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  • View in gallery

    The locations of the storms (marked by “x”s) for the 78 cases from the 2004 season for which summary maps were produced.

  • View in gallery

    The 78-case average of the adaptive observing guidance maps (shaded) for (a) ECMWF TESVs and (b) NRL TESVs. The locations of the summary map maxima for each of the 78 cases are indicated by the numbers. A number greater than 1 indicates that the maximum occurred in that location more than once.

  • View in gallery

    Same as in Fig. 2 but for (a) ECMWF ETKF and (b) NCEP ETKF.

  • View in gallery

    The 78-case average deep-layer mean wind ensemble variance (m2 s−2) at the target time for (a) the ECMWF ensemble and (b) NCEP ensemble.

  • View in gallery

    The 78-case average of the normalized ensemble variance valid at the target time (shaded) for (a) the ECMWF ensemble and (b) the NCEP ensemble. The location of the maximum variance in each of the 78 cases is given by the numbers. A number greater than 1 indicates the maximum occurred in that location more than once.

  • View in gallery

    Composite of the summary maps about the storm center (shaded) for cases when the summary map maximum lies within 15° of the storm (“near” cases) for (a) ECMWF TESV, (b) NRL TESV, (c) ECMWF ETKF, and (d) NCEP ETKF. The location of the maximum variance in each of the 78 cases is given by the numbers. The 500-hPa streamlines of the background flow, also composited about the storm center, are superimposed on this field.

  • View in gallery

    Composite of the summary maps about the storm center (shaded) for cases when the summary map maximum lies outside 15° of the storm (“far” cases) for (a) ECMWF TESV, (b) NRL TESV, (c) ECMWF ETKF, and (d) NCEP ETKF. The location of the maximum variance in each of the 78 cases is given by the numbers. The 500-hPa streamlines of the background flow, also composited about the storm center, are superimposed on this field.

  • View in gallery

    The 78-case average analysis error variance estimates of the deep-layer mean wind, normalized by the largest value in the domain, for estimates based on (a) NRL NAVDAS, (b) ECMWF ETKF, and (c) NCEP ETKF.

  • View in gallery

    The analysis error variance estimate of the deep-layer mean wind valid at 0000 UTC 14 Aug 2004, normalized by the largest value in the domain, for estimates based on (a) NRL NAVDAS, (b) ECMWF ETKF, and (c) NCEP ETKF.

  • View in gallery

    The 78-case average of the adaptive observing guidance maps (shaded) for (a) NAVDAS VARSV, (b) ECMWF VARSV, and (c) NCEP VARSV. The locations of the summary map maxima for each of the 78 cases are indicated by the numbers. A number greater than 1 indicates that the maximum occurred in that location more than once.

  • View in gallery

    Composite of the summary maps about the storm center (shaded) for cases when the maximum lies outside 15° of the storm (far cases) for (a) NAVDAS VARSV, (b) ECMWF VARSV, and (c) NCEP VARSV. The location of the maximum variance in each of the 78 cases is given by the numbers. The 500-hPa streamlines of the background flow, also composited about the storm center, are superimposed on this field.

  • View in gallery

    Summary maps for Hurricane Charley (shaded), for targets valid at 0000 UTC 14 Aug 2004, for (a) ECMWF TESV and (b) NRL TESV. The forecast position of Charley at the target time is denoted by the X. The deep-layer mean wind streamlines valid at that time are superimposed on this figure.

  • View in gallery

    Summary maps for Hurricane Charley (shaded), for targets valid on 0000 UTC 14 Aug 2004, for (a) ECMWF ETKF and (b) NCEP ETKF. The forecast position of Charley at the target time is denoted by the X. The deep-layer mean wind stream lines valid at that time are superimposed on this figure.

  • View in gallery

    Summary maps for Hurricane Charley (shaded), for targets valid at 0000 UTC 14 Aug 2004, for (a) NAVDAS VARSV, (b) ECMWF VARSV, and (c) NCEP VARSV. The forecast position of Charley at the target time is denoted by the X. The deep-layer mean wind streamlines valid at that time are superimposed on this figure.

  • View in gallery

    The singular value, or linear amplification, as measured by the square root of total energy, corresponding to the leading singular vectors for each of the 78 cases. TESVs given by filled squares, NAVDAS VARSVs given by clear squares, ECMWF VARSVs given by filled triangles, and NCEP VARSVs given by clear triangles.

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Interpretation of Adaptive Observing Guidance for Atlantic Tropical Cyclones

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  • 1 Naval Research Laboratory, Monterey, California
  • | 2 Division of Meteorology and Physical Oceanography, RSMAS, University of Miami, Miami, Florida
  • | 3 NOAA/AOML/Hurricane Research Division, Miami, Florida
  • | 4 Naval Research Laboratory, Monterey, California
  • | 5 European Centre for Medium-Range Weather Forecasts, Reading, United Kingdom
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Abstract

Adaptive observing guidance products for Atlantic tropical cyclones are compared using composite techniques that allow one to quantitatively examine differences in the spatial structures of the guidance maps and relate these differences to the constraints and approximations of the respective techniques. The guidance maps are produced using the ensemble transform Kalman filter (ETKF) based on ensembles from the National Centers for Environmental Prediction and the European Centre for Medium-Range Weather Forecasts (ECMWF), and total-energy singular vectors (TESVs) produced by ECMWF and the Naval Research Laboratory. Systematic structural differences in the guidance products are linked to the fact that TESVs consider the dynamics of perturbation growth only, while the ETKF combines information on perturbation evolution with error statistics from an ensemble-based data assimilation scheme. The impact of constraining the SVs using different estimates of analysis error variance instead of a total-energy norm, in effect bringing the two methods closer together, is also assessed. When the targets are close to the storm, the TESV products are a maximum in an annulus around the storm, whereas the ETKF products are a maximum at the storm location itself. When the targets are remote from the storm, the TESVs almost always indicate targets northwest of the storm, whereas the ETKF targets are more scattered relative to the storm location and often occur over the northern North Atlantic. The ETKF guidance often coincides with locations in which the ensemble-based analysis error variance is large. As the TESV method is not designed to consider spatial differences in the likely analysis errors, it will produce targets over well-observed regions, such as the continental United States. Constraining the SV calculation using analysis error variance values from an operational 3D variational data assimilation system (with stationary, quasi-isotropic background error statistics) results in a modest modulation of the target areas away from the well-observed regions, and a modest reduction of perturbation growth. Constraining the SVs using the ETKF estimate of analysis error variance produces SV targets similar to ETKF targets and results in a significant reduction in perturbation growth, due to the highly localized nature of the analysis error variance estimates. These results illustrate the strong sensitivity of SVs to the norm (and to the analysis error variance estimate used to define it) and confirm that discrepancies between target areas computed using different methods reflect the mathematical and physical differences between the methods themselves.

