To efficiently and effectively prioritize resources, adaptive observations can be targeted by using some objective criteria to estimate the potential impact an initial condition perturbation (or analysis increment) in a specific region would have on the future forecast. Several objective targeting guidance techniques have been developed, including total-energy singular vectors (TESV), adjoint-derived sensitivity steering vectors (ADSSV), and the ensemble transform Kalman filter (ETKF), all of which were tested during the 2008 The Observing System Research and Predictability Experiment (THORPEX) Pacific Asian Regional Campaign (T-PARC) and the Office of Naval Research Tropical Cyclone Structure-2008 (TCS-08) field experiments. An intercomparison between these techniques is performed in order to find underlying physical mechanisms in the respective guidance products, based on four tropical cyclone (TC) cases from the T-PARC/TCS-08 field campaigns. It is found that the TESV energy norm and the ADSSV response function are largely indirect measures of the TC track divergence that can be produced by an initial condition perturbation, explaining the strong correlation between these products. The downstream targets routinely chosen by the ETKF guidance system are often not found in the TESV and ADSSV guidance products, and it is found that downstream perturbations can affect the steering of a TC through the development of a Rossby wave in the subtropics that modulates the strength of the nearby subtropical ridge. It is hypothesized that the ubiquitousness of these downstream targets in the ETKF is largely due to the existence of large uncertainties downstream of the TC that are not taken into consideration by either the TESV or ADSSV techniques.
Tropical cyclones (TCs) represent a unique forecasting challenge, because while they have the capacity to pose significant threats to lives and property when they approach landfall, they exist largely over the open ocean where few in situ observations are collected. This dearth of in situ information potentially allows for large uncertainties in model analyses, which can be detrimental to the subsequent forecast of the TC in numerical weather prediction (NWP). Enhanced observations, for example from satellites or reconnoitered aircraft, can be deployed with the explicit purpose of reducing these uncertainties. In such a situation, the observations can be thought of as adaptive, and targeted (in space and time) based on some objective criteria meant to determine regions in the initial conditions for which the TC forecast is most sensitive.
An objective targeting guidance system is designed to estimate the potential impact that additional observations, assimilated into the initial conditions of an NWP model, will have on some particular metric or aspect of the forecast (Langland 2005; Majumdar et al. 2011a). These techniques are typically applied to reconnaissance campaigns aimed at improving predictions of wintertime midlatitude systems (Langland et al. 1999; Szunyogh et al. 2002) or tropical cyclones (Wu at al. 2005; Aberson et al. 2011), where they have to date been aimed at improving track forecasts.
Several techniques have been developed to provide this estimated-forecast-impact guidance. Some, like the total-energy singular vector (TESV) or adjoint-derived sensitivity steering vector (ADSSV) techniques, rely purely on the dynamics of perturbation growth in order to determine where observations are expected to have the most impact. These techniques do not directly attempt to estimate the size of the perturbation a new observation will produce in the analysis, only that a perturbation, if present, will have some effect on the chosen metric. In contrast, the ensemble transform Kalman filter (ETKF) combines both the dynamics of perturbation growth and an estimate of the variance of the impact that a new observation would impose on the analysis, if it were to be assimilated using an ensemble-based scheme.
Previous intercomparison studies have focused on qualitative or quantitative analyses of the similarities and differences between various guidance products (Majumdar et al. 2006; Reynolds et al. 2007; Wu et al. 2009a), rather than an attempt to link the products to specific physical mechanisms in the model. In this study, the physical mechanisms driving the targeting guidance of three different techniques are investigated. The focus is on four west Pacific typhoon cases from the 2008 The Observing System Research and Predictability Experiment (THORPEX) Pacific Asian Regional Campaign (T-PARC) and the Office of Naval Research Tropical Cyclone Structure-2008 (TCS-08) reconnaissance experiments: Typhoon Nuri, Typhoon Hagupit, Typhoon Sinlaku, and Typhoon Jangmi. Similarities and differences between these guidance products, both within the four case studies as well as those observed in previous intercomparison studies, are related to specific physical mechanisms driving these targeting systems. A description of the guidance products and how they were created is provided in section 2. In section 3, an analysis of physical mechanisms responsible for observed guidance product behavior is performed. A suggested modification to the ADSSV technique, based on these results, is provided in section 4. Conclusions and directions for further research are discussed in section 5.
2. Guidance products
a. Total-energy singular vector guidance
Techniques that make use of TESVs seek to find the most rapidly growing perturbations in the model trajectory for a given trajectory length and verifying area (Reynolds et al. 2010); this technique typically defines as its metric for perturbation growth a total-energy norm incorporating squared values of perturbation fields (e.g., squared perturbation wind components in the kinetic energy term, squared perturbation temperature in the available potential energy term). The perturbation that maximizes the ratio of initial condition total energy to final state total energy (integrated within a specific three-dimensional volume at the end of a chosen trajectory length) is the leading singular vector. While this technique in actuality creates a hierarchy of modes (each maximizing the energy ratio while constrained to be orthogonal to the modes above it), only the first few leading TESVs are typically used for guidance (Peng and Reynolds 2006; Chen et al. 2009; Reynolds et al. 2010).
The calculation of TESVs for the four T-PARC/TCS-08 cases under consideration was similar in methodology to the TESVs generated in real time during the field campaigns (Reynolds et al. 2010), but included some significant changes. Like the real-time TESVs, the product used in this study was produced using the tangent linear and adjoint models of the Navy Operational Global Atmospheric Prediction System (NOGAPS; Rosmond 1997) and subject to a total (dry) energy metric at initial and final times, constituting the sum of perturbation kinetic and available potential energy summed over all resolved wavenumbers and the depth of the atmosphere.
