability. For example, Palmer et al. (1998) showed that the spectra of the leading total energy SVs (TESVs) of the European Centre for Medium-Range Weather Forecasts (ECMWF) model are consistent with the spectra of estimates of analysis error variance in that both have significant power in the same wavenumber range. Gelaro et al. (1998) applied a pseudoinverse operator to the Northern Hemisphere 48-h forecast error in the ECMWF model to show that the analysis error projects approximately uniformly, on average, onto the 30 leading TESVs. Reynolds and Palmer (1998) showed that the differences between analyses produced by various versions of the ECMWF data assimilation scheme are approximately isotropic with respect to the growing, and decaying, kinetic energy based SVs of the Marshall and Molteni (1993) quasigeostrophic model. Despite these results, and the successful application of energy-based SVs to a broad range of problems, it is clear that highly simplified approximations of the analysis error covariance matrix are not only suboptimal from a theoretical standpoint, but may significantly limit the utility of the resulting SVs in certain applications. In the adaptive observation problem, for example, information (or a lack thereof) about the distribution of analysis errors associated with the routine observing network may have a significant impact on both the observing locations and optimal sampling strategy (Baker 2000; Baker and Daley 2000). Recent advances in atmospheric data assimilation, combined with increased computational capability, now make it feasible to compute SVs based on more realistic estimates of the analysis error covariance matrix. Barkmeijer et al. (1998, 1999) have shown that the Hessian (second derivative) of the cost function in a variational data assimilation scheme can be used to compute SVs that incorporate an estimate of the full analysis error covariance at initial time. The resulting SVs, referred to as Hessian SVs (HSVs), are consistent with the data assimilation scheme used to construct the forecast initial conditions. For example, compared with TESVs, HSVs tend to have smaller (greater) amplitude over well (poorly) observed areas of the globe. The HSVs also reflect the background and observational error correlations assumed by the data assimilation scheme. While, in principle, this approach provides the correct framework for computing optimal descriptions of forecast error variance, it has been difficult as of yet to demonstrate any significant improvement over the TESVs in this regard. This is most likely due to deficiencies in the formulation of the background error covariances used in current three-dimensional data assimilation schemes, which may be appropriate in a time-mean sense but not for the instantaneous atmosphere. In addition, the computational cost of the HSVs is several times greater than that of the TESVs because they require solution of a generalized eigenvalue problem (Davidson 1975), which precludes the use of a Lanczos algorithm. In this paper, we present an alternative to the total energy and full Hessian norms for computing SVs appropriate for studying atmospheric predictability. The SVs of the navy s global forecast model are computed using an initial norm based on the analysis error variance (vice covariance) estimate produced by a three-dimensional variational data assimilation scheme. A local approximation of analysis error variance is produced by performing a Choleski decomposition of the block diagonal preconditioner for the conjugate gradient descent algorithm used in the Naval Research Laboratory s (NRL) Atmospheric Variational Data Assimilation System (NAVDAS; Daley and Barker 2001). The local approximation method used to compute the variance follows naturally from the observation-space form of NA- VDAS; its implementation requires only minor modification of the same software used in NAVDAS itself. Estimates of analysis error variance are produced for temperature, horizontal wind, and geopotential, whose inverse square roots are easily computed. This enables the leading variance-based singular vectors (VARSVs) to be computed in an efficient manner using a Lanczos algorithm. Thus, the additional cost of computing VARSVs compared with TESVs is essentially that of the variance estimate alone, which, because of the highly parallel nature of the algorithm, is less than the cost of producing the analysis itself. We begin section 2 with a description of the procedure used to estimate the analysis error variance in NAVDAS. We then present results of the variance calculations for the month of February 1998, based on a 6-h NAVDAS data assimilation cycle run in conjunction with the navy s global forecast model. In section 3, we use the