Corresponding author address: Carolyn Reynolds, Naval Research Laboratory, 7 Grace Hopper Avenue, Monterey, CA 93943-5502. Email: carolyn.reynolds@nrlmry.navy.mil

Abstract

Adaptive observing guidance products for Atlantic tropical cyclones are compared using composite techniques that allow one to quantitatively examine differences in the spatial structures of the guidance maps and relate these differences to the constraints and approximations of the respective techniques. The guidance maps are produced using the ensemble transform Kalman filter (ETKF) based on ensembles from the National Centers for Environmental Prediction and the European Centre for Medium-Range Weather Forecasts (ECMWF), and total-energy singular vectors (TESVs) produced by ECMWF and the Naval Research Laboratory. Systematic structural differences in the guidance products are linked to the fact that TESVs consider the dynamics of perturbation growth only, while the ETKF combines information on perturbation evolution with error statistics from an ensemble-based data assimilation scheme. The impact of constraining the SVs using different estimates of analysis error variance instead of a total-energy norm, in effect bringing the two methods closer together, is also assessed. When the targets are close to the storm, the TESV products are a maximum in an annulus around the storm, whereas the ETKF products are a maximum at the storm location itself. When the targets are remote from the storm, the TESVs almost always indicate targets northwest of the storm, whereas the ETKF targets are more scattered relative to the storm location and often occur over the northern North Atlantic. The ETKF guidance often coincides with locations in which the ensemble-based analysis error variance is large. As the TESV method is not designed to consider spatial differences in the likely analysis errors, it will produce targets over well-observed regions, such as the continental United States. Constraining the SV calculation using analysis error variance values from an operational 3D variational data assimilation system (with stationary, quasi-isotropic background error statistics) results in a modest modulation of the target areas away from the well-observed regions, and a modest reduction of perturbation growth. Constraining the SVs using the ETKF estimate of analysis error variance produces SV targets similar to ETKF targets and results in a significant reduction in perturbation growth, due to the highly localized nature of the analysis error variance estimates. These results illustrate the strong sensitivity of SVs to the norm (and to the analysis error variance estimate used to define it) and confirm that discrepancies between target areas computed using different methods reflect the mathematical and physical differences between the methods themselves.

Corresponding author address: Carolyn Reynolds, Naval Research Laboratory, 7 Grace Hopper Avenue, Monterey, CA 93943-5502. Email: carolyn.reynolds@nrlmry.navy.mil

1. Introduction

Objective adaptive observing techniques have been applied for a decade to identify tropospheric regions in which additional observations are expected to improve forecasts of midlatitude winter storms (e.g., Langland et al. 1999; Majumdar et al. 2002b). In contrast, the National Oceanic and Atmospheric Administration’s (NOAA’s) adaptive observing strategy for aircraft synoptic surveillance missions around tropical cyclones (TCs) has been a subjective combination of uniform sampling around the storm and sampling of areas of large ensemble variance of the mass-weighted average of the 850–200-hPa horizontal wind valid at the adaptive observing time (Aberson 2003). It is plausible that objective methods that include information of forecast error growth may result in further forecast improvements than those presented in Aberson (2003). To study the feasibility of applying the objective techniques used in the midlatitudes to the TC problem, Majumdar et al. (2006, hereafter M2006) compared the ensemble wind variance with these objective techniques. Their findings indicated lower similarity between the guidance maps produced using different methods than between those produced using the same method but different forecast systems. In this follow-on study, we compare these methods in another manner, using composite techniques that allow us to quantitatively examine differences in the spatial structures of the guidance maps, and relate these differences to the approximations and constraints of the different methods. The degree to which these differences result from differing analysis error variance assumptions is also examined.

In M2006, five adaptive observing guidance products based on three different techniques for 78 two-day tropical cyclone forecasts from the 2004 Atlantic hurricane season were compared. The logistics associated with planning the aircraft synoptic surveillance missions require that deployment decisions be made 36 h in advance. Thus, products designed to guide aircraft deployment to take observations to improve a particular 2-day forecast are based on forecasts starting at least 48 h before the surveillance observation time, and at least 96 h before the verification time. The adaptive observing products considered include the aforementioned deep-layer mean (DLM) wind variance using the ensemble produced by the National Centers for Environmental Prediction (NCEP), two ensemble transform Kalman filter (ETKF; Bishop et al. 2001) products based on NCEP and European Centre for Medium-Range Weather Forecasts (ECMWF) ensembles and two total-energy singular vector (TESV; Palmer et al. 1998; Buizza and Montani 1999) products computed by ECMWF and the Naval Research Laboratory (NRL), using their respective global models. M2006 is similar to, and broader in scope than, its midlatitude counterpart (Majumdar et al. 2002a), in which ETKF and TESV guidance products for 1- and 2-day winter weather forecasts were compared. The ETKF and TESV guidance products were found to exhibit similar characteristics on large scales, but small-scale aspects often differed considerably. The target regions often corresponded to baroclinic zones in which upper-level wave amplification is likely to occur, or regions of large low-level vorticity.

M2006 calculated the similarity between the various products quantitatively on both a large scale (0°–60°N, 120°–20°W) and a small scale (within 1500 km of the storm center). Using a modified equitable threat score, M2006 found that the similarity between the methods was considerably smaller than the similarity between products from the same method but different forecast systems. The ECMWF and NRL TESV guidance products were similar on large (small) scales in 90% (56%) of the 78 cases, while ECMWF and NCEP ETKF guidance products were similar on large (small) scales in 76% (64%) of the cases. In general, the similarities were lowest for the weakest storms. In contrast to the relatively high agreement between guidance produced using the same method but different systems, agreement between guidance produced using different methods was considerably smaller. Agreement between ETKF and TESV methods ranged between 31% and 41% on the large scale and 26% and 46% on the small scale. The NCEP ETKF targets were similar to the NCEP DLM wind variance in 96% (87%) of the cases on the large (small) scales. (See Tables 2–5 in M2006 for more details.)

A few case studies in M2006 indicated structural differences between the different guidance products. When the targets were close to the storm, both TESV products produced structures in an annulus around the storm, which may be associated with vortex Rossby wave dynamics (Peng and Reynolds 2006). Both ETKF products produced maxima near the center of the storm itself. When there were significant targets that were remote from the storm, the TESV and ETKF target locations could be very different. These differences may be due to the fact that the TESV technique considers “dynamics only,” ignoring the expected analysis error covariances at the adaptive observing time. The ETKF technique considers the routine estimated analysis error covariance at the observing time, the dynamic evolution of perturbations, and the observation error covariance of adaptive observations. However, its utility is limited by the size and quality of the ensemble to which it is applied and by the dissimilarity of the ensemble covariances to the covariances assumed by the data assimilation scheme that assimilates the adaptive observations.

