• Ancell, B., , and G. Hakim, 2007: Comparing adjoint- and ensemble-sensitivity analysis with application to observation targeting. Mon. Wea. Rev., 135, 41174134.

    • Search Google Scholar
    • Export Citation
  • Anderson, J., 2007: Exploring the need for localization in ensemble data assimilation using a hierarchical ensemble filter. Physica D, 230, 99111.

    • Search Google Scholar
    • Export Citation
  • Barnett, T., , and R. Preisendorfer, 1987: Origins and levels of monthly and seasonal forecast skill for United States surface air temperatures determined by canonical correlation analysis. Mon. Wea. Rev., 115, 18251850.

    • Search Google Scholar
    • Export Citation
  • Bishop, C., , B. Etherton, , and S. Majumdar, 2001: Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129, 420436.

    • Search Google Scholar
    • Export Citation
  • Bougeault, P., and Coauthors, 2010: The THORPEX Interactive Grand Global Ensemble. Bull. Amer. Meteor. Soc., 91, 10591072.

  • Davis, C., , and K. Emanuel, 1991: Potential vorticity diagnostics of cyclogenesis. Mon. Wea. Rev., 119, 19291953.

  • Ertel, H., 1942: Ein neuer hydrodynamischer wirbelsatz. Meteor. Z., 59, 271281.

  • Etherton, B., 2007: Preemptive forecasts using an ensemble Kalman filter. Mon. Wea. Rev., 135, 34843495.

  • Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99, 10 14310 162.

    • Search Google Scholar
    • Export Citation
  • Gombos, D., 2009: Ensemble regression: Using ensemble model output for atmospheric dynamics and prediction. Ph.D. thesis, Massachusetts Institute of Technology, 190 pp.

  • Gombos, D., , and J. Hansen, 2008: Potential vorticity regression and its relationship to dynamical piecewise inversion. Mon. Wea. Rev., 136, 26682682.

    • Search Google Scholar
    • Export Citation
  • Hakim, G., , and R. Torn, 2008: Ensemble synoptic analysis. Synoptic-Dynamic Meteorology and Weather Analysis and Forecasting: A Tribute to Fred Sanders, Meteor. Monogr., No. 55, Amer. Meteor. Soc., 147–162.

  • Hawblitzel, D., , F. Zhang, , Z. Meng, , and C. Davis, 2007: Probabilistic evaluation of the dynamics and predictability of the mesoscale convective vortex of 10–13 June 2003. Mon. Wea. Rev., 135, 15441563.

    • Search Google Scholar
    • Export Citation
  • Holton, J., 1992: An Introduction to Dynamic Meteorology. Academic Press, 511 pp.

  • Hoskins, B., , M. McIntyre, , and A. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877946.

    • Search Google Scholar
    • Export Citation
  • Jazwinski, A., 1970: Stochastic Processes and Filtering Theory. Academic Press, 276 pp.

  • Molteni, F., , R. Buizza, , T. Palmer, , and T. Petroliagis, 1996: The new ECMWF Ensemble Prediction System: Methodology and validation. Quart. J. Roy. Meteor. Soc., 122, 73119.

    • Search Google Scholar
    • Export Citation
  • Nakagawa, M., 2009: Outline of the High Resolution Global Model at the Japanese Meteorological Agency. RSMC Tokyo-Typhoon Center Tech. Rev. 11, 13 pp.

  • Nicholls, N., 1987: The use of canonical correlation to study teleconnections. Mon. Wea. Rev., 115, 393399.

  • Penland, C., 1989: Random forcing and forecasting using principal oscillation pattern analysis. Mon. Wea. Rev., 117, 21652185.

  • Penland, C., , and P. Sardeshmukh, 1995: The optimal growth of sea surface temperature anomalies. J. Climate, 8, 19992024.

  • Penland, C., , and L. Matrosova, 1998: Prediction of tropical Atlantic sea surface temperatures using linear inverse modeling. J. Climate, 11, 483496.

    • Search Google Scholar
    • Export Citation
  • Rabier, E., , H. Järvinen, , E. Klinker, , J. Mahfouf, , and A. Simmons, 2000: The ECMWF operational implementation of four-dimensional variational assimilation. I: Experimental results with simplified physics. Quart. J. Roy. Meteor. Soc., 126, 11431170.

    • Search Google Scholar
    • Export Citation
  • Sampson, C., , and A. Schrader, 2000: The Automated Tropical Cyclone Forecasting System (version 3.2). Bull. Amer. Meteor. Soc., 81, 12311240.

    • Search Google Scholar
    • Export Citation
  • Sippel, J., , and F. Zhang, 2008: A probabilistic analysis of the dynamics and predictability of tropical cyclogenesis. J. Atmos. Sci., 65, 34403459.

    • Search Google Scholar
    • Export Citation
  • Torn, R., , and G. Hakim, 2008: Ensemble-based sensitivity analysis. Mon. Wea. Rev., 136, 663677.

  • Velden, C., , and L. Leslie, 1991: The basic relationship between tropical cyclone intensity and the depth of the environmental steering layer in the Australian region. Wea. Forecasting, 6, 244253.

    • Search Google Scholar
    • Export Citation
  • Wilks, D., 2006: Statistical Methods in the Atmospheric Sciences. 2nd ed. Elsevier, 672 pp.

  • Zhang, F., 2005: Dynamics and structure of mesoscale error covariance of a winter cyclone estimated through short-range ensemble forecasts. Mon. Wea. Rev., 133, 28762893.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    The Nens = 50 JMA ensemble of forecast tracks of Supertyphoon Sepat, initialized at 1200 UTC 14 Aug 2007. Gray lines depict the forecast tracks of individual ensemble members out to 114 h. The black line traces the ensemble mean of the forecast track position. The black dots represent the actual best track of Sepat every 12 h. The dashed line shows the official Joint Typhoon Warning Center (JTWC) forecast (Sampson and Schrader 2000).

  • View in gallery

    Line contours depict the 6-h forecast ensemble mean 500-hPa Z (in m) at 1200 UTC 14 Aug 2007. Filled contours depict the 6-h forecast ensemble mean 1000-hPa PV at 1200 UTC 14 Aug 2007. The large box identifies the domain of Fig. 4 (and all subsequent map figures) and the small box identifies the domain of Fig. 3. Note, however, that Fig. 3 uses data at 300 hPa, not 500 hPa. For clarity, PV values are set to zero for the region outside of the large box. Black letters identify the locations of key synoptic features: S is the ensemble mean signature of Sepat, H1 is the ridge WSW of Sepat, H2 is the strong subtropical high NW of Sepat, L1 is an embedded low in the trough to the NW of Sepat, T is an easterly wave that rides the southeastern perimeter of H2, and L2 is an extratropical low pressure system to the NE of H2. Here and in subsequent figures with color shading, small absolute values are white to highlight the important signals and irregular thin black lines represent the coastline.

  • View in gallery

    An illustration of inferring geostrophy using ER. Filled contours depict a JMA 300-hPa Z perturbation (in m) and line contours depict the associated zonal wind anomalies (in m s−1).

  • View in gallery

    (a)–(c) The effective PPs in physical space (in K m2 kg−1 s−1), defined as the filtered (as described in section 5) and resolved [viz., Eq. (4)] leading three eigenvectors of the JMA 72-h forecast 1000-hPa PV initialized at 1200 UTC 14 Aug 2007. The magenta lines in (a) and (c) depict the orientations of the axes of the dipoles used to define the AT and CT positions, respectively.