3 1168 MONTHLY WEATHER REVIEW analysis error variance estimates to compute VARSVs during a 3-week period of February We then compare their location, structure, growth rate, and subspace similarity with those of the leading TESVs during the same period. In section 4, we examine the amount and distribution of forecast error variance explained by the leading TESVs and VARSVs. In section 5, we explore the implications of this work for adaptive observing. Targeted VARSVs are computed in a real-time configuration for several cases from the 1998 North Pacific Experiment (NORPEX; Langland et al. 1999), and the impact of the analysis error variance information on the target locations is assessed. A discussion and conclusions are presented in section Estimation of the analysis error variance a. The local approximation The procedures used to obtain analysis error estimates for NAVDAS are mentioned in Daley and Barker (2001) and described in detail in Daley and Barker (2000). The features of this procedure that are relevant to the calculation of analysis error variance SVs (section 3) are summarized briefly here. NAVDAS is an observational-space, three-dimensional variational data assimilation system, in which an analysis estimate vector, x a, of length I is obtained from a background (forecast) estimate vector, x b, (also of length I) and an observation vector, y, (of length K) by solving, (HP HT b R)z y Hx b, (2) for z, and then applying, x x PHz. T b a b (3) Here, P b is an I I symmetric, positive-definite background error covariance matrix, R is a K K symmetric, positive-definite observation error covariance and H is a K I forward operator. The vector z is an intermediate vector of length K. It is assumed here that H is a linear operator. If the background and observation error covariances P b and R are correct, then the I I symmetric, positivedefinite analysis error covariance matrix, P a, is given by, T T 1 Pa Pb PH b (HPbH R) HP b. (4) The second term in (4) is the effect of the observations and, in general, trace(p a ) trace(p b ). However, we do not usually know the error covariances P b and R very accurately, so (4) tends to underestimate the true analysis error. Equations (2) and (3) describe a matrix vector problem, but (4) describes a matrix matrix problem, which is much more computationally intense. Consequently, for operational use, we are forced to approximate (4), particularly the second term. Fisher and Courtier (1995) introduced global approximations to the second term of (4) based on reduced-rank procedures. Such procedures follow naturally from the use of analysis grid-space solution methods (as opposed to the observation-space solution method of NAVDAS) for the three-dimensional variational data assimilation problem. For observationspace systems, it is more appropriate to use local approximations to (4). The local approximation used in the NAVDAS analysis error estimate is based on the block diagonal preconditioner for the conjugate-gradient descent algorithm used in obtaining the vector z in (2). In this preconditioner, the observations are divided up into local prisms over the domain. The size of the prisms is based on the local observation density, with larger prisms where the observation density is low. There are N such observation prisms and, for the nth prism, there are K n observations. Let us denote the nth diagonal block of the matrix HP b H T R as (HP b H T R) n, which is a K n K n symmetric, positive-definite submatrix consisting only of the interactions between the K n observations in the nth prism. In NAVDAS, the diagonal blocks are Choleski-decomposed, that is, we write, (HP H T T b R) n, n n (5) where n is a lower triangular matrix. We then associate each analysis grid point (including all the grid points in a vertical column) with one and only one observation prism. This is an important constraint and must be followed rigorously. Daley and Barker (2000), using the simple one-dimensional system of Fisher and Courtier (1995), demonstrate that the local approximation to (4) can become singular if this constraint is ignored. Consequently, we followed it rigorously when constructing the present algorithm. For the nth observation prism, we will suppose there are I n such grid points, and they have been assigned because they are closer to the geographic center of the nth observation prism than to any other observation prism. Now define the I n K n matrix, (P b H T ) n, whose elements involve the interactions between the K n observations and the I n grid points. Then define, T 1 Gn (PbH) n n, (6) which is an I n K n matrix. We then approximate the second term in (4) by G n GT n (which is an I n I n symmetric, positive-definite matrix) for 1 n N. The use of (5) and (6) turns the matrix matrix operations of (4) into a series of matrix vector operations that use the same software developed for NAVDAS itself. We can find the analysis error variances from the diagonal elements of G n GT n, and the off-diagonal elements can be used to find the vertical analysis error covariances (though we do not use the off-diagonal elements in this study). The local approximation, (5) and (6), like the global approximation, is conservative in that the diagonal elements of (6) are smaller than the corresponding diagonal elements of the second term of (4), which

4 MAY 2002 GELARO ET AL means that the local approximation overestimates the analysis error variance. Riishojgaard (2000) has derived a variant of (5) and (6) in which the Choleski decomposition is replaced by an eigenvector decomposition. Both methods have been tested in NAVDAS, but the Choleski procedure seems to be more computationally efficient. The algorithm is highly parallel, so that the total cost of the variance estimate is less than that of the analysis itself. Experiments with a simple one-dimensional data assimilation system comparing the global approximation of Fisher and Courtier (1995) with the local approximation of (5) and (6) are described in Daley and Barker (2000). It would appear that as the observations become more accurate, the local approximation improves, while the global approximation degrades. As the background error horizontal correlation scale increases, the global approximation improves, while the local approximation degrades. The local approximation is better with winds, while the global approximation is better with the mass field. For the purpose of generating variance-based SVs, only the analysis error variance estimates from NA- VDAS are used. b. Results from an extended data assimilation cycle NAVDAS estimates of analysis error variance were produced for the month of February 1998 using a 6-h data assimilation cycle run in conjunction with the Navy Operational Global Atmospheric Prediction System (NOGAPS; Hogan and Brody 1993) at T79L18 resolution. The observation set includes radiosondes, winds, and temperatures from automated aircraft observations, surface observations, radiances from polar orbiting satellites, wind speeds from the Special Sensor Microwave/ Imager (SSM/I) and derived winds from geostationary satellites. To improve efficiency without compromising the objectives of this study, the total number of observations was reduced to approximately 75% of that used operationally at the navy s Fleet Numerical Meteorology and Oceanography Center (FNMOC). The specification of the background error covariances has a large impact on both the analysis and estimate of analysis error variance. In NAVDAS, the background error covariances are separated into the background error variances and the background error correlations, based on an extended formulation of the method used in multivariate optimal interpolation (MVOI). The reader is referred to Daley and Barker (2000) for a complete description of this formulation. Here, we describe only some general characteristics of the background error variances, which are more relevant to the present study. Background error variances are computed for the zonal and meridional wind components, temperature, and geopotential based, to a large extent, on information from the conventional observing network and from recent field studies such as NORPEX (A. Van Tuyl 2001, personal communication). The variances are permitted to vary by latitude, longitude, and height (see below), and have a weak time-dependence based on the season and the time of day. 1 The temperature and geopotential background error variances are in hydrostatic balance, while the wind and geopotential variances in the extratropics are related geostrophically. It is emphasized that the background error variances used in the present study are experimental and, while adequate for achieving the objectives of the study, will likely undergo further modification, or tuning, before and after NAVDAS is implemented operationally in Below, we present results of the NAVDAS analysis error variance calculations for February Unless otherwise indicated, the displayed values are the square root of the expected values of the background or analysis error variances, which have the same units as the meteorological fields themselves. For convenience, we refer to these as simply the background error or analysis error, respectively. Figure 1 shows the background and analysis error estimates for 0000 UTC 4 February 1998 in terms of geopotential height at 500 hpa. The background error varies mostly on the large scales, exhibiting a strong meridional gradient and local maxima over oceanic regions as well as central Asia. In the northern midlatitudes, the largest background errors exceed 25 m over the data-sparse northeastern Pacific Ocean and Gulf of Alaska. The zonal variations in the background error are considerably smaller in the southern midlatitudes, where there are relatively few radiosonde stations. The analysis error differs significantly from the background error, both in terms of amplitude and scale, especially in the Northern Hemisphere. Note that the analysis error in the northern midlatitudes has much more variability at smaller scales, indicative of the spectral whitening induced by the observations during assimilation (see also Daley 1991 and Palmer et al. 1998). The greatest error reductions occur over land in response to radiosonde observations, but there are also significant reductions over the northern oceans in response to observations from both polar orbiting and geostationary satellites. Over the northeast Pacific, for example, the analysis error is as much as 40% smaller than the background error in some locations. The effects of isolated radiosonde stations are clearly evident in some locations (e.g., over Hawaii). In the Southern Hemisphere, the analysis error bears a much stronger resemblance to the background error especially poleward of 40 S although there are isolated minima along the coast of Antarctica in response to radiosonde stations there. 1 The analysis error variance depends on the time of day for several reasons including the number of radiosondes and related amplitude of the background error at synoptic and off-synoptic times, the locations of sun-synchronous polar orbiting satellites and the large diurnal variation in the number of visible cloud-track winds from geostationary satellites.