The goals of this study are to 1) identify and interpret systematic structural differences between these techniques and 2) examine the impact of the inclusion of error statistics into the SV formulation. Average and storm-relative composite techniques allow us to go beyond the general statements of similarity in M2006 and relate the structural differences in the adaptive observing guidance to specific constraints and limitations of the methods. In addition, we consider the impact of constraining the SVs by estimates of the analysis error variance, and examine the sensitivity of the results to the use of different analysis error variance estimates. Particular questions we address include the following: 1) How can the differences between the target products be linked to the mathematical and physical differences in the methods used to compute them? 2) If the methods are made more consistent with each other, for example, through inclusion of ensemble-based analysis error variance estimates in the SV calculation, do the targets become more similar?

The ETKF and SV techniques are described in section 2. Average and composite results are shown in section 3. A case study is presented in section 4. Concluding remarks follow in section 5.

2. Adaptive observing techniques

The five adaptive observing guidance products for TCs considered in M2006, based on different methods and different forecast systems, as well as the three additional guidance products considered in this study, are summarized in Table 1. The ensemble DLM wind variance only considers the adaptive observing (analysis) time ta, whereas the ETKF and SV techniques consider error propagation from time ta into a given forecast verification region at a verification time tυ, 2 days after ta. All sets of guidance use ensembles (ETKF/DLM) or nonlinear trajectories (SVs) initialized at an initialization time ti, at least 48 h prior to ta. This value of tati is selected to be consistent with the logistics of planning synoptic surveillance missions. M2006 provides a brief description of the DLM variance method (see Aberson 2003 for more detail) and provides detailed descriptions of the ETKF and TESV methods, summarized briefly below.

a. Ensemble transform Kalman filter observing guidance technique

The ETKF (Bishop et al. 2001) is similar to that run operationally at NCEP (Majumdar et al. 2002b) during Winter Storm Reconnaissance Programs. The ETKF uses operational ensemble forecast perturbations (initialized at time ti and listed in matrix 𝗭i) to predict the reduction in forecast error variance produced by adaptive observations. First, the analysis error covariance matrix 𝗣r(ta) pertaining to the routine observational network is found by solving the Kalman filter error statistics equations:
i1520-0493-135-12-4006-e1
where 𝗛r and 𝗥r are the observation operator and error covariance matrices, respectively, for the routine observational network and 𝗣i(ta) = 𝗭i(ta) 𝗭iT(ta) gives the ensemble-based estimate of forecast/background error covariance that will be used to assimilate data at the adaptive observing time. This estimate should account for routine observations collected at intervening times between ti and ta. However, in the version of the ETKF described here, the effects of routine observations between ti and ta are not accounted for. Note that (1) implies that the difference between the variances given by the diagonal elements of 𝗣i(ta) and 𝗣r(ta) are determined by the location and accuracy of routine observations.
The equation describing the propagation of the covariance of changes to the forecast due to the qth hypothetical “test probe” of adaptive observations with operator 𝗛q and error covariance matrix 𝗥q is
i1520-0493-135-12-4006-e2
where 𝗠 propagates perturbations from ta to tυ. Here 𝗦q(tυ) is known as the “signal covariance matrix.” The computational expense of evaluating the term 𝗠𝗣r(ta) in (2) is avoided in the ETKF by noting that if 𝗣r(ta) = 𝗫(ta)𝗧𝗧T𝗫(ta)T [where 𝗫(ta) is an n × K matrix of ensemble perturbations evaluated at the targeting time and 𝗧 is a K × K transformation matrix], then 𝗠𝗣r(ta)𝗠T may be replaced by 𝗫(tυ)𝗧𝗧T𝗫(tυ)T, where 𝗫(tυ) gives the ensemble perturbations at the verification time. Since 𝗫(tυ) is obtained directly from the ensemble forecast, the need to use 𝗠 is avoided. For more details than the brief description provided here, please see Bishop et al. (2001).

The ensemble forecast (96 h for NCEP, 108 h for ECMWF) perturbations at tυ are used to rapidly compute the trace of 𝗦q(tυ) localized within the verification region. The ETKF “summary map” represents this signal variance in the verification region at the verification time as a function of the central location of adjacent observations of horizontal wind and temperature at 1° resolution. The observation location that produces the highest signal variance is deemed optimal for adaptive observing. It can be shown that the ETKF signal variance would yield accurate estimates of forecast error variance reduction provided that (a) the forecast error was spanned by the ensemble perturbations, (b) the error covariance matrices specified by the ETKF and the operational data assimilation scheme were accurate and equivalent, and (c) the ensemble perturbations were sufficiently small for linear dynamics to be obeyed. Of course, these conditions are never met in practice. Nevertheless, Majumdar et al. (2001) were able to show that, in practice, the ETKF is able to predictively distinguish sets of observations likely to make large changes to the forecast from observations likely to make small changes to the forecast.

In this study, the ETKF uses operational ECMWF and NCEP ensembles to estimate a vertically averaged “kinetic energy signal variance” valid at tυ, based on observations of horizontal wind and temperature at ta. For the ECMWF product, 50 ensemble perturbations of horizontal wind and temperature at 200, 500, and 850 hPa, generated using SVs (Buizza et al. 2003), initialized at 1200 UTC are used here. For the NCEP product, 20 GFS ensemble perturbations, generated using masked breeding (Toth and Kalnay 1997), and initialized at 0000/1200 UTC,1 are used. Both the NCEP and ECMWF ensembles were generated with symmetric perturbation pairs. Consequently, at the initial time, the number of linearly independent perturbations is equal to half the ensemble size. Given these ensembles, the performance of the ETKF will be compromised by (a) systematic differences between the variance of the ensembles and the variance of forecast and analysis errors, and (b) spurious correlations in the 𝗣r and 𝗣q matrices. Problem (a) is likely to be important for tropical cyclone targeting if, for example, the ensembles are more underdispersive in the Tropics than they are in midlatitudes. Both problems (a) and (b) can lead to poor adaptive sampling guidance. Recent work suggests that both of these problems are significantly reduced when one applies the ETKF adaptive sampling technique to ensembles of ensembles such as those that can be found in the North American Ensemble Forecasting System (NAEFS) and The Observing System Research and Predictability Experiment (THORPEX) Interactive Grand Global Ensemble (TIGGE).