Also like the real-time TESVs, the singular vectors are produced at a reduced, T79 spectral resolution with 30 unevenly spaced vertical (sigma) levels, based on a model forward trajectory run at an operational resolution of T239 with 30 levels. Both are produced using the Fleet Numerical Meteorological and Oceanographic Center (FNMOC) analyses [data for this project are from the FNMOC and the U.S. Global Data Assimilation Experiment (USGODAE) and are available online at http://www.usgodae.org/ftp/outgoing/fnmoc/models/nogaps]. The NOGAPS tangent linear and adjoint models retain diabatic effects of large-scale precipitation and are linearized about a full (moist) physics trajectory from the forward model.
Unlike the real-time TESVs, the verification region over which the final-time perturbation energy is calculated is not geographically fixed; rather, the verification region is a 15° × 15° box centered on the TC at the end of the 48-h forecast. This is done in order to allow for direct comparisons of the TESV guidance product to the ADSSV, which uses the same verification region for its own metric (see section 2b below). A similar verification region centered on the TC is prescribed in the ETKF, in order to investigate the potential for reducing forecast error variance in the area most relevant to the TC. Also, while in field operations a lead time of 48 h is used in order to provide time for mission planning and clearance (i.e., the 48-h trajectory used in the field campaigns is hours 48–96), the TESVs produced for this study include zero lead time; this is done so that the initial time TESVs can be used directly to relate to initial condition perturbations performed later (see section 3 below). Finally, while both the real-time guidance product and the product used for this study are integrations of the first three leading TESVs, the real-time product is further integrated in the vertical while the one used in this study is not. The purpose of the vertical integration used during field campaigns is to provide a single, cohesive summary of guidance, while in this study we are interested in maintaining the three-dimensional structure of TESVs for direct comparison with the three-dimensional ADSSVs.
TESVs are expected to represent small perturbations in the initial conditions that grow quickly; when the verification area is focused on a TC and its surrounding environment, it is believed that these perturbations must therefore have a large impact on the evolution of the TC (Chen et al. 2009), either through manipulation of the TC and its near environment, such as a vortex Rossby wave interaction (Peng and Reynolds 2006) or baroclinic energy conversion in the TC vortex (Yamaguchi and Majumdar 2010), or through manipulation of the upstream environment, which can then evolve over the course of the trajectory to have a significant impact on the TC by the verification time, such as the interaction of the TC with an upstream trough (Peng and Reynolds 2006; Reynolds et al. 2009) or the evolution of the downstream environment that the TC is propagating into (Reynolds et al. 2007). In addition, if the singular vectors were further constrained by an estimate of the analysis error covariance matrix, they would theoretically evolve into the EOFs of the forecast error covariance matrix at the verification time (Ehrendorfer and Tribbia 1997). Such a constraint would include analysis uncertainty into the singular vector calculation, bringing the methodology closer to that used by the ETKF signal variance product (see below).
b. Adjoint-derived sensitivity steering vector guidance
Like the TESV guidance product, the ADSSV also uses the NWP model adjoint in order to provide information about the impact of small perturbations on some function of the model verification state (Errico 1997). However, rather than searching for the fastest-growing perturbation, the ADSSV seeks to calculate the relationship between any small (but otherwise arbitrary) perturbation to the model initial conditions and the steering of the TC at model verification (Wu et al. 2007, 2009b); this is done by defining the response function as the average flow in a box surrounding the TC at model verification through some depth of the troposphere, such that the sensitivity gradients produced by the adjoint model describe how the “steering column” driving TC motion at verification (Chan and Gray 1982; Velden and Leslie 1991; Chan 2005) is affected by small initial condition perturbations. Sensitivity of both the zonal and meridional component of the steering column are computed with separate response functions:
where n is the number of grid points (indexed zonally by i, meridionally by j, and vertically by k) within the horizontal and vertical dimensions of the volume V defining the steering column. The sensitivity gradients of each response function can then be combined into a vector:
which represents the sensitivity of the steering vector to a small perturbation to the initial conditions x0. Operationally only the sensitivity with respect to vorticity, , is typically used, since vorticity is a quasi-conserved variable (as opposed to, say, temperature) and allows for a much easier physical interpretation.
The ADSSVs used in this study are calculated from the NOGAPS adjoint model, using a spectral resolution of T159 with 30 vertical levels for both the forward trajectory and adjoint integration,1 initialized with FNMOC analyses. The zonal R1 and meridional R2 steering response functions are defined as all points in a 15° × 15° box centered on the TC at 48 h, as per the TESV calculation, through a depth from near surface to roughly 300 hPa. Also like the TESV calculation, the ADSSVs used in this study are produced with zero lead time.
As opposed to the TESV, where the expected impact of a perturbation that projects onto the TESV is difficult to physically interpret by the TESV alone but anticipated to be focused on the final-time state of the TC, the expected impact of a perturbation projecting onto the ADSSV is explicit. The inner product between the ADSSV and any initial condition perturbation is a quantitative estimate of the vector change in steering that should result by the verification time, represented by the average flow in the three-dimensional volume surrounding the TC:
where is the two-dimensional vector of the change in zonal and meridional steering observed at model verification. Other changes to the TC as a result of the initial condition perturbations (e.g., an intensity change) are not considered. Therefore, it is expected that errors that project onto the ADSSV are errors that have the greatest potential to specifically affect the steering of the TC at verification. ADSSV guidance has been routinely observed on the west side of the subtropical ridge presumably steering the TC (Wu et al. 2007) and within or near upstream troughs when a trough–TC interaction takes place (Wu et al. 2009b).