  • View in gallery

    (a) Results of the ER of the effective PP in Fig. 4a using the 1200 UTC 14 Aug 2007 6-h forecast 500-hPa Z predictand ensemble and ny = 20 and np = 10 (viz., Fig. B1). Color shading shows ER anomaly patterns (in m). Labeled line contours depict ensemble mean 1200 UTC 14 Aug 2007 6-h forecast 500-hPa Z values every 3 dam. Additional contours are drawn at 583 and 586 dam for clarity. Numbered green boxes in this and subsequent figures are identical and serve to isolate specific patterns of interest. The yellow line identifies the location of the cross section described in Fig. 11. (b) Color shading depicts point correlations between the forecast AT position of the 1000-hPa PV signature of Sepat at 72 h and the 6-h 500-hPa Z ensemble forecast. Line contours show the 6-h 500-hPa Z ensemble forecasts corresponding to the AT+ ensemble members with zAT > 0.5 (magenta) and to the AT− ensemble members with zAT < −0.5 (black). The ensemble mean (in dam) is depicted by the gray line contours. Note that the 585-dam contour is clipped to the east of box 4 for illustrative clarity.

  • View in gallery

    (a) As in Fig. 5a, but the ER is applied to the effective PP in Fig. 4c. (b) As in Fig. 5b, but the color shading depicts point correlations between the forecast CT position of the 1000-hPa PV signature of Sepat at 72 h and the 6-h 500-hPa Z ensemble forecast. Line contours show the 6-h 500-hPa Z ensemble forecasts corresponding to the CT+ ensemble members with zCT > 0.5 (magenta) and to the CT− ensemble members with zCT < −0.5 (black).

  • View in gallery

    ER-estimated 500-hPa Z sensitivity at (a) 6-, (b) 24-, (c) 42-, and (d) 60-h lead times to the 72-h AT location. Contours depict the ER predictand perturbation plus the respective ensemble mean when the predictand ensemble is the 500-hPa Z at the indicated forecast lead time and the predictor ensemble is the 72-h forecast 1000-hPa PV, for the two separate cases when the predictor perturbation is the 72-h 1000-hPa PV perturbation depicted in Fig. 4a (magenta ATP+ contours) and the 72-h 1000-hPa PV perturbation in Fig. 4a multiplied by −1 (black ATP− contours). Contoured values are equal to those of the magenta line contours in Figs. 5a, 6a. Gray shading represents land.

  • View in gallery

    As in Fig. 7, but for the ER-estimated Z sensitivities to the 72-h CT location for the two separate cases when the predictor perturbation is the 72-h 1000-hPa PV perturbation depicted in Fig. 4c (magenta CTP+ contours) and the 72-h 1000-hPa PV perturbation in Fig. 4c multiplied by −1 (black CTP− contours).

  • View in gallery

    Results of performing ER using bootstrap resampled ensemble members to estimate the variability of the ER results for the (a) AT and (b) CT sensitivities. Color shading depicts the mean of 100 separately computed ER patterns each computed using 25 randomly selected ensemble members (cf. with Fig. 5a). Line contours depict the variance of these 100 ER patterns. Black dots depict grid points where the null hypothesis that the mean value of the ER signal is equal to zero cannot be rejected, as determined by a two-sided t test at the 10% significance level.

  • View in gallery

    Sepat AT and CT position persistence. The black (gray) lines show the correlations between the AT (CT) positions of the 72-h 1000-hPa PV maxima and the AT (CT) positions of the 500-hPa Z minima at earlier lead times. Note that the axes depicted in Figs. 4a,c are used to define the AT and CT anomalies for all lead times.

  • View in gallery

    (a) Cross section of the 6-h 500-hPa Z along the AT axis depicted by the yellow line in box 1 of Fig. 5a. The x axis indicates the distance (in km) from the ESE end point of the axis. Magenta (black) lines depict AT+ (AT−) members and the thick lines depict the respective means of these two ensemble member groupings. (b) Cross section of the 6-h 500-hPa Z along the CT axis depicted by the yellow line in box 1 of Fig. 6a. The x axis indicates the distance (in km) from the SSW end point of the axis. Magenta (black) lines depict CT+ (CT−) members and the thick lines depict the respective means of these two ensemble member groupings.

  • View in gallery

    Filled contours depict the point correlations of the 6-h 500-hPa Z at the point of the maximum ER signal in box 8 of Fig. 6a with the 72-h 1000-hPa PV at all points in the domain.

  • View in gallery

    Boxplots of ensemble ACCs as a function of predictand lead time for the optimal values of ny and np at each lead time (in parentheses, ordered as ny and then np). The plot presents the median ACC (denoted by the large dot) and the variability of the ACCs (illustrated by the extent of the boxplot) at each lead time when the ER is performed using the optimal number of PCs. See text for more details.

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Ensemble Statistics for Diagnosing Dynamics: Tropical Cyclone Track Forecast Sensitivities Revealed by Ensemble Regression

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  • 1 Atmospheric and Environmental Research, Lexington, Massachusetts
  • | 2 Naval Research Laboratory, Monterey, California
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Abstract

Ensemble regression (ER) is a simple linear inverse technique that uses correlations from ensemble model output to make inferences about dynamics, models, and forecasts. ER defines a multivariate regression operator in the principal component subspaces of ensemble forecasts and analyses of atmospheric fields. ER uses the ensemble members of a predictor and a predictand field as training samples to compute the ensemble anomaly (with respect to the ensemble mean of the predictand field) with which a dynamically relevant ensemble anomaly (with respect to the ensemble mean of the predictor field) is linearly related. Specifically, an ER operator defined by the Japan Meteorological Agency’s ensemble forecast 500-hPa geopotential height and 1000-hPa potential vorticity is used to show that Supertyphoon Sepat’s (2007) track strongly covaried with the position and strength of the antecedent steering subtropical high to its northeast and the trough to its northwest. The case study illustrates how ER can identify, in real time, the dynamical processes that are particularly relevant for operational forecasters to make specific forecasting decisions and can help researchers to infer physical relationships from multivariate statistical sensitivities.

Corresponding author address: Dr. Daniel Gombos, Atmospheric and Environmental Research, 131 Hartwell Avenue, Lexington, MA 02421. E-mail: dgombos@aer.com

Abstract

Ensemble regression (ER) is a simple linear inverse technique that uses correlations from ensemble model output to make inferences about dynamics, models, and forecasts. ER defines a multivariate regression operator in the principal component subspaces of ensemble forecasts and analyses of atmospheric fields. ER uses the ensemble members of a predictor and a predictand field as training samples to compute the ensemble anomaly (with respect to the ensemble mean of the predictand field) with which a dynamically relevant ensemble anomaly (with respect to the ensemble mean of the predictor field) is linearly related. Specifically, an ER operator defined by the Japan Meteorological Agency’s ensemble forecast 500-hPa geopotential height and 1000-hPa potential vorticity is used to show that Supertyphoon Sepat’s (2007) track strongly covaried with the position and strength of the antecedent steering subtropical high to its northeast and the trough to its northwest. The case study illustrates how ER can identify, in real time, the dynamical processes that are particularly relevant for operational forecasters to make specific forecasting decisions and can help researchers to infer physical relationships from multivariate statistical sensitivities.