Another route to alleviate these spurious correlations in data assimilation is via covariance localization (Houtekamer and Mitchell 2001; Hamill et al. 2001). Khare and Anderson (2006) document the negative impact of spurious correlations in ETKF adaptive observing techniques using small ensembles in a simplified setting. They show improvements to the technique when implementing covariance localization both in space and time, based on the group velocity of the most unstable mode in their simple model. Consistent with their results, these spurious long-distance correlations may be significant in adaptive observing applications studied here as well, as described below. However, it is presently unclear as to how these localized covariance structures may be maintained as they are propagated from the assimilation time to a future verification time within the complex subtropical environment studied here.

b. Singular vector observing guidance technique

Localized SVs, introduced by Buizza (1994) in the ECMWF ensemble prediction system, and used by Palmer et al. (1998) and Buizza and Montani (1999) to identify observation target areas, are a particular type of the general class of targeted analysis error covariance (AEC) optimals. The targeted AEC optimals vi of the tangent forward propagator 𝗟(ta; tυ) (with corresponding adjoint 𝗟T) sample the phase space directions of maximum growth during a finite optimization time interval (ta, tυ), evolving into the leading eigenvectors of the forecast error covariance matrix 𝗣(tυ). Given an analysis error covariance metric ||v||2r = 〈v(ta); (𝗣a)−1v(ta)〉 and verification-time metric ||v||2υ = 〈v(tυ); (𝗣υ)−1v(tυ)〉, the analysis-time optimals vi(ta) are computed via the eigenvalue equation
i1520-0493-135-12-4006-e3
where the operator 𝗚 localizes perturbations within the verification region. The square roots of the eigenvalues, σi, are the singular values, and the eigenvectors vi(ta) are the (right) SVs of 𝗚𝗟 with respect to the metrics (Noble and Daniel 1988).
The class of targeted AEC optimals in which 𝗣a and 𝗣υ are identical, diagonal, and set equal to the fixed total-energy weights, such that (𝗣a)1 = (𝗣υ)1 = 𝗘 and ||v||2E = 〈v; 𝗘v〉, are known as TESVs. A quadratic “perturbation total energy” is computed, and each “SV summary map” is then given by the weighted average of the leading SVs:
i1520-0493-135-12-4006-e4
where ej(x, t) is the vertically integrated total energy of the jth SV at latitude–longitude grid location x, and Nsv is the number of SVs used in the calculation. Both ECMWF and NRL SVs have been computed using tangent linear and adjoint models that include surface drag and horizontal and vertical diffusion, but not moist processes.

If the inverse of the true analysis error covariance is used as the initial-time metric, then the final-time SVs will be the leading eigenvectors of the forecast error covariance (Ehrendorfer and Tribbia 1997). Barkmeijer et al. (1999) describe a technique employed at ECMWF to calculate Hessian SVs (so named because the Hessian, or second derivative, of the cost function of the variational data assimilation system is an approximation to the inverse analysis error covariance metric). Leutbecher et al. (2002) showed that the Hessian SVs can be superior to TESVs for adaptive observing applications in idealized experiments.

Gelaro et al. (2002) proposed “VARSVs” as an alternative to the full covariance (Hessian SV) technique. A local approximation of the analysis error variance is produced through a Choleski decomposition of the block diagonal preconditioner for the conjugate gradient descent algorithm used in the NRL Atmospheric Variational Data Assimilation System (NAVDAS; Daley and Barker 2000; Daley and Barker 2001). Because a generalized eigenvalue solver is not needed for the VARSV calculation, the SV calculation itself is only minimally more expensive than the TESV calculation. Gelaro et al. (2002) and Reynolds et al. (2005) found that use of the initial-time variance constraint modulates SVs away from areas where the analysis error variance is relatively small and can have a significant impact on adaptive observing products.

To examine the impact of using analysis error variance estimates on SV guidance, we calculate VARSV summary maps and compare these to the TESV and ETKF summary maps used in M2006. These VARSVs will be referred to as the NAVDAS VARSVs when the NAVDAS analysis error variance estimate is used as the initial-time constraint. We also constrain the SVs using the ensemble-based ETKF estimate of the analysis error variance at ta [i.e., the diagonal of 𝗣r(ta)] referred to as the ECMWF VARSVs and NCEP VARSVs (see Table 1). These three estimates of analysis error variance differ considerably, so one may expect significant differences in the resulting VARSV structures and target regions.

c. Key differences between the ETKF and the SV observing guidance techniques

As noted in M2006, there are several reasons as to why these techniques produce different targets. The ETKF technique combines information on expected perturbation evolution into the region of interest with information on where errors in the analysis at ta are expected to be large (dynamics plus error statistics). The TESV calculation only considers what perturbations will grow the fastest into the verification region (dynamics only), while VARSVs include information on estimated error statistics at target times (VARSVs structures will depend on the methods used to generate the estimated error statistics). There are also differences due to the different forecast systems employed (i.e., forecast model and resolution; see Table 1). In addition, the summary maps represent different quantities. The TESV and VARSV summary map based on (4) gives a vertically integrated, weighted sum of SVs pertaining to the routine observational network, whereas the ETKF produces a map of reduction in forecast error variance associated with augmentations to the routine observational network. We will explore how the inclusion of the analysis error variance constraint modulates the SV guidance maps in sections 3c and 4.

3. Composite results

We first summarize the differences between the techniques by averaging the summary maps, followed by composites of the summary maps about the storm center, for the different products. All 78 cases in which the NOAA/National Hurricane Center (NHC) issued forecasts at 0000 UTC that existed at 48 h (ta) and 96 h (tυ) were selected for this study. These are summarized in the appendix of M2006. The verification region is centered on the NHC 96-h forecast position of the TC valid at tυ. All cases are at least 24 h apart, eliminating serial correlations (Aberson and DeMaria 1994), and there are multiple cases from the same storm on consecutive days.

a. TESV and ETKF targets

We first present the locations of the storms for the 78 cases considered, so as to allow for comparison with the average summary maps that follow. The observed locations of the storms at ta are presented in Fig. 1. The locations of the storms from the forecasts at ta (not shown) are, in general, similar. The storms often occur over the Caribbean and eastern and central Gulf of Mexico, as well as extending up the U.S. eastern seaboard. There is also a cluster of storm locations in the central Atlantic, associated with Hurricanes Danielle, Karl, and Lisa.

The average summary maps for the ECMWF and NRL TESVs, along with the locations of the 78 maxima of the individual summary maps, are shown in Fig. 2. There are strong similarities between the ECMWF and NRL TESVs, both in the averaged fields and the individual maxima. Both TESV products show maxima in the tropical storm track region of the subtropical Atlantic and Caribbean. But there is also significant signal to the northwest of this region, extending over the central and western United States and Canada.

As with the TESVs, the average summary maps for the ETKF methods (Fig. 3) show large average values and frequent maxima in the tropical storm track region. However, outside this region there are striking differences between the ETKF and TESV summary maps. The ETKF maxima often occur in the North Atlantic. Whereas both TESV summary maps have frequent targets over central and western North America, the NCEP ETKF has only four target maxima located over North America outside of the southeast, and the ECMWF ETKF has no target maxima over the western two-thirds of North America.2 As will be discussed below, it appears that some of the distant maxima are spurious, due to limited ensemble size and ensemble construction techniques that produce large variances in regions not necessarily associated with TCs, such as the midlatitude storm track in the North Atlantic. Note that, in many cases, when the ETKF summary map had a maximum far from the storm, there was a secondary maximum much closer to the storm. Users would often recognize that the distant maxima ought to be ignored in favor of those closer to the storm. Bishop and Toth (1999) discuss how time series of summary maps may be used to objectively distinguish relevant maxima from spurious maxima.