While the ADSSV technique endeavors to find the sensitivity of the environmental flow steering the TC, a perturbation analysis (see section 3, below) reveals that this response function is actually strongly affected by any change in the TC's position at the verification time, and does not appropriately represent sensitivity of the TC steering. This problem with the ADSSV and a proposed solution are discussed in section 4.
c. Ensemble transform Kalman filter signal variance guidance
The version of the ETKF in this paper seeks to estimate the potential impact of a new observation on the forecast error variance of total energy, as defined by a squared energy norm similar to the TESV, using both dynamical information from an ensemble forecast as well as information about how observations are assimilated in an ETKF system (Bishop et al. 2001). This reduction in forecast error variance is theoretically equal to the variance of the forecast impact, represented by the signal, defined as the difference between a forecast (or an analysis) state initialized with routine and targeted observations and that same state initialized with routine observations only. The conventional ETKF guidance map presents the vertically averaged signal variance within a chosen verification region, plotted as a function of the location in which a hypothetical observation is to be assimilated in the ETKF system (Majumdar et al. 2002a, 2011b).
This technique differs from the previous two in several ways. First, rather than using the adjoint of the NWP model to identify a dynamical sensitivity, the ETKF uses an ensemble of forecasts to predict forecast error variance reduction based on any hypothetical configuration of observations of wind and temperature. The focus on error variance reduction means the ETKF is not as concerned with initially small perturbations that have the fastest growth (like TESVs), but more usually initially large but not necessarily rapidly growing errors (Majumdar et al. 2002b).
Second, the introduction of aspects of assimilation into the metric, in addition to the dynamics of perturbation growth that forms the basis of the TESV and ADSSV techniques, effectively introduces a further constraint on the guidance product. The ETKF seeks to estimate the potential impact that a new assimilated observation will have on the forecast error variance, not just the impact of an arbitrarily prescribed perturbation to the analysis. As a result, the ETKF signal variance is informed by a hypothetical observation's ability to produce a change to the analysis, in addition to the dynamical sensitivity of the forecast error variance to that increment, which means ETKF guidance is drawn toward areas of high dynamical sensitivity and high (estimated) analysis uncertainty. This additional constraint is responsible for some large disagreements between ETKF and TESV/ADSSV guidance (see section 3, below).
Third, the ETKF explicitly seeks to focus on the variance of the background flow steering the TC; in order to do so, the symmetric circulation of the TC is removed from each analysis member prior to computing the signal variance. The resulting product is less focused on the precise location of the center of the TC's symmetric circulation, and more focused on how an additional observation would influence the variance in the background flow of the TC's environment, as well as the asymmetric portion of the TC flow.
It is worth noting, however, that the ETKF can be applied to any set of initial and forecast perturbations, and the theoretical equivalence between forecast error variance reduction using singular vectors with an analysis error covariance metric and the ETKF estimation of signal variance has been demonstrated (Leutbecher 2003; Majumdar et al. 2006). In practice, singular vectors are not computed using this metric, and the ETKF uses evolved nonlinear perturbations from an operational ensemble forecast, thereby leading to significant differences between the respective sets of guidance. In this study, the ETKF guidance is produced from the European Centre for Medium-Range Weather Forecasts (ECMWF) 50-member global ensemble at 1° × 1° resolution, courtesy of the THORPEX Integrated Global Grand Ensemble (TIGGE) archive (data are available from http://tigge-portal.ecmwf.int). The 48-h lead time is retained for these calculations, because the ensemble perturbations must be large enough to approximate actual analysis errors, making a zero lead-time calculation unfeasible. The ETKF summary map is a single-level product summarizing the signal variance within the verification region centered on the TC at +96 h, based on hypothetical observations to be taken at +48 h.
Maxima in ETKF guidance tend to collocate with regions of strong uncertainty, though mere uncertainty alone is not sufficient to be considered a sensitive target region. In a case study of Typhoon Sinlaku (2008), it was found that the ECMWF ensemble's ETKF signal variance highlighted several nearby synoptic features including the northwest side of the subtropical ridge and a region of confluent flow between the ridge and the upper-level low, with secondary remote targets in the midlatitude jet north of the ridge and over the Chinese mainland (Majumdar et al. 2011b).
3. Case studies
a. T-PARC/TCS-08 cases
Four significant cases from T-PARC/TCS-08 were chosen for this study: Typhoon Nuri, Typhoon Hagupit, Typhoon Sinlaku, and Typhoon Jangmi. The 48-h period examined for each case is provided in Table 1. Typhoons Nuri and Hagupit moved along predominantly zonal tracks, while Typhoons Sinlaku and Jangmi underwent recurvature (Fig. 1). NOGAPS simulations (T159 resolution) appear to correctly differentiate the predominantly zonal characteristic of Nuri's and Hagupit's tracks, as well as adequately resolve the tracks of Jangmi and Sinlaku as they enter recurvature.