Corresponding author address: Dr. Daniel Gombos, Atmospheric and Environmental Research, 131 Hartwell Avenue, Lexington, MA 02421. E-mail: dgombos@aer.com

1. Usefulness of ensembles for dynamics and sensitivities

Because ensemble forecasts are designed to sample the probability density functions of potentially realizable model states, it is natural and effective to use them to quantify weather forecast uncertainty. Unfortunately, however, uncertainty estimation is virtually the exclusive use of ensemble model output by the operational sector at this time. This article demonstrates that the utility of ensembles transcends simply measuring forecast spread by developing an ensemble regression [ER; Gombos and Hansen 2008 (GH08); Gombos 2009] technique to extract currently underutilized information supplied by ensembles and then using ER to study the sensitivity of ensemble Japan Meteorological Agency (JMA; Nakagawa 2009) tropical cyclone (TC) track forecasts to geopotential heights Z.

ER is one promising application of ensemble synoptic analysis (ESA; Hakim and Torn 2008)—the use of ensemble analysis and forecast covariances to make inferences about the atmosphere. By treating individual ensemble members as independent samples, ESA employs standard statistical techniques to detect sensitivities, infer dynamical couplings, and aid forecasters in identifying dynamical processes that are particularly relevant for specific weather predictions. Using correlations between individual ensemble data points, for example, pioneering ESA applications have unveiled otherwise obscured linkages between a midlatitude cyclone and the subtropical jet stream (Hakim and Torn 2008) and have defined optimal climatological (Torn and Hakim 2008) and adaptive (Ancell and Hakim 2007) observing sites.

ESA is not the first approach that enables dynamical inferences about the atmosphere–ocean system from statistical sensitivities; for example, field covariability techniques such as canonical correlation analysis (CCA; e.g., Barnett and Preisendorfer 1987) and linear inverse modeling (LIM; e.g., Penland 1989; Penland and Sardeshmukh 1995) have been used for decades by the climate community to study Southern Oscillation teleconnections (Nicholls 1987) and to forecast interannual tropical Atlantic sea surface temperatures (Penland and Matrosova 1998). However, it is inappropriate to use the stationary climatic time series statistics employed by CCA and LIM for modeling the “errors of the day.” ESA, along with the related ensemble-based probabilistic analysis (PA) techniques used in Zhang (2005), Hawblitzel et al. (2007), and Sippel and Zhang (2008), is the first technique that facilitates dynamical inferences about flow-dependent mechanisms from statistical sensitivities. By using ensemble statistics derived from state estimation techniques with evolving covariances [including the ensemble Kalman filter (e.g., Evensen 1994), extended Kalman filter (e.g., Jazwinski 1970), and 4D variational assimilation (e.g., Rabier et al. 2000)] to model specific synoptic dynamics (and not just to quantify forecast uncertainty), ESA and PA have shown how to extend climatologists’ field covariability techniques to issues of concern to synopticians.

The technique used here, ER, determines multivariate linear relationships between forecast and/or analysis fields through the use of an ensemble-derived covariance-based mapping matrix that propagates a perturbation in one field (the predictor) into a perturbation in the second field (the predictand). In the current study, the relationship between predictand 500-hPa heights Z and subsequent predictor 1000-hPa potential vorticity (PV) is examined. ER can be considered a multivariate extension of the point-correlation ESA and PA sensitivity techniques. Whereas point correlations compute field sensitivities by independently and iteratively computing ensemble correlations between individual elements of the state vector, ER computes field sensitivities by computing multivariate regression operators between portions of the (or of the entire) state vector, which may be valid at different times. The use of a multidimensional operator enables inferences about how entire fields jointly relate, rather than just how scalars individually relate.

Previously, GH08 verified the validity of using ensemble covariances to form a regression operator by comparing the statistically predicted state using ER to the physically determined state using a dynamical model. By defining an operator with the ensemble covariances of PV, Z, and potential temperature, GH08 showed that, if the same effective perturbation is applied to the ER (or inverted, in the case of the dynamical model), piecewise PV ER and Davis and Emanuel (1991) dynamical piecewise PV inversion yield a nearly identical Z perturbation.

Under balanced, adiabatic, and inviscid conditions, atmospheric inversion theory (e.g., Ertel 1942; Hoskins et al. 1985; Davis and Emanuel 1991) asserts that PV and potential temperature are the necessary and sufficient set of predictors to fully predict collocated Z anomalies (GH08). However, most ERs involve imperfect predictor–predictand relationships and/or nonlinearities that potentially invalidate the use of covariance-defined operators. The current study extends the work of GH08 by analyzing the utility of ER when the predictor–predictand relationship is imperfect. More specifically, via the ER of antecedent midtropospheric Z with a PV perturbation associated with a TC, this article analyzes the utility of ER as a method of dynamical inference, as a means for forecasters to identify relevant forecast-specific dynamical processes, as a way to identify elements of the state on which a TC’s track is highly dependent, and as a tool for researchers to understand TC track model dynamics.

The remainder of this article is organized as follows: section 2 describes the synoptic setting and the data used. Section 3 previews the ER dynamical analysis technique with a simple example pertaining to the familiar geostrophic wind relationship. Section 4 more formally defines the ER mathematics and notation. Section 5 motivates the experimental approach and details the application of this approach to the Sepat case study. The results are then presented in section 6 and analyzed in section 7, which is divided into subsections that discuss the sensitivity of Sepat’s track to the key antecedent synoptic features identified in section 2. Section 8 provides conclusions and ideas for future work. The ER regularization algebra required to reduce the dimensionality of the ER is detailed in appendix A. Appendix B presents an ER error analysis using leave-one-out cross validation (LOOCV; e.g., Wilks 2006) and examines the impact of the domain chosen for the analysis by comparing the results of section 6 to results for an inner domain with roughly half the number of grid points.

2. Data and synoptic overview

The case studies analyzed in this article make use of JMA ensemble data retrieved from The Observing System Research and Predictability Experiment (THORPEX) Interactive Global Grand Ensemble (TIGGE) data archive (Bougeault et al. 2010). JMA generates its initial time ensemble perturbations using the singular vector method (e.g., Molteni et al. 1996). This implies that the initial ensemble members are optimized to sample the most dynamically sensitive regions of the analysis distribution, making them ideal candidates for this dynamical sensitivity study. The ensemble data, which were originally on a 1° latitude by 1° longitude grid, have been interpolated to a ⅓° latitude by ⅓° longitude grid.

The synoptic-scale event on which this article focuses is Supertyphoon Sepat, the TC that devastated parts of the Philippines, Taiwan, and China with sustained winds over 260 km h−1 (Sampson and Schrader 2000) in August 2007. The primary case study examines the sensitivity of the JMA track forecast at 72 h (approximate time of ensemble mean landfall in Taiwan) to the 500-hPa Z forecast conditions at 6 h (about 66 h before landfall). Figure 1 depicts the highly dispersed 72-h JMA ensemble track forecasts of Sepat, initialized at 1200 UTC 14 August 2007 and based on tracking the 1000-hPa PV signature of the storm. Note that the surface pressure for some areas in the domain is above 1000 hPa so there may be minor extrapolation errors in these areas (see Fig. 1 caption for more details).

Fig. 1.
Fig. 1.

The Nens = 50 JMA ensemble of forecast tracks of Supertyphoon Sepat, initialized at 1200 UTC 14 Aug 2007. Gray lines depict the forecast tracks of individual ensemble members out to 114 h. The black line traces the ensemble mean of the forecast track position. The black dots represent the actual best track of Sepat every 12 h. The dashed line shows the official Joint Typhoon Warning Center (JTWC) forecast (Sampson and Schrader 2000).