There are also significant differences between the NCEP and ECMWF ETKF guidance. While the averaged summary map fields have a maximum in both ETKF products over the northern North Atlantic, the maximum in the ECMWF ETKF extends farther to the west over eastern Canada than does the maximum in the NCEP ETKF field. Aside from the relatively large values in both the ECMWF and NCEP ETKF fields over the Caribbean, the patterns over the Tropics and subtropics are very different. ECMWF ETKF relative maxima occur over the eastern Pacific and the eastern Atlantic off of Cape Verde. NCEP ETKF relative maxima occur in the central and eastern Atlantic between 20° and 35°N, with a minimum in the eastern Pacific.

The differences between the ECMWF and NCEP ETKF products are largely reflected in the DLM wind variance of the ECMWF and NCEP ensembles valid at ta (Fig. 4). This is because the spatial structure of the routine analysis error variance at ta is similar to the ensemble variance, which differs considerably between ECMWF and NCEP, reflecting differences in the ensemble construction methodologies. NCEP bred vectors used to construct the initial ensemble perturbations are truly global, while the ECMWF initial perturbations include SVs computed poleward of 30°N and 30°S, in addition to tropical SVs that have been optimized for regions that include tropical storms at the analysis time (Barkmeijer et al. 2001). Buizza et al. (2005) discuss these differences, pointing out the larger perturbations equatorward of 30°N in the NCEP ensemble compared to the ECMWF ensemble at initial time and 48 h. This would account for the relatively small values of ensemble variance in the central and eastern subtropical North Atlantic in the ECMWF ensemble as compared to the NCEP ensemble.

While the estimated analysis error variance is used in the ETKF calculation, the ETKF does not simply put targets where this estimated error variance is large. This is illustrated by Fig. 5, which shows the ensemble variance at ta, normalized by the largest value in the domain, averaged over the 78 cases, and the locations of the maxima from each case. Comparison with Fig. 3 shows how the ETKF modulates the targets away from the regions of largest variance in the North Atlantic extratropical storm track.

To explore the systematic differences between the adaptive observing methods in more detail, we form composites of the summary maps about the storm center. We divide the cases into those where the maximum summary map value is within 15° of the storm center (“near” cases), and those where the maximum summary map value lies outside 15° of the storm center (“far” cases). Figure 6 shows the average storm-relative composite summary maps, as well as the locations of the individual maxima, for the near cases. Superimposed on this field are the 500-hPa streamlines of the forecasts valid at that time, also composited about the storm center, added to provide a perspective on the structure of the circulation in the immediate storm vicinity. Both TESV summary map composites (Figs. 6a,b) have a maximum in an annulus about 5° around the center of the storm, as noted in M2006. Peng and Reynolds (2005, 2006) find similar annular structures and hypothesize that this is indicative of rapid perturbation growth associated with vortex Rossby wave dynamics. The annulus in the NRL composite is slightly more compact than in the ECMWF composite. This may have to do with the representation of the storm in the different nonlinear trajectories used in the SV calculations (the NRL trajectory is of finer resolution than the ECMWF trajectory, and the NRL forecast system includes assimilation of TC synthetic observations; Goerss and Jeffries 1994). It is not clear why the ECMWF SV maximum occurs to the northeast of the storm, while the NRL SV maximum occurs to the east of the storm. In contrast, the ETKF maxima (Figs. 6c,d) occur right at the storm center, indicative of large ensemble variance in the location of the storms themselves.

The storm-relative composites based on the far cases are shown in Fig. 7. Both TESV products (Figs. 7a,b) show maxima within a 40° radius of the storm in the northwest quadrant (consistent with Peng and Reynolds 2006). In contrast, the far targets for the ETKF products (Figs. 7c,d) are more scattered. The maximum can occur in any quadrant relative to the storm, although it occurs to the southwest only once in both the ECMWF and NCEP ETKF products. The ECMWF ETKF composite has a broad maximum to the north of the storm, while the maximum in the NCEP ETKF composite is to the northeast of the storm. In several cases, particularly for the NCEP ETKF, the targets can occur more than 40° to the east of the storm. While meaningful targets may be found to the east of the storms, especially for the storms embedded in the deep easterlies, these very distant targets may well be due to spurious ensemble correlations. There is still a maximum in the average field near or at the center of the plots, indicating that even when the absolute maximum value in the summary plot domain is remote from the storm, there is still often a significant signal associated with the storm itself.

b. Analysis error variance estimates

One of the most important differences between the ETKF and TESV methods is that the TESV method considers “dynamics only.” Thus the TESVs are far more likely to have targets over well-observed regions such as the United States, reflecting, for example, the dynamic connection between the storm and an approaching trough, than are the ETKF products. Constraining the SVs at initial time with analysis error variance instead of TE, as discussed in section 2b, should modulate the SV summary maps toward regions where the variance is relatively large. Before showing VARSVs results, it is instructive to examine different analysis error variance estimates.

The average analysis error variances of the DLM wind based on the NAVDAS three-dimensional variational data assimilation (3DVAR) system and the ECMWF and NCEP ETKF are shown in Fig. 8. Values in all plots have been normalized by the largest value in the domain. The TESVs would be equivalent to VARSVs for the case where the analysis error variances are spatially uniform. Larger spatial differences in the variance estimate may result in a potentially larger modulation of the SVs. For the NAVDAS estimate (Fig. 8a), the highest analysis error variances occur in the northern part of the domain. The smallest analysis error variances occur over the well-observed eastern United States, where values are between 30% and 40% of the maxima in the northern part of the domain. The ETKF analysis error variance patterns (Figs. 8b,c) are very similar to the ensemble forecast variances shown in Fig. 4 (although they are modulated by statistics reflecting regional differences in the observing network). Both show maxima associated with the North Atlantic midlatitude storm track, and the NCEP estimate shows significant maxima in the North Atlantic between 15° and 35°N. It is not surprising that these ETKF analysis error variance estimates are considerably different, given the differences in the ensemble construction methodology as discussed in the previous section. These differences are consistent with the results in Buizza et al. (2005), who show larger variance equatorward of 30°N in the NCEP ensemble than in the ECMWF ensemble. The NCEP ensemble method operational during 2004 had resulted in large perturbations to the tropical storm positions at initial time, resulting in ensemble track spread larger than the ensemble mean track error. For more recent tropical cyclone seasons, this erroneously large spread has been reduced through vortex relocation of the perturbed storms in the initial fields.