Typhoon Sinlaku has been a topic of especially intense study (Weissmann et al. 2012; Wu et al. 2012; Huang et al. 2012), owing largely to the unprecedented opportunities for aircraft sampling as the typhoon became motionless for an extended period of time before finally recurving away from land. It was also a relatively low-predictability case, with considerable nonlinear sensitivity to initial environmental perturbations (Yamaguchi and Majumdar 2010; Komaromi et al. 2011). Typhoon Jangmi has also been referenced often in previous studies as a companion to Sinlaku as a major typhoon case from T-PARC/TCS-08. It has been observed that dropsonde observations assimilated into forecasts for Jangmi produced some noteworthy forecast degradations in the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS) relative to those assimilated for Sinlaku (Chou et al. 2011). These are presented alongside Nuri and Hagupit for comparison with typhoons in the “zonal motion” regime.
b. Similarities: TESV and ADSSV guidance
Guidance from TESVs (Fig. 2) and ADSSVs (Fig. 3) exhibit a high degree of similarity, especially in local maxima. This strong correlation (Table 2) has been observed in previous work (Wu et al. 2009a); while it may not at first seem surprising (since both products are purely defined by the dynamics of perturbation growth), it is important to recognize that both of these products are looking at very different metrics. The ADSSV seeks to determine how much the average flow in the verifying area will change as a result of an initial condition (vorticity) perturbation, while the TESV seeks to determine what initial condition total-energy perturbation will produce the highest total perturbation energy in the verifying area. Comparing Figs. 2 and 3, it appears that the vast majority of the salient features in the TESV are replicated in the ADSSV, including the upstream targets over mainland China in the Sinlaku case (Figs. 2c and 3c) and, to a lesser extent, in the Jangmi case (Figs. 2d and 3d).
One can reasonably assume that a perturbation that projects strongly onto initial condition TESVs will also project strongly onto initial condition ADSSVs in these cases. If the same perturbation can excite both the TESV and ADSSV, that means that either (i) the same physical mechanism is controlling both metrics, or (ii) one of the metrics is strongly controlled by the other. A simple way to determine this is to introduce a perturbation into the model initial conditions that reasonably projects onto both the TESV and ADSSV guidance products, and observe the resulting perturbation fields at 48 h.
The focus is kept on Jangmi, for which a significant maximum in TESV and ADSSV guidance exists southeast of the TC vortex in the model initial conditions (see Figs. 2d and 3d). While Fig. 3 only displays the magnitude of the sensitivity vectors, vectors in this region indicate that a positive (negative) vorticity perturbation should yield a stronger southerly (northerly) component to the steering flow (not shown). A vorticity perturbation is inserted in this region, projecting specifically onto the meridional component of the ADSSV; this is done by directly prescribing a vorticity perturbation by the meridional component of ADSSV scaled by an appropriate value:
where is the maximum value of the meridional component of ADSSV and S is a scaling factor. For this experiment, S was set to be either 1.0 × 10−5 or −1.0 × 10−5, such that the maximum expressed value of the vorticity perturbation was equal to S at the maximum (absolute) value of sensitivity, and perturbations in less sensitive areas were scaled accordingly (see Fig. 5a). In addition, vorticity perturbations were only allowed within a circle of radius 10° between 600 and 400 hPa.2 The theory behind the ADSSV predicts that the perturbations with S > 0 (S < 0) should produce a perturbation to the average flow that has a strong southerly (northerly) component.
Finally, we note that this perturbation technique differs slightly from that introduced by Komaromi et al. (2011), in which a multiplicative perturbation of relative vorticity within a local area and layer is created, and the flow is rebalanced prior to forward integration of the numerical model. Here, a perturbation is directly prescribed by the observed sensitivity with respect to vorticity; the flow is not rebalanced in order to maintain a consistent projection of the perturbation onto the sensitivity gradient, and for this reason care is taken to insure that initial condition perturbations remain small. We wish to observe the direct, literal interpretation of sensitivity with respect to vorticity, which requires a sacrifice of realism in the initial condition perturbation in order to maintain the relevance of the perturbation to the interests of the study. However, we expect that the qualitative nature of the results presented here will be consistent with those of Komaromi et al. (2011).
To facilitate better understanding of the results, a conceptual diagram of the physical process responsible for perturbation flow in the verification region is provided (Fig. 4). A verification region is defined by a 15° × 15° box centered on the location of the TC at the end of a 48-h simulation (Fig. 4a); this means that the average flow in this box can represent the steering of the TC, since the TC's own circulation would cancel out in the averaging. However, once an initial condition perturbation is introduced, the track of the TC can change, placing the TC out of the center of the verification region (Fig. 4b). The perturbation vorticity (Fig. 4c) and perturbation flow (Fig. 4d) are dominated by a dipole of vorticity representing the translation of the TC from its position in the unperturbed simulation to its new position.
It is this perturbation flow that defines the perturbation flow in the verification region, which is predominantly defined by a small track divergence. Once the TC is no longer in the middle of the box at verification time, its own flow (which constitutes the vast majority of the flow) can contribute to the mean flow because it no longer perfectly cancels out in the averaging. This flow unfortunately is not directly related to the steering of the TC at the final time, but rather is an indirect measure of the accumulated track divergence over 48 h that has appeared as a result of the initial condition perturbation. As a result, the ADSSV does not appear to predict the change in steering due to an initial condition perturbation, which was the original goal. This problem has been discussed in previous work (Hoover 2009; Hoover and Morgan 2010), and an examination of a possible modification to the technique will be discussed (see section 4).
One can imagine that the kinetic energy of the perturbation winds is maximized between the positive and negative lobes of the vorticity dipole (Fig. 4d); this is the kinetic energy of the perturbation field that contributes to the total-energy norm that defines the TESV. In fact, since the remaining terms in the total-energy norm contain squared values of perturbation temperature and moisture, and these fields, like vorticity, are characterized primarily by a maximum in the TC with a circular gradient, these remaining terms are likewise predominantly dictated by any small displacement of the TC from its unperturbed 48-h position as well. So it is found that TESVs and ADSSVs produce similar guidance because they are both primarily driven by small track divergences brought on by initial condition perturbations, and both the ADSSV “steering” response function and the TESV energy norm are essentially indirect functions of the TC's displacement.