Citation: Monthly Weather Review 140, 8; 10.1175/MWR-D-11-00002.1

The 6-h forecast ensemble mean 500-hPa Z field (line contours) and the 6-h forecast ensemble mean 1000-hPa PV field (filled contours) are shown in Fig. 2. The sensitivity of the 72-h forecast position of Sepat to the position and strength of several synoptic features (labeled in black letters in Fig. 2) is studied in sections 57. The low pressure center (in the small box labeled L2 in Fig. 2) is used to introduce the ER inference in terms of the geostrophic relationship in section 3. Note, however, that the geostrophy example uses data at 300 hPa, not 500 hPa, but Fig. 2 still serves to identify the gross features of the fields at 300 hPa.

Fig. 2.
Fig. 2.

Line contours depict the 6-h forecast ensemble mean 500-hPa Z (in m) at 1200 UTC 14 Aug 2007. Filled contours depict the 6-h forecast ensemble mean 1000-hPa PV at 1200 UTC 14 Aug 2007. The large box identifies the domain of Fig. 4 (and all subsequent map figures) and the small box identifies the domain of Fig. 3. Note, however, that Fig. 3 uses data at 300 hPa, not 500 hPa. For clarity, PV values are set to zero for the region outside of the large box. Black letters identify the locations of key synoptic features: S is the ensemble mean signature of Sepat, H1 is the ridge WSW of Sepat, H2 is the strong subtropical high NW of Sepat, L1 is an embedded low in the trough to the NW of Sepat, T is an easterly wave that rides the southeastern perimeter of H2, and L2 is an extratropical low pressure system to the NE of H2. Here and in subsequent figures with color shading, small absolute values are white to highlight the important signals and irregular thin black lines represent the coastline.

Citation: Monthly Weather Review 140, 8; 10.1175/MWR-D-11-00002.1

3. ER inference 101: Geostrophy via anomaly patterns

The method employed here to analyze TC track sensitivities to geopotential heights in section 6 is to analyze predictor and predictand ER anomaly patterns. Before developing the complete mathematical machinery of ER (in section 4) and applying it to Sepat (in sections 57), this section outlines the use of ER anomaly patterns to infer physical relationships using the well-known example of geostrophy. Mathematical details will follow.

The geostrophic zonal wind relationship in isobaric coordinates is given by
e1
(e.g., Holton 1992), where ug is the geostrophic zonal wind, g is the gravitational constant, f is the Coriolis parameter, and y denotes the meridional distance; under geostrophic conditions of an approximately equal Coriolis and pressure gradient force, a westerly wind blows at the location of a negative Z gradient.

Here, it is shown via ER anomaly patterns that, without any prior knowledge of this dynamical relationship, geostrophy can potentially be inferred using only ensemble model output statistics. The first step is to define the predictor and the predictand. Here, the predictor is the 300-hPa geopotential height field Z300, and the predictand is the zonal wind field u300, also at 300 hPa. Both Z300 and u300 are 6-h ensemble forecasts from JMA initialized at 1200 UTC 14 August 2007. Note that the date and time of the JMA ensemble data are the same as those used in sections 57, although the domain used in this section (small box in Fig. 2, except at 300 hPa, not 500 hPa) is to the northeast of the domain used in sections 5 and 6. The overall synoptic situation is dominated by a low pressure center near the center of the domain and, as described in section 2, the strong subtropical high to the southwest of the domain at H2 of Fig. 2.

The second step is to determine the ER operator, which predicts u300 given Z300 using a least squares mapping matrix. Here, and in what follows, all of the Z300 values for each ensemble member are concatenated into one long vector and all of the u300 values for each ensemble member are concatenated into a separate vector. The ensemble mean of each vector is removed so that there is no constant term in the linear mapping; the ER matrix operator maps Z300 ensemble anomalies to u300 ensemble anomalies. Furthermore, since the Z300 and u300 are long vectors and the number of ensemble members Nens is only 50, the dimensionality of the least squares problem is reduced by converting the ensemble data into the space of the empirical orthogonal function (EOF) coefficients [i.e., principal components (PCs)]. This requires calculating the eigenvectors (i.e., EOFs) of the ensemble covariance matrices of the Z300 and u300 anomalies, and determining the appropriate number of EOFs to retain for the ER analysis. In this example, 10 PCs are retained each for Z300 and u300.

The third step is to prescribe a user-chosen ensemble perturbation for the predictor Z300. Here, this perturbation is chosen to be a north–south dipole that yields a Z300 center located south of the ensemble mean Z300 maximum. This perturbation is depicted by filled contours in Fig. 3. For the purposes of this ER, the anomaly pattern is dynamically representative (Gombos 2009) of a negative meridional gradient of Z300.

Fig. 3.
Fig. 3.

An illustration of inferring geostrophy using ER. Filled contours depict a JMA 300-hPa Z perturbation (in m) and line contours depict the associated zonal wind anomalies (in m s−1).

Citation: Monthly Weather Review 140, 8; 10.1175/MWR-D-11-00002.1

The fourth step is to apply the ER mapping operator to the Z300 perturbation to obtain the corresponding u300 perturbation. The resulting u300 perturbation is the most probable given a negative meridional Z300 gradient and the flow-dependent dynamics implied by the ensemble. Note that the ER is not a one-to-one mapping: all Z300 perturbations that make the same projection onto the 10 leading EOFs of the ensemble covariance matrix will yield the same u300 perturbation.

The last step is to interpret the predicted u300 perturbation given the prescribed perturbation of a negative meridional gradient of Z300. Here, determined purely from ensemble statistics, the line contours of Fig. 3 depict zonal westerly u300 anomalies whose maximum value is collocated with the strongest Z300 gradient. With prior knowledge of geostrophy, Fig. 3 simply illustrates that the ensemble statistics corroborate the known physical relationship given by Eq. (1). More interestingly, without this prior knowledge, the relationship depicted by Fig. 3 statistically suggests the relationship given by Eq. (1); this suggestion can later be tested and potentially verified using appropriate physical approaches. Although it is strictly inappropriate to conclude from this statistical evidence that the state of the zonal winds is dynamically linked to the Z300 gradient, ER results are supportive of this claim and can provide a dynamically based first guess to infer new relationships and to quantify linkages in particular synoptic cases.

4. ER

Now that the goals and mechanics of ER have been outlined in section 3, the notation and mathematical machinery of ER are detailed in this section.

a. ER notation

Let x be a column vector representing an arbitrary model state, be an initial or forecast model state for one ensemble member, and be the ensemble average model state. Lowercase letters with a prime indicate a perturbation with respect to the ensemble average. In particular, and serve to define and x′ as perturbations for an ensemble member and for an arbitrary model state, respectively. The matrix formed by concatenating the column vectors is denoted . Where subscripts are necessary, the elements of x are indexed by k = 1, … , K and the ensemble members by n = 1, … , Nens, making a K × Nens matrix.