5 1170 MONTHLY WEATHER REVIEW FIG. 1. Square root of expected value of (a) background and (b) analysis error variance in terms of 500-hPa geopotential height for 0000 UTC 4 Feb 1998, taken from a NAVDAS 6-h data assimilation cycle. The contour interval is 2 m. Some discontinuities in the analysis error field are evident as a result of the local approximation method used in this calculation (section 2a). These occur primarily in the Southern Hemisphere, where the observation density is low, but are not severe. These results are representative of those for other days and analysis variables. The local approximation method allows for significant day-to-day variability in the analysis error estimate, especially over oceanic regions, in response to transient components of the conventional observing network (primarily satellites). Figure 2 shows the estimated analysis errors over the Northern Hemisphere in terms of geopotential height at 500 hpa for 4 consecutive days during the study period. Over the North Pacific and, to a lesser extent, the North Atlantic, the patterns and locations of the maxima vary substantially each day. Again, these variations occur on much smaller scales than the land sea differences in the analysis error distribution based on the density of the radiosonde network alone. Close inspection reveals weaker day-to-day variations over land areas as well, due, primarily, to variations in the numbers and locations of aircraft observations. The vertical structure of the background and analysis errors are presented in Figure 3, which shows regional

6 MAY 2002 GELARO ET AL FIG. 2. Square root of expected value of analysis error variance in terms of 500-hPa geopotential height for 0000 UTC 5 8 Feb 1998, (a) (d), respectively. The contour interval is 2 m. profiles of the errors in temperature and meridional wind, averaged for the month of February 1998 at 0000 UTC. The errors in the wind field increase with height in accordance with the natural variance of the wind field itself, reaching a relative maximum at the level of the upper-tropospheric jet near 300 hpa before increasing again through the stratosphere. The errors in temperature exhibit near-maximum values close to the surface due to uncertainty in the boundary layer structure, then decrease with height through the lower and middle troposphere before increasing again up to jet level. It is clear that the background error is the controlling factor in determining the analysis error estimate at all levels and locations. However, as expected, over data-rich areas such as North America, the analysis errors depart most from the background errors (Fig. 3d). Note also

7 1172 MONTHLY WEATHER REVIEW FIG. 3. Vertical profiles of expected value of background (solid) and analysis (dashed) error variance in terms of temperature (T) and meridional wind ( ), averaged for Feb Regional profiles correspond to (a) Northern Hemisphere: N; (b) Southern Hemisphere: S; (c) North Pacific: N, 150 E 130 W; and (d) North America: N, 130 W 60 W. The units are K for temperature and m s 1 for meridional wind. that the magnitudes of the errors differ from one region to another in accordance with the observation density. Notwithstanding the inevitable degree of subjectivity in the background error specification, the previous results indicate that the associated estimates of analysis error variance are reasonable and, in the present context, are an appropriate candidate metric for computing SVs suitable for studying atmospheric predictability. We explore this issue in the remaining sections of this paper. 3. Analysis error variance singular vectors a. The total energy and analysis error variance metrics The evolution of infinitesimal perturbations in a nonlinear forecast model is governed by the tangent linear system x(t ) L(t, t )x(t ), (7)

8 MAY 2002 GELARO ET AL FIG. 4. Vertically integrated values of the average energy function, E(s), for the 10 leading (a) TESVs and (b) VARSVs at initial time for the period 31 Jan 20 Feb The contour interval is 2 J kg 1. See text for details. linearized about the nonlinear forecast trajectory. Here, L is the forward tangent propagator, x is the perturbation state vector and t 2 t 1 is an interval on the forecast trajectory, with t 1 t 2. Within this tangent linear context, the SVs maximize the ratio Px(t 2); EPx(t 2), (8) x(t 1); Cx(t 1) where E and C are positive-definite, symmetric operators that induce a norm at the initial (t 1 ) and final (t 2 ) time, respectively, and the angled brackets denote the Euclidean inner-product. The operator P is a local projection operator (Barkmeijer 1992; Buizza 1994), which sets the perturbation state vector to zero outside a prescribed region of interest (specified below). The SVs that maximize (8) are solutions of the eigenvalue problem LTPTEPLx Cx. (9) The leading SV (SV1) maximizes (8) over the interval t 2 t 1, while SV2 maximizes (8) over this interval in the space C-orthogonal to SV1, and so on (Strang 1976, p 272). At final time, the evolved SVs form an E-orthogonal set, y PLx. If the same norm is used at both the initial and final time (C E), then the eigenvalues,, measure the growth of the norm associated with the corresponding SVs over the interval t 2 t 1. At present, NOGAPS SVs are computed using perturbation total energy as the norm at initial and final time. This is done, in part, because total energy has been shown to be an appropriate metric for studying atmospheric predictability on timescales of a few days (section 1). In addition, this choice has the advantage that the operator, C, (as well as E) takes the form of a diagonal matrix in model spectral space whose inverse square root is easily computed. This makes it convenient to use the transformation x C 1/2 u to express (9) in a form suitable for solution with a Lanczos algorithm (Strang 1986). The resulting SVs are referred to as total energy SVs or TESVs, and the eigenvalues,, measure the growth of total energy associated with these SVs over the interval t 2 t 1. As an alternative to using the total energy norm at initial time, we use the inverse estimate of analysis error variance provided by NAVDAS (section 2). In addition to providing a more realistic (data assimilation-based) estimate of the analysis error distribution, the use of the variance-based norm preserves the ability to solve (9) with the Lanczos algorithm because its inverse square root is easily computed. In this case, however, calculation of the product C 1/2 u involves a series of transformations between the spectral forms of vorticity, divergence, and potential temperature used in the NO- GAPS tangent models and the physical-space forms of the horizontal wind components and temperature used to produce the NAVDAS estimate of the analysis error variance. The procedure used to apply the analysis error variance norm is described in detail in the appendix. The resulting SVs are referred to as analysis error variance SVs or VARSVs. Because the VARSVs optimize total energy at final time, we have, in this case, C E, so that the eigenvalues of (9) do not have the same physical interpretation as for the TESVs. However, it is straightforward