Comparison of the analysis error variance estimates valid at one time highlights these differences. Figure 9 shows the analysis error variance estimate valid at 0000 UTC 14 August 2004. The NAVDAS analysis error variance (Fig. 9a) on that day looks similar to the time average (Fig. 8a). The NAVDAS estimate, although modulated by changes in the observing network (such as satellite passes, the availability of feature-track winds, etc.), is dominated by the static background error covariance term. In contrast, the ensemble-based ETKF estimates (Figs. 9b,c) exhibit strong flow dependence. On this day, the ETKF estimates are both dominated by uncertainties in the southwest flow on the eastern side of a trough over the eastern United States (trough shown in Fig. 12), along with some tropical features and, in the case of the NCEP estimate, large variance over western Europe. The maxima are highly localized. The analysis error variance estimates from the ETKF methods over much of the domain are less than 20% of the value of the domain maximum. In this case, due to the large variance associated with the trough over the well-observed eastern United States, the NAVDAS and ETKF variance estimates are strongly out of phase.

One reason the ETKF analysis error variance estimates are so highly localized is that the estimate is based on ensemble forecasts with integration times of 48–60 h. Over this time period, the ensemble perturbations have had time to amplify in a few rapidly growing directions (probably only a small subset of all growing directions). As the assimilation of observations will preferentially reduce large-scale errors (Palmer et al. 1998; Daley 1985, 1991; Hollingsworth 1987), it is expected that forecast error variances for short times (e.g., over a 6-h data assimilation cycle) would be more isotropic and spatially uniform than the forecast error variance valid at 48 h. Therefore it is expected that consideration of the assimilation of routine observations taken between ti and ta would result in a more spatially uniform error field. Given the large differences in the analysis error variance estimates, it is expected that SVs constrained by these different estimates would themselves be very different. This is investigated in the next subsection.

c. VARSV targets

As discussed in the second section, if the true analysis error covariance matrix is used as the initial-time metric, then the final-time SVs will be the leading eigenvectors of the forecast error covariance (Ehrendorfer and Tribbia 1997). Here we constrain the initial-time SVs by the variance only (ignoring off-diagonal terms). This will have the effect of modulating the SVs away from areas where the analysis error variance is small, and toward regions where it is large.

Figure 10 shows the average summary maps and individual summary map maximum locations for VARSVs constrained by the NAVDAS analysis error variance estimate (NAVDAS VARSVS) and the ECMWF and NCEP ETKF analysis error variance estimates (ECMWF VARSVS and NCEP VARSVS, respectively). Because the 3DVAR analysis error variance estimate is relatively smooth (Fig. 8a), the NAVDAS VARSV average summary map (Fig. 10a) is similar to that of the NRL TESVs (Fig. 2b). The major difference is a reduction in the average values over the eastern United States, coincident with the region of lowest analysis error variance, and a reduction in the number of individual maxima over the central United States.

The results for the ETKF-based VARSVs (Figs. 10b,c) do differ considerably from the TESVs. For example, while the TESVs have maxima occurring west of 95°W in 15 cases, this occurs for the ECMWF (NCEP) VARSVs in only 2 (6) cases. In general, the ETKF-based VARSV average summary maps look more like the ETKF summary maps (Figs. 3a,b) than the TESV summary map (Fig. 2b), and correspond more closely to the TC tracks.

These qualities are also reflected in the summary maps composited about the storm center in the far cases. Figure 11 shows the VARSV composite summary maps that can be compared with those shown in Fig. 7. As with the TESVs (Fig. 7b), the NAVDAS VARSVs are clustered to the northwest of the storm and the patterns are quite similar. The targets occur to the northwest of the storm 24 times for the TESVs and 32 times for the NAVDAS VARSVS. When the SVs are constrained by the ETKF variance estimates (Figs. 11b,c), they occur more often to the east of the storm than the TESVs (Fig. 7b), but not as often as the ETKF targets (Figs. 7c,d). ECMWF VARSV targets occur to the northwest of the storm 24 times as compared to 16 times for the ECMWF ETKF targets. NCEP VARSV targets occur to the northwest of the storm 20 times as compared to 13 times for the NCEP ETKF targets. Many of the most distant northeast targets in the NCEP ETKF summary maps are gone from the NCEP VARSV summary maps. In short, the ETKF-based VARSV composites lie somewhere between the TESV and ETKF composites.

Clearly the target guidance provided by VARSVs is strongly influenced by the analysis error variance estimate used. While the NAVDAS analysis error variance suffers from the fact that it does not include any flow dependence, the ETKF analysis error variance estimates suffer from latitudinally dependent underdispersion in the ensembles, neglect of the impact of observations taken between ti and ta, and small ensemble size. We examine the impact of these constraints in one case study and consider the impact on perturbation growth in the next section.

4. Case study and impact on perturbation growth

The case we examine in detail, Hurricane Charley, was chosen because the NRL TESVs have a maximum over the eastern United States, which is a minimum in NAVDAS analysis error variance (Fig. 9a). We therefore expect the NAVDAS variance constraint to have a significant impact on the target guidance. We then examine the impact on perturbation growth for all cases.

a. Hurricane Charley: 0000 UTC 14 August 2004

Summary maps (Figs. 12 –14) are shown for Hurricane Charley for the ta of 0000 UTC 14 August 2004. At this point Charley has begun its northeast turn toward the Florida peninsula and is forecast to be off the New England coast in 2 days. Both TESV summary maps show significant sensitivity in the trough with which Charley is interacting (Fig. 12). The NRL TESVs show maximum sensitivity over the central United States (centered over Missouri), with a secondary maximum in the vicinity of the storm, whereas the ECMWF TESVs show maximum sensitivity farther upstream, west of the Great Lakes, and over the central eastern seaboard. A third local maximum is located over western Hudson Bay. The two ETKF summary maps (Fig. 13) are far less localized than the TESV summary maps shown in Fig. 12, and emphasize the eastern side of the trough over the eastern United States, as well as the environment of the storm. The NCEP ETKF has additional maxima over the Atlantic and western Europe. If we constrain the NRL SVs with the NAVDAS variance estimate (Fig. 14a) we can see that, as expected, the signal over Missouri has been considerably reduced, as that is where the estimated analysis error variance is smallest. Now the strongest signal is over Hudson Bay.