The conceptual model is verified with perturbation experiments in the case of Typhoon Jangmi (Fig. 5). The initial condition perturbation places a small amount of kinetic energy southeast of the TC (Fig. 5a), projecting strongly onto the meridional component of the ADSSV (by design) as well as projecting at least partially onto the TESV maximum (Fig. 2d).3 By the verification time, the perturbation height field in the verification region is dominated by a dipole caused by a slight translation of the cyclone to the west for a positive vorticity perturbation (Fig. 5b) or east for a negative perturbation (Fig. 5c). Likewise the perturbation wind field is described by the dipole, with southerly flow for the positive perturbation case and northerly flow for the negative perturbation case. The major salient feature at 48 h is a small displacement of the final time position of Jangmi caused by an accumulated track divergence over the 48-h trajectory (Fig. 5d).
Recall that this result, an average perturbation flow that is southerly (northerly) for a positive (negative) initial condition vorticity perturbation southeast of Jangmi, was precisely what was predicted by the ADSSV. The change in the average meridional flow in the response function box was 0.573 m s−1 for the positive perturbation experiment and −0.648 m s−1 for the negative perturbation experiment. Similarly, the adjoint-derived sensitivity gradients predicted a change of 0.529 m s−1 (−0.528 m s−1) for the positive (negative) perturbation experiment [see Eq. (4)], accounting for 92.3% (81.4%) of the observed change.
The error is in the assumption that this perturbation flow must translate directly to a steering flow, brought on by an assumption about the relationship between the average flow and the steering flow that is valid for the simulation with unperturbed initial conditions, but not necessarily valid once a perturbation has been prescribed. The strength of this dipole is dependent on how far the perturbed TC track has diverged; therefore, the ADSSV is better thought of as an indirect measure of sensitivity of accumulated track divergence than it is a measure of sensitivity of steering at the final time.
Likewise, the observed growth4 of perturbation kinetic energy is derived from the perturbation wind field and is, therefore, also dominated by the translation-induced dipole (Fig. 5). Therefore the TESV energy norm is also an indirect measure of the sensitivity of the accumulated track divergence, and the observed correlation between TESV and ADSSV guidance is predictably caused by the fact that both guidance measures are essentially looking at the same thing. By itself, this is not necessarily a problem; the accumulated track divergence (relative to an analysis best track) is typically used as a standard by which to measure the success of a TC forecast, so a guidance product that is principally focused on the track divergence is addressing a relevant concern. However, the original intent of the ADSSV was to estimate the sensitivity of the final-time steering of the TC to initial condition vorticity perturbations, not the sensitivity of the accumulated track divergence.
c. Differences: ETKF signal variance guidance and TESV/ADSSV guidance
The ETKF signal variance is provided for each of the four T-PARC/TCS-08 cases (Fig. 6). The amount of sensitivity to new observations varies strongly between cases, with Jangmi displaying significantly more sensitivity than the other cases. It has also been observed in a previous study that Jangmi displayed more erratic forecast impact for assimilated T-PARC dropsondes than did Sinlaku (Chou et al. 2011), which may indicate that strong sensitivity to new observations may not necessarily translate to strong, positive improvement from their assimilation. A number of problems plague the assimilation of dropsonde data in or near tropical cyclones (Aberson 2008), and it may be possible that heightened sensitivity to targeted observations indicates the potential for the exaggeration of those issues.
A comparison of ETKF signal variance to the analysis ensemble variance estimate (Fig. 7) reveals a strong correlation between these fields (Table 2), consistent with the results of Majumdar et al. (2011b). This is not surprising, considering that the signal variance is concerned with the three following characteristics: (i) the dynamical sensitivity of an analysis perturbation in a given region, (ii) the potential for a new observation to produce an increment in that region once it has been assimilated, and (iii) the potential for a new observation to produce a broadly distributed analysis increment due to long-distance correlations that may be real or spurious. Therefore, the sensitivity defined by the signal variance is informed by the dynamical sensitivity but also constrained to regions of large analysis error variance and covariance where a new observation can produce a significant analysis increment both locally and remotely.
Comparing the signal variance to TESV (Fig. 2) and ADSSV (Fig. 3) guidance, a major disagreement appears: remote downstream targets in the subtropics northeast of the TC. Significant local maxima in signal variance appear in all four cases, collocated with regions of maximum analysis error variance. These downstream targets appear as only faint sensitivity in the ADSSV for Nuri and Jangmi, and do not appear in any of the TESV guidance plots. The existence of similar downstream targets in the ADSSV guidance suggests that a purely physical mechanism exists to allow a downstream perturbation to affect the tropical cyclone, while the absence of such targets in TESV guidance suggests this physical mechanism has a low growth rate as defined by the singular vector total-energy metric. While other possible sources for the disagreement about downstream targets are available because of inconsistencies between the calculation of the ETKF signal variance and the other guidance products (including differences arising from the linearity and reduced moisture physics of the adjoint methods, possible differences in the evolution of the ECMWF ensemble mean forecast relative to a deterministic NOGAPS forecast over the same trajectory period, and the inclusion of stochastic physics in the ECMWF used to generate a wider ensemble spread), the correlation of these downstream targets with regions of large estimated analysis error variance makes the inclusion of analysis uncertainty in the ETKF a prime suspect.
This raises the question of how a downstream perturbation can affect the TC. It is known that an initial perturbation can propagate both upstream and downstream in baroclinic flows (Simmons and Hoskins 1979), and features downstream of a TC such as a jet can have strong influences on TC intensity and/or the possibility of extratropical transition (McTaggart-Cowan et al. 2003). While there is always the possibility that these remote, downstream targets are due to spurious correlations in the ensemble, physical mechanisms do appear to exist to potentially link a downstream perturbation with the error variance surrounding the TC upstream, and the covariance between a TC and a downstream wave trough feature can be significant (Majumdar et al. 2011b). Here, a test of this downstream sensitivity can be performed to determine what, if any, physical mechanism can link a downstream perturbation to the TC.