9 1174 MONTHLY WEATHER REVIEW to compute an equivalent growth rate in terms of total energy for comparison purposes. b. Comparison of the TESV and VARSV subspaces TESVs and VARSVs based on the NOGAPS adjoint modeling system (Rosmond 1997) at T47L18 resolution and a 48-h optimization interval were computed for each day of the 3-week period 31 January 20 February The optimization area encompasses the Northern Hemisphere from N, and extends vertically from approximately 150 hpa to the surface. The forward tangent and adjoint models are dry with simple boundary layer physics. The basic state trajectory for the SV calculations is the moist, nonlinear NOGAPS forecast produced at T79L18 resolution and then truncated to T47L18 resolution. Our experience shows that the trajectory produced this way is of better quality than that produced at T47L18 resolution directly. The TESV and VARSV calculations begin with the analyzed initial conditions at 0000 UTC produced from a 6-h multivariate optimal interpolation (MVOI) data assimilation cycle. At the time of this writing, the MVOI scheme was still being run operationally at FNMOC. The decision to use the MVOI, rather than NAVDAS, analyses for computing the trajectories was based on this and other considerations, including the fact that the MVOI analyses were known to be of equal or superior quality to those of the still-experimental version of NA- VDAS at that time, and our extensive experience with using the MVOI trajectories for computing TESVs for other studies and applications. While the NAVDAS estimate of analysis error may differ from the errors in the analyses produced by MVOI, this discrepancy is of secondary importance to the objectives of the present study. We emphasize that the primary objective here is to study the first-order differences between SVs based on an energy metric and those based on a realistic estimate of analysis error variance. To examine the time-averaged perturbation fields for the leading TESVs and VARSVs, we compute an average energy function r q 1 E(s) e (s), (10) i,j qr j 1 i 1 where s is a three-dimensional model grid-point index and e i,j (s) is the grid-point value of the total energy of the ith SV on the jth day of the study period. Here, E(s) combines in a physically meaningful way the contributions of the mass and velocity fields to the total perturbation amplitude. We note that (10) is similar to the measure introduced by Buizza and Montani (1999) for identifying regions of atmospheric sensitivity suitable for collecting targeted observations, except that the contributions from the individual SVs are weighted according to their amplification rates (see section 4). Figure 4 shows vertically integrated values of E(s) for the 10 leading TESVs and VARSVs at initial time FIG. 5. Vertical cross sections of the average energy function, E(s), for the 10 leading (a) TESVs and (b) VARSVs at initial time for the period 31 Jan 20 Feb The cross sections represent latitudinal averages from N. The contour interval is 0.3 J kg 1. for the period 31 January 20 February 1998 (i.e., the contours represent vertically integrated averaged values based on SVs). For comparison purposes, we ensure that individual SVs of both types are normalized to have unit total energy prior to computing E(s). The results in Fig. 4 show significant differences between the geographical distributions of the TESVs and VARSVs, especially over the Pacific. The TESVs are strongly concentrated over Japan, in the baroclinic region associated with the east Asian jet and the entrance region of the Pacific storm track. In contrast, the VARSVs are shifted away from the coast of east Asia and have their maximum concentration over the central Pacific where the analysis error variance is much larger (cf. Fig. 2). Over the Atlantic, the VARSVs are also shifted away from the well-observed North American continent toward the midoceanic region. Differences in the vertical distribution of the perturbations are illustrated in Fig. 5, which shows cross sections of E(s) over the Pacific region for the 10 leading TESVs and VARSVs at initial time. The cross sections represent latitudinal averages from N for the period 31 January 20 February For the TESVs, the maximum in the western Pacific is concentrated in the