Let p and y define transformations of the model state x. In general, these transformations may subset x and/or calculate diagnostic quantities from x at the initial time and/or at one or more forecast times. Given the ensemble members and , and the ensemble averaged quantities and , then , , p′, y′, , and are defined in the same way as the x variables. The symbols and refer to specially chosen values estimated by ER. Where subscripts are necessary, the elements of p are indexed by i = 1, … , I and the elements of y are indexed by j = 1, … , J, making an I × Nens matrix and a J × Nens matrix. In what follows, for example, the elements of p equal the gridpoint values of 1000-hPa PV and the elements of y equal the gridpoint values of 500-hPa Z in a specific domain.

b. ER definition

Let be a linear operator that maps the ensemble anomalies of p into ensemble anomalies of y such that
e2
(Hakim and Torn 2008; GH08; Gombos 2009). The operator is a statistical approximate transition matrix that maps a state in the span of ensemble to a state in the span of ensemble . Assuming a sufficiently large ensemble and a linear relationship between the predictor ensemble and the predictand ensemble , given any perturbation p′, is a Green’s function that approximates the most highly statistically associated ensemble anomaly of y, , via
e3
Note that effectively projects arbitrary p′ states onto the span of the ensemble . That is, although p′ is the “prescribed” predictor perturbation (PP), the estimated anomaly is in fact statistically associated with
e4
the projection of p′ onto the subspace spanned by ; is the “effective” predictor perturbation (GH08) that is resolved by linear combinations of the columns of in Eq. (3) (GH08).1

ER uses the ensemble members of the fields p and y as training samples to compute , the anomaly (with respect to the ensemble mean of y) to which (the effective PP with respect to the ensemble mean ) is linearly related. That is, the Green’s function predicts the most probable state of field given the state of field that is resolved by the ensemble.

In realistic applications, I and J are typically much greater than Nens, making and rank-deficient and not strictly invertible. Moreover, multicollinearities due to geographic proximity are likely to render ill-conditioned. To alleviate problems relating to multicollinearity, inflated regression parameter variances, and computational tractability, the ER machinery is regularized by projecting and onto their leading np and ny PCs, respectively (Gombos and Hansen 2008; Gombos 2009; see appendix A for details).

5. Sepat track ER experimental motivation and design

The method of ER inference via anomaly patterns described in section 3 and the ER machinery outlined in section 4 are used to study the dynamics and track sensitivities of Sepat in section 6. Section 5 describes the predictors, predictands, and prescribed predictor perturbations chosen for this analysis, as well as a detailed motivation for the experimental design. All data used are from JMA, as described in section 2.

Because TCs are steered, to first order, by the large-scale flow at midtropospheric levels (e.g., Velden and Leslie 1991, their Table 1), the focus here is on Sepat’s track sensitivities to antecedent midtropospheric Z. More specifically, the following ER applications use the 72-h forecast along-track (AT) and cross-track (CT) positions, defined below, of the 1000-hPa PV field as the predictor and the 6-h forecast 500-hPa Z field as the predictand. Note that the predictor is the 1000-hPa PV at the latter time because the goal is to predict the pattern of antecedent 500-hPa Z associated with a specified PV perturbation. Also note that the 6-h Z forecasts are used rather than the analyses because forecast guidance is typically not available to a forecaster for about 6 h, so 6-h forecasts are the most useful for estimating current sensitivities. Despite the likelihood that additional inferences can be made by using additional state elements to define the ER operator, the choice of using only 1000-hPa PV and 500-hPa Z is made because 1) PV inversion theory (e.g., Ertel 1942; Hoskins et al. 1985; Davis and Emanuel 1991) states that PV and Z are dynamically linked, increasing the likelihood of the strong ensemble covariances required for ER, and 2) choosing two specific fields simplifies the presentation and analysis of ER results. It is crucial to note, however, that although only two variables are directly involved in the ER, all variables correlated to the 1000-hPa PV are effectively involved (GH08).2 Note that only the relevant portions of the fields (i.e., the portions plotted in Figs. 5a, 6a) are used in this analysis, so that I = J = 22 444.

Considering the approximately east-southeast (ESE)–west-northwest (WNW) orientation of the track forecasts in Fig. 1, the dynamically representative prescribed PP used to study the sensitivity of the AT (CT) position is a WNW–ESE [north-northeast (NNE)–south-southwest (SSW)]-oriented dipole of 1000-hPa PV centered on the ensemble mean location of the 72-h forecast 1000-hPa PV landfall location. A WNW–ESE-oriented dipole with the positive (negative) anomaly lobe positioned WNW (ESE) of the ensemble mean position dynamically represents a PV center location at 72 h WNW of the ensemble mean location (or, equivalently, farther AT than the ensemble mean); likewise, a NNE–SSW-oriented dipole with the positive (negative) anomaly lobe positioned NNE (SSW) of the ensemble mean position dynamically represents a PV center location at 72 h NNE of the ensemble mean location (or, equivalently, farther CT than the ensemble mean). Applying ER to these prescribed PPs will yield as resulting predictand perturbations the most likely configuration of the antecedent 500-hPa Z field given that the subsequent 1000-hPa PV location at 72 h is AT and CT of the ensemble average.

These dynamically representative prescribed PPs can be prescribed in an ad hoc manner or constructed from the PV ensemble in any number of ways. Here, an eigenvector approach is used to ensure the independence of and to maximize the resolvability of the prescribed PPs. The chosen method is to 1) filter out other potentially obfuscating dynamical features by zeroing out all elements of the predictor ensemble outside of a 5° radius of the TC (note that the predictor ensemble used to form is not filtered in this way), 2) find the eigenvectors of this filtered ensemble, 3) use Eq. (4) to resolve these eigenvectors using the predictor ensemble , and then 4) project the resulting effective PPs into PC space using Eq. (A3).

It certainly need not be the case, but it is expected that the leading three eigenvectors of the filtered predictor ensemble are related to the storm position and amplitude variability. This is in fact the case here, as the first and third eigenvectors appear to express the variability of the AT and CT positions, respectively, and the second eigenvector appears to be related to the intensity variability of the 72-h landfall forecast 1000-hPa PV ensemble. After being projected into the subspace spanned by the predictor ensemble as described above, the first and third eigenvectors (effective PPs) take the form of the dipoles in Figs. 4a,c, respectively, and are consistent with the desired dynamically representative prescribed PPs described above. That is, for the purposes of an eigenanalysis, these dipoles respectively imply that two of the three most variable directions in the filtered ensemble state space correspond with the uncertainty in the WNW–ESE and NNE–SSW positions of the TC. But, for the purposes of ER, the dipoles simply respectively represent effective PP TCs located WNW and NNE of the ensemble mean location at 72 h (denoted by the black dots in Fig. 4). The magenta lines in Figs. 4a,c depict the respective orientations of the axes of these dipoles. Herein, the AT position is defined as the position along the axis in Fig. 4a, where points WNW (ESE) of the ensemble mean along the axis are positive (negative) anomalies. Likewise, the CT position is defined as the position along the axis in Fig. 4c, where points NNE (SSW) of the ensemble mean along the axis are positive (negative) anomalies.

Fig. 4.
Fig. 4.

(a)–(c) The effective PPs in physical space (in K m2 kg−1 s−1), defined as the filtered (as described in section 5) and resolved [viz., Eq. (4)] leading three eigenvectors of the JMA 72-h forecast 1000-hPa PV initialized at 1200 UTC 14 Aug 2007. The magenta lines in (a) and (c) depict the orientations of the axes of the dipoles used to define the AT and CT positions, respectively.