A perturbation is inserted into the middle troposphere in the Typhoon Nuri simulation (Fig. 8a). Like the Typhoon Jangmi experiment, a positive vorticity perturbation5 is inserted into the model, scaled by the meridional component of the ADSSV,6 which is positive for these downstream targets (not shown). The initial condition perturbation translates upstream as a Rossby wave (Fig. 8b); a positive vorticity perturbation therefore imposes a largely northerly and westerly perturbation flow on Nuri throughout the 48-h trajectory, pushing Nuri slightly eastward relative to the unperturbed simulation (Fig. 8d). It is very unlikely that this track displacement has anything to do with the ETKF signal variance, like it does with ADSSV or TESV guidance. As described in section 2c (above), before the ensemble fields are supplied to the ETKF signal variance routine, the axisymmetric component of the TC circulation is removed; this is explicitly done because it has been observed that when the full fields are used, the ETKF signal variance is almost entirely a function of the ensemble disagreement in the TC location at the initial and final times (Majumdar et al. 2011b). Therefore, the ETKF focuses less on the final-time position of the TC and more on changes to the surrounding environment.
To observe the perturbations to Nuri's environment, perturbation fields are computed with the vorticity of Nuri's TC vortex removed; this is done by zeroing out the vorticity and divergence in the 15° × 15° verification area before computing the perturbation vorticity and divergence. The dipole representing the track divergence of Nuri is eliminated, and the remaining perturbation vorticity/divergence is inverted to recover the changes to Nuri's environment. The streamfunction of this perturbation flow reveals enhanced anticyclonic circulation in the subtropical ridge to the east of Nuri (Fig. 8c), a feature that is maximized at ∼700 hPa (not shown). The nondivergent flow described by this perturbation streamfunction implies an environmental flow steering Nuri slightly more to the northeast, consistent with a rough calculation7 of the perturbation to Nuri's motion at this time.
This perturbation streamfunction maximum in the subtropical ridge east of Nuri is an expression of an accumulation of negative perturbation vorticity in this region. This is achieved through a southerly perturbation flow east of Nuri induced by the perturbation Rossby wave (Fig. 8b). Through conservation of absolute vorticity, this southerly flow creates more anticyclonic vorticity and increases the strength of the subtropical ridge. A similar result is observed for a perturbation introduced into the Sinlaku8 simulation (Fig. 9), consistent with results from Komaromi et al. (2011). The predominant feature that develops as a result of the downstream perturbation to the northeast of Sinlaku is also an enhanced anticyclonic circulation in the (sub)tropics east of the TC (Fig. 9b). Throughout a 96-h simulation covering both the 48-h of interest as well as the next 48 h, the perturbed simulations express a track that is displaced north and east of the unperturbed simulations, resulting in a difference of several hundred kilometers in the 96-h position of both TCs (Fig. 9c).
This is a more subtle change to the TC's environment than the formulation of the Rossby wave itself; since the majority of the effect of the initial condition perturbation exists along the subtropical/midlatitude potential vorticity waveguide, the majority of perturbation energy produced by the initial condition perturbation exists outside of the verification region. It is expected that this is the reason the ADSSV assigns very low sensitivity to these remote downstream targets, and why the growth rate of such a perturbation is not large enough for these targets to be expressed in TESV guidance for most cases. However, it should be noted that these downstream targets do occasionally appear in TESV guidance, especially for cases involving extratropical transition. In addition, when singular vectors are constrained by estimated model analysis uncertainty (“VAR-SVs”; Reynolds et al. 2007), these downstream targets become more likely.
4. Modified ADSSV approach
The sensitivity of the ADSSV response function to small track divergence poses a problem even if sensitivity of track divergence may contain desirable information, because the ADSSV intends to be explicitly interpreted as a sensitivity of TC steering at the verification time. A relatively simple modification is described here. Further details can be found in Hoover and Morgan (2010, 2011).
Recall that the perturbation to Nuri's near environment due to a downstream initial condition perturbation (Fig. 8c) was defined by first removing the component of the wind field induced by the vorticity and divergence within the 15° × 15° verification box, and then recovering the residual flow induced by vorticity/divergence features that lie outside of the box. It was the changes to this flow, rather than the full flow, which was evaluated in order to determine that the steering change to the northeast brought about by the downstream initial condition perturbation was the result of an enhanced anticyclonic flow in the subtropical ridge east of Nuri.
The calculation of this “environmental” flow from the full flow is defined by a series of linear operators. As a result, the sensitivity of this environmental flow can be defined in the adjoint model through the transpose of this series of operators. The environmental flow is defined by first defining the vorticity and divergence from the full wind field (F1), then applying a local projection operator (F2) to mask the vorticity and divergence in the verification box, and then performing an inversion of the remaining vorticity and divergence (F3) to recover the environmental flow:
where , , , and are the zonal flow, meridional flow, vorticity, and divergence, respectively, of the environmental fields. The adjoint model is initialized with the gradient of the response function with respect to the full-field variables of the verification state xf, which can be computed from the gradient with respect to the environmental fields by working backward through the transpose of each of these three operators (T3, T2, and T1):
In this way, response functions defined by the environmental flow can be converted into sensitivity gradients with respect to the full flow. Thus, for response functions defining the average environmental zonal and meridional wind in the verification box:
one can compute the sensitivity of this environmental steering with respect to initial condition perturbations as before. Since vorticity and divergence within the verification box is removed before calculating the response function, the dipole structure in the vorticity perturbation field that is the result of a TC track divergence (Fig. 4c) is effectively removed (provided the TC does not translate so far that it ends up outside of the verification box). This environmental steering flow may better approximate the actual flow steering the TC; it is in many respects similar to the “hurricane advective flow” wherein the potential vorticity (PV) of the TC is removed and the remaining PV is inverted in order to recover a balanced, environmental flow presumably steering the vortex (Wu and Emanuel 1995).