10 MAY 2002 GELARO ET AL FIG. 6. As in Fig. 5, except at final time ( 48 hrs). The contour interval is 0.2 J kg 1. lower troposphere in the layer hpa, with strong vertical gradients above and below this layer. This is consistent with the results of Gelaro et al. (2000), who examined the structure and locations of TESV-based targets for adaptive observations during NORPEX. In contrast, the VARSV maxima in the central and eastern Pacific have greater vertical depth, with significant amplitude extending as high as 400 hpa. A relative maximum occurs near 140 W, where the analysis error variance is largest, at a noticeably higher vertical level than the maxima to the west. Both the TESVs and VARSVs have a relative maximum near 100 E in the layer hpa, whose existence appears to have some relationship to the Himalayan plateau (although this has not been confirmed). Figure 6 shows the corresponding cross sections of E(s) at final time ( 48 hrs). Both the TESVs and VARSVs evolve into deep structures with maxima in the upper troposphere. For the TESVs, the primary maximum at final time is clearly located in the central Pacific, downstream from the primary TESV maximum at initial time in the vicinity of 140 W (Fig. 5a). The TESVs have a weaker, secondary maximum at final time near 130 W. Compared with the TESVs, the VARSVs at final time have less amplitude over the central Pacific and larger amplitude over the eastern Pacific and western North America. Again, the results are consistent with the eastward shift of the VARSVs at initial time (Fig. 5b) in response to the larger values of analysis error variance over the central Pacific. The results are qualitatively similar for the Atlantic basin (not shown). A more detailed look at the structural characteristics of the perturbations is presented in Fig. 7, which shows the mean vertical energy profiles and total wavenumber spectra of the leading TESVs and VARSVs at initial and final time. As shown in previous studies, the vertical profile for the TESVs at initial time peaks sharply in the lower troposphere in the layer hpa. The initial profile for the VARSVs, though not dramatically different from that of the TESVs, is clearly broader and obtains its maximum values in the middle troposphere in the layer hpa. Note that the VARSV profile does not have the same structure as the analysis error variance itself, which peaks near 300 hpa (Fig. 3), but does reflect an increased likelihood that the perturbations will occur at levels above those favored by the dynamic forcing alone. At final time, both the TESVs and VARSVs peak strongly near the level of the uppertropospheric jet, where the forecast errors tend be largest, but also exhibit a weak secondary maximum near the surface. The total wavenumber spectra for the TESVs and VARSVs are virtually identical, implying that both types of SVs have the same horizontal structure. The spectra show the strong upscale (nonmodal) growth of the perturbations, which have a broad spectral peak at wavenumbers at initial time and a much sharper peak at the synoptic scales at final time (see also Buizza and Palmer 1995; Buizza et al. 1997). The similarity of the spectra is not surprising given the lack of covariance information in both norms. The mean amplification rates for the leading TESVs and VARSVs in terms of total energy are shown in Fig. 8. The leading VARSVs grow only slightly more slowly than the TESVs in terms of this measure. Note that the energy growth rates for the VARSVs do not necessarily decrease monotonically because of the different norms at initial and final time. The similarity of the growth rates for the TESVs and VARSVs is, in part, a reflection of the fact that their structures do not differ significantly (Fig. 7). While the growth rates also depend on the locations of the SVs through their interaction with the background state, we note that, during February 1998, a strong baroclinic zone extended across most of the North Pacific (see Gelaro et al. 2000). This too may be partially responsible for the comparable growth rates of the TESVs and VARSVs over the western and central Pacific, respectively, during this period. The differences between the subspaces spanned by the leading TESVs and VARSVs can be quantified by computing a similarity index (Buizza 1994), which measures the amount of variance of one subspace explained by a linear combination of SVs in the other.

11 1176 MONTHLY WEATHER REVIEW FIG. 7. Mean (a),(b) vertical energy profiles and (c),(d) total wavenumber spectra for the 10 leading (a),(c) TESVs and (b),(d) VARSVs for the period 31 Jan 20 Feb Dashed (solid) curves correspond to SVs at initial (final) time. For display purposes, the values at initial time have been multiplied by 100. Parallel subspaces have a similarity index of 1, while orthogonal subspaces have a similarity index of 0. Figure 9 shows the similarity index (based on the total energy inner-product) of the leading TESVs and VARSVs at initial and final time. Results are presented for the leading 5, 10, and 20 SVs for each day of the study period. At initial time, the similarity index ranges from an average value of 0.66 for the 5 leading SVs to 0.75 for the 20 leading SVs. That is, on average, between two-thirds and three-quarters of the variance of the leading VARSVs can be explained by a linear combination of the leading TESVs (and vice versa), though the day-to-day variability in similarity increases significantly as the number of SVs is decreased. We note that these values are significantly larger than those reported by Barkmeijer et al. (1999) in their comparison of likesize ensembles of TESVs and Hessian SVs (HSVs), where the metric for the latter includes the full covariance information from a three-dimensional variational data assimilation scheme. This is because the predominantly large-scale, isotropic structure of the covariances used in these schemes penalizes heavily the tilted, baroclinic structures identified by the TESVs at initial time. At final time, the similarity between the leading TESVs and VARSVs increases, ranging from, on average, 0.75 for the 5 leading SVs to 0.85 for the 20 leading SVs. The convergence of the subspaces at final time is also evident from the results in Figs. 6 and 7 in the present study, and is a characteristic that has been noted in this and other contexts (e.g., Gelaro et al. 1998;