The ECMWF VARSV summary map (Fig. 14b) highlights similar regions as the ECMWF ETKF (Fig. 13a), that is, the eastern side of the trough over the mid-Atlantic states. Although this region is downstream of the current hurricane position, this is the region through which the TC will propagate in 2 days. Thus it is plausible that this downstream target area has a dynamic connection to the tropical cyclone through modification of the environment into which the tropical cyclone propagates. The NCEP VARSV summary map (Fig. 14c) highlights similar regions as the NCEP ETKF (Fig. 13b), that is, the storm itself, and the region into which it will propagate, over the Mid-Atlantic States. This demonstrates that given the strong constraint of the ETKF variance estimate, with variances that are highly localized, the SVs will highlight similar regions as the ETKF targets. This is not surprising, given that a perturbation in a particular region with analysis error variance that is 10 times smaller than another region would have to have energy growth that is 10 times larger to result in a more dominant SV. That noted, there must be some connection between the SV target region and the verification region through the (dry linear) dynamics. For example, the VARSV summary map (Fig. 14c) does not have significant amplitude over southern Britain and northwestern France, even though the domain maximum in the NCEP ETKF occurs here (Fig. 13b). This maximum, likely due to spurious correlations given its distance downstream of the storm, is ignored by the SV calculation. This case illustrates how inclusion of the variance constraint will modulate targets toward regions where the analysis error variance is large, but it also, in a certain sense, can act as a dynamic filter, reducing the importance of potentially spurious correlations. This effect is also illustrated in the reduction in the number of targets that are very far from the storm, particularly to the northeast, when comparing the NCEP VARSV composite (Fig. 11c) with the NCEP ETKF composite (Fig. 7d).

Constraining the SVs with the NAVDAS variance reduces the linear perturbation growth in this case (measured in terms of the square root of the total energy) from 7.72 to 5.05. The ETKF variance constraints reduce growth to 1.47 and 1.77 for the ECMWF and NCEP estimates, respectively. The general reduction in perturbation growth is discussed in the next subsection.

b. Linear perturbation growth

Figure 15 shows the amplification, measured in terms of the square root of the perturbation total energy, of the leading SV, for each of the 78 cases, for the TESVs, NAVDAS VARSVs, ECMWF VARSVs and NCEP VARSVs (for the TESVs, this is equivalent to the leading singular value). The average values for all 78 cases are 5.76 and 5.21 for the TESV and NAVDAS VARSV, respectively, and 1.54 and 1.44 for the ECMWF VARSV and NCEP VARSV, respectively. The NAVDAS constraint results in a slight decrease in energy perturbation growth from the optimal total-energy growth, reflecting the rather small modification of the SVs by this smooth variance field, in most cases. The Charley case considered above (case 6) has one of the largest drops in perturbation growth between TESVs and NAVDAS VARSVs. In contrast, the decrease in perturbation growth when the SVs are constrained using the ETKF variance estimates is substantial, such that in some cases the amplification is less than 1 (i.e., the perturbations decay). The very large decrease in perturbation growth for the ETKF estimates reflects the highly localized nature of the analysis error variance estimates (i.e., the analysis error variance estimate over most of the domain is usually less than 20% as large as the localized maximum). These results highlight the impact of the analysis error covariance estimates for adaptive observing applications, and thus the importance of accurate analysis error statistics.

Peng and Reynolds (2006) found that, on average, larger amplification rates were related to larger track errors. We did not find a similar correlation here, which may be due to the incorporation of a 2-day lead time in the current study. We did find that the amplification rates were inversely correlated with the forecast sea level pressure, indicating larger potential for perturbation growth associated with stronger storms.

5. Conclusions and discussion

In M2006, the similarities between TESV and ETKF adaptive sampling guidance products were quantitatively compared using statistics such as a modified equitable threat score. M2006 noted that target summary maps produced using the same method but different forecast systems were more similar than those produced using different methods. In the current study, we use averaging and storm-relative composite techniques that allow us to go beyond the general similarity analysis in M2006, to document in detail these structural differences and to relate them to the mathematical and physical differences in the methods used to compute the summary maps. In addition, we investigate the impact of including error statistics in the SV calculation, in effect making the SV and ETKF calculations more similar.

Results show that systematic differences between the TESV and ETKF guidance maps are mainly due to the fact that the TESVs consider the dynamics of perturbation growth only, whereas the ETKF considers dynamics of perturbation evolution and analysis error covariance statistics. In fact, when TESV maxima are remote from the storm, they occur almost exclusively to the northwest of the storm, often over the well-observed United States, reflecting the fact that the calculation is not constrained by analysis error statistics. In contrast, the ETKF remote targets rarely occur over the western and central United States, and are far more likely to occur over the North Atlantic, reflecting the large ensemble variance and large ETKF analysis error variance in this region.

The systematic differences between the ECMWF ETKF and NCEP ETKF products were found to be mainly due to differences in the ensemble spread at ta, which is 48–60 h into the forecast, and arise from the differences in the ensemble construction methods. For example, the NCEP ensemble has far more variance in the central and eastern subtropical North Atlantic, reflecting unrealistically large ensemble spread in tropical cyclone track forecasts. (This has since been corrected through the use of vortex relocation.) In contrast, ECMWF tropical SVs are regularly calculated for the Caribbean area, but are only calculated for other areas after a storm reaches hurricane strength. The NCEP initial perturbations are truly global and produce larger ensemble variance than the ECMWF ensemble in the Tropics and subtropics (Buizza et al. 2005).

Inclusion of analysis error statistics in the SV calculation brings the SV and ETKF guidance maps closer together. These experiments illustrate the SV sensitivity to metric, specifically the estimate of analysis error variance used to replace TE as the initial-time norm. The use of the NAVDAS variance constraint has a small impact on the SV summary maps and results in a relatively small diminishment in perturbation growth (a 10% reduction, on average). In contrast, the use of the ETKF analysis error variance estimates can result in large changes to the SV summary map and a large reduction in perturbation growth. Because the SV calculation is based on a (linear, dry) dynamic connection between initial and final time, the SV calculation acts as a way to filter out spurious correlations in the ETKF. That is, the ETKF VARSV will produce patterns that look like the ETKF summary map to the extent that a dynamic connection can be found between the ETKF summary map maxima at ta and the verification region at tυ. The ETKF constraint produces VARSV maxima there are often collocated with ETKF maxima because the estimated analysis error variances based on the 48–60-h ensembles produced at NCEP and ECMWF are very localized. Should the impact of observations taken between ti and ta be accounted for, it is assumed that the estimated analysis error variance field would be more spatially uniform. This strong constraint is also reflected in the strong reduction in perturbation growth when using the ETKF analysis error variance estimate (74% on average). The small perturbation growth (sometimes decay) reflects the fact that most of the domain has analysis error variance values less than 20% of the localized maximum. It also may reflect the fact that the ECMWF and NCEP ensembles do not necessarily have the largest variance where the NOGAPS forecast has the storm.

In addition to consideration of the observations taken between ti and ta, it is anticipated that the ETKF technique may be improved through larger ensembles, which should reduce spurious correlations. The application of space–time covariance localization (Khare and Anderson 2006) may also result in improvements, but it is unclear how to apply such a technique in complex flows. Potential improvements to the SV technique include the consideration of moist processes in the SV calculation, and the consideration of different final-time metrics. A possible way of combining the strengths of the SV technique (guaranteed absence of spurious targets, rapid error growth) with the strengths of the ETKF (quantitative estimates of reduction in forecast error variance due to adaptive observations) would be to apply the ETKF to VARSVs based on the best available estimate of analysis error variance, as in Leutbecher (2003).