The ADSSV guidance for a positive vorticity perturbation inserted in the remote, downstream target region of Nuri much more closely matches the observed steering change when the modified ADSSV response functions are used (Fig. 10), with the ADSSV pointing to the northeast in the modified guidance, while the original guidance appears to completely miss the meridional component. It is interesting to note that the TC track divergence accumulated over the entire 48-h trajectory places Nuri slightly east of its unperturbed 48-h position (Fig. 8d); such a track divergence would allow Nuri's own symmetric circulation to contribute a northerly component to the original meridional steering response function, in effect cancelling out the environmental flow component that is actually steering Nuri in a more southerly direction. The same perturbation was inserted into the Typhoon Hagupit simulation, which was similar to Nuri in many respects though ADSSV guidance yielded no downstream targets. A similar (though weaker) response was observed, with enhanced anticyclonic flow in the subtropical ridge at the same location. However, Hagupit was a much faster-moving storm, and by verification time it had outpaced these changes to the subtropical ridge and experienced no significant track change as it moved toward landfall over the coast of China (not shown).
The ADSSV guidance with the environmental steering response functions (Fig. 11) can be compared against the original guidance (Fig. 3) as well as TESV (Fig. 2) and ETKF (Fig. 6) guidance. Major features in the original ADSSV metric are conserved using the modified metric (yielding a high correlation between the fields, see Table 2), with the exception of the development of downstream targets northeast of Sinlaku (Fig. 11c), and an increase in upstream sensitivity to the northwest of Jangmi (Fig. 11d). It is also important to recognize that while the magnitude of the ADSSV may be similar between the original and modified metrics, the direction of the ADSSV may vary greatly, as illustrated in Fig. 10. With the changes in Jangmi's ADSSV guidance, a precedent appears whereby recurving typhoons (Sinlaku and Jangmi) have stronger sensitivity to vorticity perturbations in the northwest quadrant of their near environment than do zonally moving storms (Nuri and Hagupit).
Recall that the high correlation between ADSSV and TESV targets was reason to suspect that either 1) both metrics are controlled by the same physical mechanism, or 2) one metric controls the other (see section 3b). In that case, it was argued that both ADSSV and TESV metrics are controlled by the same physical mechanism—the divergence of the TC track as a result of an initial condition perturbation. When examining the high correlation between the two ADSSV metrics, it is recognized that the track divergence has no impact on the modified ADSSV targets by design, so this is not the most likely scenario. However, it seems plausible that a perturbation that affects the steering of a TC at verification time has affected the steering at previous times, thus affecting the track as well. In this case, scenario 2 appears to be more likely.
Through the use of prescribed perturbations to model initial conditions informed by objective targeting guidance, some of the physical mechanisms that provide the underpinning for these guidance products have been investigated. Unlike previous intercomparison studies, the focus has been maintained on the dynamical processes most important for defining the various forms of objective targeting guidance, rather than a comprehensive quantitative analysis of the degree of agreement or disagreement between various products. These previous intercomparison studies provide detailed information on several aspects of targeting guidance upon which it would be wise to focus our attention (such as the strong agreement between TESV and ADSSV guidance and the insistence of ETKF guidance on downstream targets), while this study seeks to provide some necessary physical understanding of those phenomena. In addition, while previous studies have focused on possible physical mechanisms, which may explain the structure of these guidance products (e.g., initial singular vectors) and speculated on their impact at the verification time, this study has maintained the focus on the evolved perturbations in a simulation perturbed in regions of strong sensitivity, and the evolved perturbations have been used to inform our understanding of the importance of various physical processes (such as track divergence) on the metrics used to define TC sensitivity.
The correlation between TESV and ADSSV guidance can be understood from the perspective that the metrics used to define both products are most sensitive to the same physical process: an accumulated track divergence that results from an initial condition perturbation. Seemingly more than any other physical process, even a small track divergence plays havoc with the resulting perturbation wind field in a verification region defined by a tropical cyclone, because the wind field is so strongly determined by the precise location of the TC center. Perturbations southeast of Jangmi meant to modulate the meridional steering 48 h later had the principal effect of inducing a west/east track divergence by the verification time; this track divergence overwhelmingly defined the perturbation meridional flow in the verification box. While this process does not provide useful information about the steering of the TC at the final time (which the ADSSV explicitly seeks to define), these metrics, functioning primarily as indirect measures of TC track divergence, appear to still provide practical information since track error is a major concern in TC forecasting.
Downstream targets found ubiquitously by the ETKF northeast of the TC only occasionally appear as regions of weak ADSSV sensitivity, and do not appear in any of the TESV guidance for the cases considered here. Vorticity perturbations placed in this downstream location appear to have the principal effect of creating a Rossby wave that travels upstream. Meridional flow in the (sub)tropics induced by this perturbation modifies the strength of the subtropical ridge through accumulation of relative vorticity via conservation of absolute vorticity, which in turn modifies the steering flow of the TC by the verification time. In the Nuri perturbation experiment, the modification to the subtropical ridge by a downstream, positive vorticity perturbation strengthened the southwesterly flow over the TC at verification time, correlating with a change in steering toward the northeast. This physical mechanism requires a substantial initial condition perturbation in order to have any practical effect, meaning that this downstream region is typically defined by low (but nonzero) sensitivity in the ADSSV. Likewise, since the majority of the impact of the initial perturbation exists as a Rossby wave that follows the subtropical/midlatitude PV gradient waveguide, the majority of the perturbation's impact exists outside of the verification box, which renders its effective growth rate too small to be included in the leading singular vectors. The ETKF guidance downstream of the TC was found to typically be coincident with a local maximum in estimated analysis uncertainty.