12 MAY 2002 GELARO ET AL FIG. 8. Average 48-h amplification rates in terms of total energy for the 20 leading TESVs (bold) and VARSVs (thin) for the period 31 Jan 20 Feb Barkmeijer et al. 1999; Reynolds and Errico 1999; Errico 2000; Gelaro et al. 2002). 4. Explanation of forecast error variance In many applications of SVs to problems in numerical weather prediction, the underlying objective is to compute optimal estimates of forecast error variance. Here, we examine the forecast error variance explained by the leading TESVs and VARSVs in both a linear and nonlinear context. In doing so, we note that neither the TESVs or VARSVs are truly optimal in this respect owing to the simplifications inherent in both the total energy and inverse analysis error variance norms. For the VARSVs, the use of different data assimilation schemes to compute the analysis error variances and SV trajectories (see section 3b) introduces additional uncertainty. Nonetheless, the results presented here shed light on a number of issues relevant to this problem. a. Linear results The forecast error projection onto n leading SVs can be written as n nsv 1/2 t i i i 1 e E d v, (11) where v i is a (normalized) singular vector at final time, d i v i ; E 1/2 e t is a projection coefficient and e t is the state-vector forecast error. If n K N, where N is the nsv dimension of the model state vector, then e t represents the n fastest growing components of e t with respect to total energy in a tangent linear sense. For a perfect model and linear error growth, it also straightforward nsv to write the inverse of as e t n nsv 1/2 1/2 0 i i i i 1 e E d u, (12) where u i is a singular vector at initial time and i is FIG. 9. Subspace similarity for the leading 5 (bold solid), 10 (thin solid), and 20 (dashed) TESVs and VARSVs at (a) initial and (b) final time for each day of the period 31 Jan 20 Feb See text for details. nsv the corresponding singular value. The quantity e 0 is referred to as the pseudoinverse estimate of the analysis error (e.g., Penrose 1955) in that only the fastest growing components of the analysis error are represented. Because the SVs at final time have been orthonormalized, the sum of the squared projection coefficients, d 2 i, measures the fraction of the variance of e t explained nsv by e t in terms of total energy. Figure 10 shows the fraction of the variance of the Northern Hemisphere 48- FIG. 10. Fraction of NOGAPS Northern Hemisphere 48-h forecast error variance explained by the 20 leading TESVs (bold) and VARSVs (thin) for each day of the period 31 Jan 20 Feb Values are based on the (linear) projection of forecast error onto SVs at final time. See text for details.

13 1178 MONTHLY WEATHER REVIEW FIG. 11. (a) NOGAPS Northern Hemisphere 48-h forecast error for 0000 UTC 13 Feb 1998 in terms of 10SV 500-hPa geopotential height; (b),(c) corresponding projection, e 48, onto 10 leading TESVs and VARSVs, 10SV respectively; (d),(e) pseudoinverse projection, e 0, onto 10 leading TESVs and VARSVs, respectively. The contour interval is 30 m in (a), 15 m in (b),(c) and 1.5 m in (d),(e), with negative values dashed.

14 MAY 2002 GELARO ET AL has been made to distinguish between model error and analysis error in the (perfect model) context of the pseudoinverse perturbations shown here. Further analysis would be required to determine whether additional observations, or improvements to the model itself, would produce the greatest benefit to the initial conditions. Nevertheless, even a simple reordering of a few leading, but geographically separated, SVs could have significant implications for applications such as targeted observing (section 5), in which the locations of these leading SVs are of paramount importance. FIG. 12. Time series of NOGAPS Northern Hemisphere 48-h forecast errors in terms of total energy of the period 31 Jan 20 Feb 1998 based on the control analyses (bold solid), and analyses perturbed 20SV with the pseudoinverse correction, e 0, using 20 leading TESVs (thin solid) and VARSVs (dashed). See text for details. h forecast error explained by the 20 leading TESVs and VARSVs on each day of the study period. Despite their different geographical distributions, particularly at the initial time (Fig. 4), both subsets of SVs explain nearly the same fraction of the 48-h forecast error variance, approximately 12%, on average. This is consistent with the high similarity index for the two subspaces (Fig. 9). While the geographical patterns of the error projections also tend to be quite similar for the TESVs and VARSVs, there can be significant differences in individual cases. These differences become more apparent as the number of SVs is decreased, as noted by Barkmeijer et al. (1999) in their comparison of TESVs and HSVs. As an example, we show in Fig. 11 the 48-h forecast error, e 48, for the case 13 February 1998 in terms of 500-hPa height and, for both the TESVs and VARSVs, the forecast error projection onto the 10 leading SVs, 10SV 10SV e48, and the pseudoinverse, e0.the forecast error projections for the TESVs and VARSVs (Figs. 11b,c) have similar structure and amplitude over the North Atlantic and northern Europe, but differ almost entirely over the Pacific. The VARSVs explain a significant amount of the 48-h forecast error over the northeastern Pacific and western North America, but explain little or none of the error over the western Pacific. The situation is reversed for the TESVs. The differences can be interpreted in terms of the pseudoinverse patterns (Figs. 11d,e), which show that the TESVs identify initial errors over east Asia as being among the most rapidly growing, while the VARSVs give more weight to growing initial errors over the central Pacific where the analysis error variance is larger. Overall, the 10 leading VARSVs explain over 15% of the Northern Hemisphere forecast error variance for this case, while the TESVs explain just under 11%. These differences are reduced significantly when the number of SVs included in the projections is increased to 20 (not shown). In that case, both the TESVs and VARSVs explain approximately 20% of the Northern Hemisphere forecast error variance, as indicated in Fig. 10. We note that no attempt b. Nonlinear results The results in section 4a apply strictly in a tangent linear context. Within this context, it is easily shown nsv nsv that et L e0,which implies that the results in Fig. 10 may be interpreted as describing the percent reduction in the Northern Hemisphere 48-h forecast error variance obtained by correcting the analysis error in the subspace of the 20 leading SVs (see also Buizza et al. 1997; Gelaro et al. 1998; Reynolds and Palmer 1998). To examine the impact of such an analysis correction in a nonlinear context, we compare the NOGAPS 48-h forecast error based on the control analysis, x a (the same analysis used to compute the SVs), with that based on the perturbed analysis, 20SV x a xa e 0, (13) which is intended to bring about the maximum forecast improvement obtainable by reducing the analysis error in the SV subspace. Figure 12 shows the total energy of the Northern Hemisphere 48-h forecast errors for the period 31 January 20 February 1998 based on the control analyses and the analyses perturbed with the TESVand VARSV-based pseudoinverse corrections. We note first that there is a high degree of consistency between the results in Fig. 12 and the linear estimates of explained error variance (Fig. 10). For example, the perturbed analyses produce large forecast improvements with respect to the control on 3, 13, 19, and 20 February, which correspond to the cases in which the linear estimates of explained error variance are largest. Conversely, both the linear and nonlinear results show very small impacts on 1 and 11 February. Comparing the perturbed nonlinear forecasts themselves, however, it is clear that the TESV perturbations lead to uniformly larger reductions in forecast error than do their VARSV counterparts. On average, the TESV perturbations reduce the control forecast error by 14%, as compared with 10% for the VARSV perturbations (although both values compare well with the linear estimate of 12% explained variance for both the TESVs and VARSVs). Several factors may contribute to the poorer performance of the VARSV perturbations in this experiment, including the use of different data assimilation schemes to compute the analysis error variances (NAVDAS) and SV trajectories (MVOI), as well as un-