Citation: Monthly Weather Review 140, 8; 10.1175/MWR-D-11-00002.1

Even though all elements of the predictor ensemble outside of a 5° radius of the TC are zeroed out when constructing the prescribed PPs, the effective PPs in Fig. 4 have considerable perturbations far from the TC. For example, although the prescribed PP in Fig. 4c consists only of Sepat’s CT dipole (i.e., only the strong NNE–SSW dipole in Fig. 4c), the ensemble is not able to perfectly resolve this prescribed PP only and therefore the ER actually involves an effective PP equal to the least squares approximation of the prescribed PP (i.e., all of Fig. 4c). This effective PP includes several other considerable unprescribed perturbations, including the strong signals to the east of the dipole at approximately 20°N, 144°E. There exist no ensemble linear combinations that construct the CT dipole independently of the other features of Fig. 4c, so these other unprescribed features in the effective PPs affect the resulting ER pattern as well, as will be explained in section 7e.

Although this article focuses on TC track sensitivities, it is worth commenting on the second eigenvector (effective PP) that appears to be related to the TC intensity amplitude variability. The resolved eigenvector effective PP, which is depicted in Fig. 4b, shows a positively signed monopole and is therefore dynamically representative of a TC whose landfall position approximately equals that of the ensemble mean at 72 h and whose intensity amplitude is greater than average.

The magnitudes of these effective PPs are also multiplied by factors (that approximately equal the square roots of the respective eigenvalues) to ensure that, when added to the ensemble mean predictor ensemble, they 1) actually represent their intended dynamically representative prescribed PP [i.e., adding the AT dipole (Fig. 4a) to the ensemble mean actually yields a TC located WNW of the average], 2) have approximately equal magnitudes to each other, and 3) have peak TC magnitudes comparable to the peak TC magnitudes of the ensemble. Here, the effective PPs are scaled so that the resulting TCs have peak magnitudes of approximately 0.65 PVU, where 1 PVU = 10−6 K m2 kg−1 s−1, which is about one standard deviation greater than the ensemble mean. The first (third) scaled effective PPs represent TCs approximately 150 (200) km west (north) of the ensemble mean. It is crucial to note, however, that the sensitivity pattern obtained when operating on the effective PP with the ER operator is independent of the magnitude of the effective PP. If one is simply interested in sensitivity patterns, the scaling of the effective PP is unnecessary.

ER sampling error is addressed by bootstrap resampling (e.g., Wilks 2006) the ensemble (e.g., Anderson 2007) to estimate the sensitivity of the ER patterns to the choice of ensemble members used to define the ER operator. Ideally, this bootstrapping procedure would sample ensemble members from the infinitely large ensemble probability density function that fully spans the model dynamics subspace. However, this grand ensemble does not exist, and so here the bootstrapping technique is approximated using the relatively small Nens = 50 members used throughout the rest of the article to identify which ER signals are prone to sampling errors.

Bootstrap resampling ER is performed using the same predictor ensemble, predictand ensemble, and prescribed PPs already described in this section and applied in section 6. To assess the sampling errors, the entire ER is independently performed 100 times, each time using a randomly sampled (with replacement) 25-member ensemble from the full Nens = 50 member ensemble. As will be explained in section 6 and analyzed in section 7, areas particularly prone to exhibiting sampling errors are identified as those where the variance of the 100 ER patterns is high, which implies that the ER signals are sensitive to the particular ensemble members chosen to form the ER operator; ER features in these regions may be spurious or too unreliable to interpret dynamically, since they may be incorrectly representing model dynamics because of a nonuniform sampling of the full ensemble subspace. Moreover, to more formally identify ER patterns that are potentially affected by sampling error, a t test (e.g., Wilks 2006), whose null hypothesis is that the mean value of the ER signal is equal to zero, is performed on each of the grid points.

In summary, using this set of predictors, predictands, and effective PPs, the sensitivity of Sepat’s track to the 500-hPa Z is explored using a backward-in-time ER. An ER operator is defined using predictor (72-h ensemble forecast 1000-hPa PV PCs) and predictand (6-h ensemble forecast 500-hPa Z PCs), both initialized at 1200 UTC 14 August 2007. The prescribed PPs (or equivalently, the effective PPs, as explained in the footnote of section 4b)—the first and third 72-h forecast landfall PC PV perturbations—are applied to this ER operator to determine the resulting predictand perturbations , which are the most probable states of the antecedent prelandfall 500-hPa Z ensemble anomalies given that the eventual TC at landfall is located AT and CT, respectively, of the ensemble mean location.

6. Sepat track ER results

The method of using perturbation anomaly patterns to make dynamical inferences used in the simple example of section 3 is applied here to an Nens = 50 operational JMA ensemble to study Supertyphoon Sepat track sensitivities. The following discussion is intended to outline dynamical inferences and sensitivity assessments that can be made using ER. These ideas can potentially be used by researchers a posteriori to understand what may have contributed to changes in forecast tracks and can also be used a priori by operational forecasters to identify the most relevant features that influence the track, thereby determining the “forecast problem of the day.” The following analysis uses only forecast information, so all necessary data are available in real time.

Figures 5a, 6a show the results of the ER of the effective PP in Figs. 4a,c, respectively, using the 1200 UTC 14 August 2007 6-h forecast 500-hPa Z predictand ensemble, where ny = 20 and np = 10 (viz., appendix B). That is, Fig. 5a (Fig. 6a) depicts the most probable state of the 500-hPa Z 6-h forecast given that the 1000-hPa PV signature of the 72-h forecast TC is located anomalously AT (CT), or equivalently, WNW (NNE) of the ensemble mean location. Note that the ER pattern magnitudes in Fig. 5a (Fig. 6a) indicate approximate expected deviations from the 500-hPa Z ensemble mean given that the 1000-hPa PV effective PP is approximately 150 km WNW (200 km NNE) of the ensemble mean with a maximum magnitude of approximately 0.65 PVU. (See Figs. 5a, 6b captions for more details.)

Fig. 5.
Fig. 5.

(a) Results of the ER of the effective PP in Fig. 4a using the 1200 UTC 14 Aug 2007 6-h forecast 500-hPa Z predictand ensemble and ny = 20 and np = 10 (viz., Fig. B1). Color shading shows ER anomaly patterns (in m). Labeled line contours depict ensemble mean 1200 UTC 14 Aug 2007 6-h forecast 500-hPa Z values every 3 dam. Additional contours are drawn at 583 and 586 dam for clarity. Numbered green boxes in this and subsequent figures are identical and serve to isolate specific patterns of interest. The yellow line identifies the location of the cross section described in Fig. 11. (b) Color shading depicts point correlations between the forecast AT position of the 1000-hPa PV signature of Sepat at 72 h and the 6-h 500-hPa Z ensemble forecast. Line contours show the 6-h 500-hPa Z ensemble forecasts corresponding to the AT+ ensemble members with zAT > 0.5 (magenta) and to the AT− ensemble members with zAT < −0.5 (black). The ensemble mean (in dam) is depicted by the gray line contours. Note that the 585-dam contour is clipped to the east of box 4 for illustrative clarity.

Citation: Monthly Weather Review 140, 8; 10.1175/MWR-D-11-00002.1

Fig. 6.
Fig. 6.

(a) As in Fig. 5a, but the ER is applied to the effective PP in Fig. 4c. (b) As in Fig. 5b, but the color shading depicts point correlations between the forecast CT position of the 1000-hPa PV signature of Sepat at 72 h and the 6-h 500-hPa Z ensemble forecast. Line contours show the 6-h 500-hPa Z ensemble forecasts corresponding to the CT+ ensemble members with zCT > 0.5 (magenta) and to the CT− ensemble members with zCT < −0.5 (black).