The focus on TC track divergence presents a problem for ADSSV guidance because an estimate of the change in final-time steering is the explicit goal. A possible solution was presented whereby the response function is replaced with the average perturbation flow once the vorticity and divergence within the verification area is removed; this is similar to both a measure of the “hurricane advection flow” as well as making the ADSSV metric more like the ETKF metric, where the axisymmetric flow of the TC is removed from the dataset at initial and verification times prior to its use. This allows both the ADSSV and the ETKF to focus on the environment rather than on the location of the TC itself. When this change is applied to ADSSV guidance for Nuri, the ADSSV in the downstream target region point to the northeast agreeing with the perturbation analysis for a positive vorticity perturbation in that region.
It is necessary to achieve a strong understanding of the primary physical mechanisms underlying various forms of objective targeting guidance if we wish to be able to intelligently discuss their relative differences, strengths, and weaknesses with regard to specific forecasting challenges. Tropical cyclones have been found to contribute to large uncertainty downstream, especially in cases of extratropical transition (Anwender et al. 2008); the possibility exists that TCs are capable of generating the very downstream uncertainty that ETKF signal variance guidance gravitates toward.
The squaring of perturbation fields within the TESV total-energy norm necessarily means that a translation-induced dipole will not cancel out in the integration of the total-energy metric in the verification box; therefore, the TESVs cannot tell the difference between perturbation energy brought about by an intensification change, or by a small translation of the TC. It may turn out that these metrics are so sensitive to this physical mechanism that any track divergence, no matter how small, will take precedence over other potentially more relevant processes, such as impacts on TC intensity. During a rapid-intensification forecast, for example, the ADSSV or TESV guidance may completely overlook potential targets that would significantly impact the final-time intensity of the TC in favor of targets that yield an arbitrarily small impact on TC track, if the perturbation growth rate gained by an arbitrarily small track divergence is greater than that of a more practically relevant but less “energetic” (from the TESV total-energy norm's perspective) intensity change. So-called evolved singular vectors have been shown to be almost entirely defined by these translational dipoles (Kim and Jung 2009), and in an idealized, barotropic model, the first two of the leading three TESVs were found to be focused on track divergences in perpendicular directions, with vortex structure changes not being relevant until the third singular vector (Yamaguchi et al. 2011). It may be found that, in order to investigate the sensitivity of TC intensity more closely, one must radically alter the TESV or choose a more focused measure of sensitivity (e.g., Doyle et al. 2012). Strategies for focusing adaptive observing guidance on nontrack-related aspects of the forecast (e.g., intensity) remain a relatively unexplored area of research and demands further study.
Finally, it is worth noting that the target regions produced by each technique investigated here are commonly fairly broad, and extend beyond the limited range of aircraft observations. As recommended in a recent comprehensive review of the field of targeted observations (Majumdar et al. 2011a), the potential utility of targeting satellite observations such as radiances and atmospheric motion vectors, using correct interpretations of the techniques investigated in this paper, requires further exploration.
The authors thank Dr. Rolf Langland at the Naval Research Laboratory for his aid with running the NOGAPS model and its adjoint and Dr. Carolyn Reynolds for her guidance with the NOGAPS singular vectors used in this study. This work was supported by the Office of Naval Research under Grants N000141110609 and N000141010123.
Unlike the TESV product, the ADSSV product was not produced by the Naval Research Laboratory, and was instead produced locally for this study. The resolution chosen for the forward and adjoint models was based on computational constraints of the computer used to produce the ADSSV product, and represents a middle ground between the higher resolution of the operational forward model and lower resolution of the operational adjoint.
The horizontal confinement of the perturbation is done to ensure that only the region of interest southeast of Jangmi is perturbed, and the vertical confinement is done to focus on the region near 500 hPa where the TESV and ADSSV have been examined.
The projection of the initial perturbation kinetic energy field onto the TESV is 0.5905.
A simple calculation of the ratio of total kinetic energy input at the initial conditions to the 48-h perturbation kinetic energy in the verification region between 1000 and 250 hPa yields a “growth rate” of 2.17 for the positive vorticity perturbation experiment. While this figure is low in the context of TESVs, it should be noted that only the kinetic energy portion of the energy norm is used, and the perturbation introduced only partially projects onto the TESV. However, one should expect that a perturbation that projects onto the TESV should be, at the very least, a growing perturbation rather than a decaying one.
The horizontal confinement of the perturbation is done to ensure that only the downstream environment near the ETKF/ADSSV target is perturbed, and the vertical confinement is done to focus on the region near 500 hPa where the TESV and ADSSV have been examined.
Nuri's motion at 48 h is defined here as the distance in meters traveled by Nuri between 45 and 51 h, divided by 6 h. Nuri's position at 45 and 51 h is calculated from a subgrid-scale interpolation of the sea level pressure minimum, and these positions are transformed into zonal and meridional components of a motion vector using the Haversine formula for computing distances on a sphere. This calculation reveals a change in Nuri's motion of 0.7 m s−1 to the northeast in the perturbed simulation.