The next stage in comparing the adaptive observing methods should be to perform data-denial experiments to compare the relative impact of observations in target regions produced using different methods. Both Buizza et al. (2007) and Aberson (2003) have used data denial experiments to show that observations taken in target regions are indeed more valuable than observations taken in random regions (the former in the context of midlatitude winter and summer forecasts using SV-based targets, and the later in the context of tropical cyclone forecasts using subjective guidance). While one may speculate about the conditions under which one target method would be superior to another (e.g., TESVs would do well where analysis error variances are spatially uniform, and ETKF would do well when based on very large ensembles), determining which method is better must await data denial experiments that explicitly compare methods. These should include not only the methods considered in this study, but other methods such as observation sensitivity (Langland and Baker 2004), and methods employed during the Dropwindsonde Observations for Typhoon Surveillance near the Taiwan Region (DOTSTAR) tropical cyclone adaptive observing field program (Wu et al. 2005). Note that the concept of adaptive observing should not be limited to dropsonde data, but should be extended to other observing system components, such as high-density satellite winds, or off-time rawinsondes.

Acknowledgments

S. J. Majumdar and S. D. Aberson acknowledge financial support from the NOAA Joint Hurricane Testbed, and Z. Toth and the Environmental Modeling Center at NCEP for making the paper possible via the provision of an account on the IBM SP supercomputer. C. H. Bishop, M. S. Peng, and C. A. Reynolds acknowledge the support of the Naval Research Laboratory and the Office of Naval Research under Program Element 0601153N, Project Number BE-033-03-4M.

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Fig. 1.
Fig. 1.

The locations of the storms (marked by “x”s) for the 78 cases from the 2004 season for which summary maps were produced.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 2.
Fig. 2.

The 78-case average of the adaptive observing guidance maps (shaded) for (a) ECMWF TESVs and (b) NRL TESVs. The locations of the summary map maxima for each of the 78 cases are indicated by the numbers. A number greater than 1 indicates that the maximum occurred in that location more than once.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 3.
Fig. 3.

Same as in Fig. 2 but for (a) ECMWF ETKF and (b) NCEP ETKF.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 4.
Fig. 4.

The 78-case average deep-layer mean wind ensemble variance (m2 s−2) at the target time for (a) the ECMWF ensemble and (b) NCEP ensemble.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 5.
Fig. 5.

The 78-case average of the normalized ensemble variance valid at the target time (shaded) for (a) the ECMWF ensemble and (b) the NCEP ensemble. The location of the maximum variance in each of the 78 cases is given by the numbers. A number greater than 1 indicates the maximum occurred in that location more than once.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 6.
Fig. 6.

Composite of the summary maps about the storm center (shaded) for cases when the summary map maximum lies within 15° of the storm (“near” cases) for (a) ECMWF TESV, (b) NRL TESV, (c) ECMWF ETKF, and (d) NCEP ETKF. The location of the maximum variance in each of the 78 cases is given by the numbers. The 500-hPa streamlines of the background flow, also composited about the storm center, are superimposed on this field.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 7.
Fig. 7.

Composite of the summary maps about the storm center (shaded) for cases when the summary map maximum lies outside 15° of the storm (“far” cases) for (a) ECMWF TESV, (b) NRL TESV, (c) ECMWF ETKF, and (d) NCEP ETKF. The location of the maximum variance in each of the 78 cases is given by the numbers. The 500-hPa streamlines of the background flow, also composited about the storm center, are superimposed on this field.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 8.
Fig. 8.

The 78-case average analysis error variance estimates of the deep-layer mean wind, normalized by the largest value in the domain, for estimates based on (a) NRL NAVDAS, (b) ECMWF ETKF, and (c) NCEP ETKF.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 9.
Fig. 9.

The analysis error variance estimate of the deep-layer mean wind valid at 0000 UTC 14 Aug 2004, normalized by the largest value in the domain, for estimates based on (a) NRL NAVDAS, (b) ECMWF ETKF, and (c) NCEP ETKF.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 10.
Fig. 10.

The 78-case average of the adaptive observing guidance maps (shaded) for (a) NAVDAS VARSV, (b) ECMWF VARSV, and (c) NCEP VARSV. The locations of the summary map maxima for each of the 78 cases are indicated by the numbers. A number greater than 1 indicates that the maximum occurred in that location more than once.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 11.
Fig. 11.

Composite of the summary maps about the storm center (shaded) for cases when the maximum lies outside 15° of the storm (far cases) for (a) NAVDAS VARSV, (b) ECMWF VARSV, and (c) NCEP VARSV. The location of the maximum variance in each of the 78 cases is given by the numbers. The 500-hPa streamlines of the background flow, also composited about the storm center, are superimposed on this field.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 12.
Fig. 12.

Summary maps for Hurricane Charley (shaded), for targets valid at 0000 UTC 14 Aug 2004, for (a) ECMWF TESV and (b) NRL TESV. The forecast position of Charley at the target time is denoted by the X. The deep-layer mean wind streamlines valid at that time are superimposed on this figure.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 13.
Fig. 13.

Summary maps for Hurricane Charley (shaded), for targets valid on 0000 UTC 14 Aug 2004, for (a) ECMWF ETKF and (b) NCEP ETKF. The forecast position of Charley at the target time is denoted by the X. The deep-layer mean wind stream lines valid at that time are superimposed on this figure.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 14.
Fig. 14.

Summary maps for Hurricane Charley (shaded), for targets valid at 0000 UTC 14 Aug 2004, for (a) NAVDAS VARSV, (b) ECMWF VARSV, and (c) NCEP VARSV. The forecast position of Charley at the target time is denoted by the X. The deep-layer mean wind streamlines valid at that time are superimposed on this figure.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Fig. 15.
Fig. 15.

The singular value, or linear amplification, as measured by the square root of total energy, corresponding to the leading singular vectors for each of the 78 cases. TESVs given by filled squares, NAVDAS VARSVs given by clear squares, ECMWF VARSVs given by filled triangles, and NCEP VARSVs given by clear triangles.

Citation: Monthly Weather Review 135, 12; 10.1175/2007MWR2027.1

Table 1.

Details of the different adaptive observing products, specifying (according to row number) 1) name, 2) technique, 3) forecast model, 4) estimate of analysis error variance used (if any), 5) number of ensembles or SVs used, 6) the resolution of the nonlinear model, 7) the resolution of the output, 8) the difference between the verification and adaptive observing times, 9) the difference between the adaptive observing and initial times, and 10) the start time and trajectory about which the perturbations are calculated.

Table 1.

1

The NCEP GFS ensembles initialized at 0600 and 1800 UTC were not archived and are not used in this study.

2

The larger values for the ETKF averages versus the TESV averages reflect the fact that the domain-averaged values in the individual ETKF summary maps are significantly larger than in the TESV summary maps.

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