Citation: Monthly Weather Review 140, 8; 10.1175/MWR-D-11-00002.1

Appendix B describes the selection of the number of PCs used in the ER and discusses some expected ER errors. The ER predictions discussed in this section are considered to be statistically significant according to the LOOCV (e.g., Wilks 2006) described in appendix B; the expected value of the anomaly correlation coefficient (ACC) of these ER perturbations with the true perturbations is approximately 0.47 when the 500-hPa Z predictand lead time is 6 h.

Whereas Figs. 5a, 6a depict multivariate ER track sensitivity patterns for the full effective PPs in Figs. 4a,c, respectively, Figs. 5b, 6b depict point-correlation sensitivity patterns of Sepat’s 72-h AT and CT positions, respectively, with the 6-h 500-hPa Z at every point in the domain. In the forthcoming analysis, the ER track sensitivity anomaly patterns in Figs. 5a, 6a are compared to these point-correlation patterns to both elucidate the meaning of the ER patterns and to illustrate the differences between point-correlation and multivariate sensitivities.

The meanings of Figs. 5, 6 are further reinforced by taking a closer look at the characteristics of the individual ensemble members that contribute to the ER and correlation patterns. Figure 5b (Fig. 6b) displays select color-coded spaghetti Z contours that are collocated with the strong ER signals in Fig. 5a (Fig. 6a) and the strong correlation signals in Fig. 5b (Fig. 6b). Letting zAT (zCT) be the z score (i.e., standardized anomaly; e.g., Wilks 2006) of the 72-h forecast AT (CT) location, the magenta contours in Fig. 5b (Fig. 6b) represent the 14 (12) ensemble members for which zAT > 0.5 (zCT > 0.5) [hereafter, AT+ (CT+) members] and the black contours in Fig. 5b (Fig. 6b) represent the 14 (12) ensemble members for which zAT < −0.5 (zCT < −0.5) [hereafter, AT− (CT−) members]. That is, the magenta AT+ (black AT−) ensemble members in Fig. 5b have a 72-h location that is WNW (ESE) of the ensemble mean and the magenta CT+ (black CT−) ensemble members in Fig. 6b have a 72-h location that is NNE (SSW) of the gray-contoured ensemble mean.

Figure 7 shows the ER-estimated time evolution of the 500-hPa Z for the case for which the PP is the AT-oriented 1000-hPa PV dipole in Fig. 4a (magenta contours; hereafter ATP+) and for which the PP is that same dipole multiplied by −1 (black contours; hereafter ATP−). Here, contours in Figs. 7a–d depict the full 500-hPa Z field (i.e., the ER predictand perturbation plus the respective ensemble mean) when the predictand ensemble is the 500-hPa Z at 6-, 24-, 42-, and 60-h lead times, respectively. For each, the predictor ensemble is the 72-h forecast 1000-hPa PV. Figure 8 is the CT counterpart to Fig. 7; for Fig. 8, the two PPs are the CT-oriented 72-h 1000-hPa PV perturbation depicted in Fig. 4c (magenta contours; hereafter CTP+) and that same dipole multiplied by −1 (black contours; hereafter CTP−). Note that linearity implies that ATP− = −ATP+ and CTP− = −CTP+, so Fig. 7 (Fig. 8) depicts the ensemble mean plus and minus the AT (CT) ER predictand perturbation.

Fig. 7.
Fig. 7.

ER-estimated 500-hPa Z sensitivity at (a) 6-, (b) 24-, (c) 42-, and (d) 60-h lead times to the 72-h AT location. Contours depict the ER predictand perturbation plus the respective ensemble mean when the predictand ensemble is the 500-hPa Z at the indicated forecast lead time and the predictor ensemble is the 72-h forecast 1000-hPa PV, for the two separate cases when the predictor perturbation is the 72-h 1000-hPa PV perturbation depicted in Fig. 4a (magenta ATP+ contours) and the 72-h 1000-hPa PV perturbation in Fig. 4a multiplied by −1 (black ATP− contours). Contoured values are equal to those of the magenta line contours in Figs. 5a, 6a. Gray shading represents land.

Citation: Monthly Weather Review 140, 8; 10.1175/MWR-D-11-00002.1

Fig. 8.
Fig. 8.

As in Fig. 7, but for the ER-estimated Z sensitivities to the 72-h CT location for the two separate cases when the predictor perturbation is the 72-h 1000-hPa PV perturbation depicted in Fig. 4c (magenta CTP+ contours) and the 72-h 1000-hPa PV perturbation in Fig. 4c multiplied by −1 (black CTP− contours).

Citation: Monthly Weather Review 140, 8; 10.1175/MWR-D-11-00002.1

Figure 9 displays the results of the ER bootstrapping experiment described in section 5. As expected, the mean ER patterns over the 100 independent ER experiments (color shading in Figs. 9a,b) are extremely similar to the respective ER patterns in Figs. 5a, 6a, which are computed using the same prescribed PPs and the full 50-member versions of the same predictor and predictand ensembles to define the ER operators. Black-dotted points represent locations where ER signals are likely negligible or significantly influenced by sampling error. (See Fig. 9 caption for more details.)

Fig. 9.
Fig. 9.

Results of performing ER using bootstrap resampled ensemble members to estimate the variability of the ER results for the (a) AT and (b) CT sensitivities. Color shading depicts the mean of 100 separately computed ER patterns each computed using 25 randomly selected ensemble members (cf. with Fig. 5a). Line contours depict the variance of these 100 ER patterns. Black dots depict grid points where the null hypothesis that the mean value of the ER signal is equal to zero cannot be rejected, as determined by a two-sided t test at the 10% significance level.

Citation: Monthly Weather Review 140, 8; 10.1175/MWR-D-11-00002.1

7. Sepat track ER analysis

A feature-based approach is taken to explain the ER patterns illustrated in Figs. 5a, 6a. Sections 7ae, respectively, describe the sensitivities of the 72-h AT and CT positions to the 6-h Sepat signature S, to the trough and embedded low L1, to the western ridge H1, to the subtropical high H2, and to the easterly wave T of the 6-h forecast 500-hPa Z. Section 7e is also used to illustrate the differences between multivariate ER sensitivities and point-correlation sensitivities.

Note that a graphical display of the results of the ER of the effective PP in Fig. 4b is omitted here for brevity. That ER pattern is dominated by persistence.

a. 72-h track sensitivities to the 6-h 500-hPa Sepat signature S

This subsection analyzes the sensitivities of the 72-h 1000-hPa PV AT and CT positions to the 6-h 500-hPa Z Sepat signature (labeled S in Fig. 2). Box 1 of Fig. 5a illustrates a straightforward statistical representation of dynamical persistence. The 581-dm contour of this box represents the ensemble mean position of the 500-hPa Z signature of Sepat. The negative anomaly to the WNW of the ensemble mean position illustrates that anomalously WNW Sepat tracks at 72 h are likely preceded by anomalously WNW 500-hPa TC signatures; ensemble members whose 6-h forecast midtropospheric Z signatures of Sepat are WNW of the average are likely to have associated surface PV features that remain WNW of the average. Similarly, box 1 of Fig. 6a is the CT-oriented counterpart to box 1 of Fig. 5a; box 1 of Fig. 6a indicates that a 6-h forecast ensemble member whose 500-hPa Z TC signature is located NNE of the average is likely to have associated surface PV features that remain NNE of the average.