15 1180 MONTHLY WEATHER REVIEW certainty in the specification of the background errors. While NAVDAS and MVOI use broadly similar background error covariances, they differ in the details of their implementations, in particular since the MVOI and NOGAPS have been mutually tuned over the years to optimize their performance as an integrated system. As noted in section 2b, the specified NAVDAS background error variances had not been tuned for this study. They have been tuned more recently, and, for example, the maximum error in the Gulf of Alaska has been reduced and extended westward across the Pacific (not shown) relative to that in Fig. 1b. This would probably cause less eastward extension of the VARSVs, as compared with the TESVs. While it is our intention to conduct a follow-up study in which the revised NAVDAS background error variances are used for both the analysis cycle and SV trajectory calculations, this has not been possible as of the time of this writing. Having said this, we must also acknowledge that the gross assumptions inherent in the total energy norm including the complete neglect of information about the background error statistics are likely to be at least as effective in making TESVs suboptimal. Moreover, it is not clear why the factors described previously should manifest themselves primarily in the nonlinear perturbation experiments, and not in the linear projection of the errors. This discrepancy may, in part, reflect a significant overestimation of the time interval in which the pseudoinverse perturbations evolve quasi-linearly, which we implicitly assume to be 2 days in these experiments. A recent study by Gilmour et al. (2001) suggests that the duration of the linear regime is often less than a day for similarly constructed perturbations used in operational ensemble weather prediction. In addition, previous experience shows that experiments of this type may be quite sensitive to even minor variations in the amplitude and structure of the initial perturbations (Buizza et al. 1997; Gelaro et al. 1998; Klinker et al. 1998). Either way, we are unable to explain conclusively this systematic difference between the impact of the TESV and VARSV pseudoinverse perturbations at the present time. In view of the suspected deficiencies in the current experimental setup, and the primary intent of this paper to report on the methodology and general characteristics of the VARSVs rather than provide a comprehensive demonstration of improvement based on this initial implementation, no further attempt was made to investigate the poorer performance of the VARSV pseudoinverse perturbations at the present time. Nonetheless, the results highlight the sensitivity to, and potential difficulties of, incorporating more realistic, but imperfect, information into the specification of the norm. 5. Implications for targeted observing The objective of targeted, or adaptive, observing is to improve weather forecasts by collecting observations in data-sensitive locations where analysis errors would have the largest impact on the forecast for a specific event or region of interest. Strategies for choosing these target locations include the SV method (Palmer et al. 1998; Gelaro et al. 1999; Buizza and Montani 1999; Bergot 1999), the gradient sensitivity method (Langland and Rohaly 1996; Baker and Daley 2000), the ensemble transform and ensemble transform Kalman filter method (Bishop and Toth 1999; Bishop et al. 2001) and the inverse method (Pu and Kalnay 1999). In the SV approach to targeted observing, the target locations are defined by the initial-time leading SVs optimized for a given forecast interval and verification area. The idea is to concentrate observations on the structures that evolve into the leading eigenvectors of the forecast error covariance matrix localized within the verification area. While, in fact, this requires an estimate of the analysis error covariance matrix associated with the routine observing network at the targeting time, most SV-based targeting experiments to date have been conducted using norms that ignore the routine observing network completely (see references above). In this section, we use the VARSVs to examine how the distribution of analysis errors associated with the routine observing network might influence the deployment of targeted observations. We present results for several cases during NORPEX (Langland et al. 1999). The forecast length and verification area are case dependent, but in general are designed to improve forecasts of h for storms making landfall on the west coast of the United States. The SVs are computed at T47L18 resolution, but in a realtime targeting mode based on a forecast trajectory begun 36 h prior to the observing time. Because the NAV- DAS estimate of analysis error variance is unavailable at this advanced time (as it depends on the observation set used in the analysis at that time), the VARSVs are computed using time-averaged values of the variance for the corresponding time of day. The target areas are defined in terms of the vertically integrated energy of the three leading SVs, defined as in (10) except that, here, r 1, q 3, and the contribution from each SV is weighted by the factor i/ 1, where 1 is the amplification rate of the leading SV. The choice of q 3 is motivated by the fact that NRL typically used three leading SVs to define real-time target locations during NORPEX field operations. Of the 10 cases examined, 3 exhibited large differences in the primary target areas produced by the TESVs and VARSVs, 2 exhibited smaller, but significant differences, and the remaining cases exhibited differences that would not have prompted a different deployment of supplemental observations. A subset of these cases is examined below. As it turns out, all cases examined were targeted for 0000 UTC on the day in question, as were most cases during NORPEX. The analysis error variances used to compute the VARSVs for these cases are the monthly mean values for February 1998.

16 MAY 2002 GELARO ET AL FIG. 13. Targeted observing locations in terms of vertically integrated energy of the three leading (a) TESVs and (b) VARSVs, weighted by their amplification rates. Targets are computed for the 24-h forecast beginning 0000 UTC 18 Jan 1998 and verifying 0000 UTC 19 Jan 1998, based on the 60-h trajectory from 1200 UTC 16 Jan Boxed outline shows the forecast verification area. Black contours in (b) show the Feb 1998 mean square root of analysis error variance used to compute the VARSV targets in real time, in terms of 500-hPa height (contour interval is 2 m). See text for details. In Fig. 13, we present a case in which the analysis error variance norm has a large impact on the SV targets. This figure shows the TESV and VARSV targets for the 24-h forecast beginning 0000 UTC 18 January 1998 and verifying 0000 UTC 19 January 1998 (outlined area), based on the 60-h trajectory from 1200 UTC 16 January For reference, the mean analysis error variance used to compute the VARSV target is plotted as a background field in Fig. 13b in terms of 500-hPa height. The TESV target exhibits two areas of nearly equal sensitivity near 40 N, 175 W and 30 N, 140 W, suggesting the need for supplemental observations in both locations. The VARSVs, in contrast, strongly emphasize only the more northern of these areas, in accordance with the larger values of analysis error variance in this location. Similarly, the targets for the 36-h forecast beginning 0000 UTC 31 January 1998 (Fig. 14) show a significant shift in emphasis from a broad, but weak, area of sensitivity easily accessible from Hawaii in the case of the FIG. 14. As in Fig. 13, except for the 36-h forecast beginning 0000 UTC 30 Jan 1998 and verifying 1200 UTC 31 Jan 1998, based on the 72-h trajectory from 1200 UTC 28 Jan TESVs, to two distinct sensitivity maxima along N in the case of the VARSVs. In the latter, both maxima lie close to maxima in the analysis error variance. Finally, as an example in which the analysis error variance norm produces a more subtle, but distinctive, change in the primary target area, we show in Fig. 15 the TESV and VARSV targets for the 36-h forecast beginning 0000 UTC 11 February In this case, there is a northeastward shift of the primary target area in response to the more moderate gradient of analysis error variance over the central Pacific. At the same time, there is a clear diminution of the sensitivity around Hawaii, where the analysis error variance obtains a local minimum. The VARSV targets in Figs. 13 and 14 also show reduced sensitivity in this area, though to a lesser degree. In summary, the VARSVs do not appear to alter significantly the general structure of the (dynamic) sensitivity pattern derived from the TESVs, but instead emphasize those features of the pattern that occur where there is an expectation that the analysis error will be large. Nevertheless, the effect on the primary regions of sensitivity may be pronounced even when the optimization is constrained to a very limited area

17 1182 MONTHLY WEATHER REVIEW FIG. 15. As in Fig. 13, except for the 36-h forecast beginning 0000 UTC 11 Feb 1998 and verifying 1200 UTC 12 Feb 1998 based on the 72-h trajectory from 1200 UTC 9 Feb as in the case of targeted observing and thus favor a rather different strategy for deploying supplemental observations. 6. Discussion and conclusions Singular vectors (SVs) of the navy s global forecast model have been computed using an initial norm that is consistent with estimates of analysis error variance provided by the NRL Atmospheric Variational Data Assimilation System (NAVDAS). The variance estimate is based on a decomposition of the block diagonal preconditioner for the conjugate-gradient descent algorithm used in NAVDAS. Because the inverse square root of the operator that defines the variance norm is readily computed, the leading SVs can be obtained using a standard Lanczos algorithm, as with more commonly used diagonal norms such as total energy. The resulting SVs, referred to as variance SVs (VARSVs), take into account not only the dynamic instability of the flow in a given location, but the a priori probability that an error will occur in that location given the quality and distribution of the observations used by the data assimilation scheme. Compared with SVs that use a total energy norm at initial time (TESVs), the leading VARSVs have, on average, significantly less amplitude over well-observed areas such as North America and Europe, and greater amplitude over data-sparse regions such as the northeastern Pacific. Because the analysis error variance exhibits significant day-to-day variability on spatial scales similar to those of the leading SVs themselves, the degree to which the locations of the leading TESVs and VARSVs differ on any given day can also vary substantially. This is particularly true over oceanic regions where transient components of the observing network have the strongest influence on the analysis error variance. In the vertical, the TESVs at initial time peak sharply in the lower troposphere ( hpa), while the VARSVs obtain a broader peak in the middle troposphere ( hpa). At final time, both the TESVs and VARSVs peak strongly near the level of the uppertropospheric jet. The VARSVs are, in the above respects, similar to the Hessian SVs (HSVs) described by Barkmeijer et al. (1998, 1999), which incorporate the complete covariance information at initial time based on the second derivative (Hessian) of the analysis cost function. This is expected in that the analysis error variance represents the diagonal component of the analysis error covariance matrix. However, unlike the Hessian norm, the variance norm does not significantly alter the horizontal scale of the SVs or the general structure of the associated sensitivity patterns relative to that obtained with the energy norm, but instead emphasizes those features that occur where there is an expectation that the analysis error will be large. For this reason, the subspace similarity between like-sized ensembles of TESVs and VARSVs is significantly greater than between like-sized ensembles of TESVs and HSVs as reported by Barkmeijer et al. (1999). Optimal descriptions of forecast error variance ultimately require the complete covariance information at initial time. In this sense, the HSV approach paves the way forward. However, as Barkmeijer et al. point out, the background error covariance formulations used in current operational data assimilation schemes lack a realistic description of small-scale error structures. As a result, the full covariance norm disproportionately penalizes small-scale baroclinic structures that occur in dynamically unstable regions. In contrast, the variance norm preserves much of the small-scale dynamic structure and large amplification rates inherent in the energy norm. Barkmeijer et al. (1998) show that the HSVs exhibit similar characteristics if the full observational error covariance information is included in the analysis cost function, but the background error covariances are replaced by a diagonal matrix whose elements are proportionate to the energy weights. Nonetheless, it should be noted that the VARSVs also depend on the background error correlations used in NAVDAS via their impact on the background error variance. The background error variance, in turn, strongly