Reinforcing this interpretation, the point-correlation signals of box 1 of Fig. 5b and, to a lesser extent, box 1 of Fig. 6b show similar negative anomaly patterns to the ER patterns depicted in box 1 of both Figs. 5a, 6a. The underlying statistics of the correlation plots are illustrated by the distribution of the spaghetti contours in box 1 of Fig. 5b (Fig. 6b), which shows that most magenta contours corresponding to AT+ (CT+) ensemble members are WNW (NNE) of the ensemble mean, whereas most black contours corresponding to AT− (CT−) ensemble members are ESE (SSW) of the mean. The ER signals, point-correlation patterns, and spaghetti contour distributions all imply that most ensemble TCs that start out farther AT (CT) at 6 h remain farther AT (CT) through 72 h.

Figure 10 illustrates the persistence of Sepat’s AT and CT positions as a function of lead time. The black (gray) lines show the correlations between the AT (CT) positions of the 72-h 1000-hPa PV maxima and the AT (CT) positions of the 500-hPa Z minima at earlier lead times. Note that the same axes depicted in Figs. 4a,c are used to define the AT and CT positions, respectively, for all lead times. Consistent with the ER signals in box 1 of both Figs. 5a, 6a, AT and CT autocorrelations are approximately 0.51 and 0.60, respectively, for the 6-h lead time and even higher for later lead times. Note that the autocorrelation between the early and late PV positions (rather than between the PV and 500-hPa Z positions) is even higher. Moreover, the persistence of the AT (CT) position can be seen in the AT (CT) ER time-evolution snapshots in Fig. 7 (Fig. 8); as the predictand lead time increases from 6 to 60 h, the magenta ATP+ (CTP+) 500-hPa TC signature remains approximately WNW (NNE) of the black ATP− (CTP−) 500-hPa TC signature. Note that, in general, the ER signals at later lead times imply dynamics similar to the 6-h patterns, but are considerably stronger (because of the decreased difference in lead time from 72 h) and more large scale (because of the considerably greater variance of the location and intensity of the more uncertain longer-lead forecasts) than their 6-h counterparts.

Fig. 10.
Fig. 10.

Sepat AT and CT position persistence. The black (gray) lines show the correlations between the AT (CT) positions of the 72-h 1000-hPa PV maxima and the AT (CT) positions of the 500-hPa Z minima at earlier lead times. Note that the axes depicted in Figs. 4a,c are used to define the AT and CT anomalies for all lead times.

Citation: Monthly Weather Review 140, 8; 10.1175/MWR-D-11-00002.1

While the negative signals in box 1 of Figs. 5a,b (Figs. 6a,b) are consistent with the spaghetti plot in Fig. 5b (Fig. 6b), the lack of a strong corresponding positive signal on the opposite side of the ensemble mean may seem to be inconsistent at first glance; a WNW (NNE)-shifted perturbation would imply both a greater than average 500-hPa Sepat signature WNW (NNE) of the ensemble mean and a strong corresponding positive dipole lobe ESE (SSW) of the mean. The lack of this strong positive signal can be explained by pointing out that the magnitudes of ER and point-correlation signals are functions of both position perturbations and intensity perturbations.

This is investigated in Fig. 11a (Fig. 11b), which depicts a cross section of the 6-h 500-hPa Z ensemble members interpolated at 50 evenly spaced points along the yellow-axis dipole drawn in box 1 of Fig. 5a (Fig. 6a; see Fig. 11 caption for more details). Figure 11a (Fig. 11b) shows that the AT+ (CT+) members have stronger 500-hPa TC signatures (i.e., lower heights) than do the AT− (CT−) members, particularly in the WNW (NNE) portion of the axis in Fig. 5a (Fig. 6a); 1000-hPa PV maxima that are anomalously WNW (NNE) at 72 h tend to be associated with 500-hPa lows that are anomalously WNW (NNE) and stronger at 6 h. As seen by the lower means for the AT+ (CT+) members at the WNW (NNE) points of this axis [i.e., distances from the ESE (SSW) end point greater than approximately 450 km (350 km) in Fig. 11a (Fig. 11b)], the difference in TC strength between the AT+ and AT− members (CT+ and CT− members) is more apparent at the WNW (NNE) points, where the AT+ (CT+) members tend to be located at 6 h (viz., Fig. 10). At the ESE (SSW) points of this axis [i.e., distances from the ESE (SSW) end point less than approximately 450 km (350 km) in Fig. 11a (Fig. 11b)], where weaker storms in the AT− (CT−) members tend to be located at 6 h, the difference in strength between the AT+ and AT− (CT+ and CT−) members is not as strong and both the ER signals and point correlations are correspondingly weak in these areas.

Fig. 11.
Fig. 11.

(a) Cross section of the 6-h 500-hPa Z along the AT axis depicted by the yellow line in box 1 of Fig. 5a. The x axis indicates the distance (in km) from the ESE end point of the axis. Magenta (black) lines depict AT+ (AT−) members and the thick lines depict the respective means of these two ensemble member groupings. (b) Cross section of the 6-h 500-hPa Z along the CT axis depicted by the yellow line in box 1 of Fig. 6a. The x axis indicates the distance (in km) from the SSW end point of the axis. Magenta (black) lines depict CT+ (CT−) members and the thick lines depict the respective means of these two ensemble member groupings.

Citation: Monthly Weather Review 140, 8; 10.1175/MWR-D-11-00002.1

Note, however, that despite this persistence in Sepat’s AT and CT positions, the ER signal in portions of box 1 of Figs. 9a,b is highly variable and statistically insignificant at the 10% level according to the t test described in section 5. In Fig. 9a, the variance peak in box 1 is collocated with the zero line of the AT dipole of the mean ER pattern (i.e., between the lobes of the ER AT dipole in box 1), indicating substantial uncertainty in the exact position of the 6-h forecast 500-hPa ER perturbation associated with a WNW 1000-hPa PV perturbation at 72 h. It is important to point out that the variance maximum in box 1 is collocated with the ER mean zero line, not with the ER mean minimum to its northwest. This is simply because small variations in the location of the ER perturbation minimum near the mean location of the ER minimum result in relatively small deviations from that mean, whereas (the small number of) perturbations that place the ER perturbation minimum closer to the location of the ER mean zero line yield very large deviations from zero, resulting in a high variance at the location of the zero line. This result illustrates that the exact locations of ER patterns are especially variable and sensitive to sampling error where the spatial gradient of the ER pattern is high.

b. 72-h track sensitivities to the 6-h 500-hPa trough and low L1

This subsection analyzes the sensitivities of the 72-h 1000-hPa PV AT and CT positions to the 6-h 500-hPa Z trough and its embedded low L1, paying particular attention to the signals in boxes 2, 3, and 4 of Figs. 5 and 6.

Box 2 relates Sepat’s 72-h forecast AT and CT positions to the position of the northeastern section of the trough and to the position of L1 (marked approximately by the ensemble mean 578-dm contour on the western side of box 2 in Figs. 5a, 6a). The negative (positive) ER signal in box 2 of Fig. 5a (Fig. 6a) indicates that an ensemble TC located anomalously AT (CT) at 72 h is likely to be associated with antecedent Z at 6 h that is anomalously low (high) in this region, such that L1 is east (west) of the average and the northeastern section of the trough is broader (narrower) and stronger (weaker) than average. This interpretation is consistent with the correlation patterns and spaghetti plots in box 2 of Fig. 5b (Fig. 6b), which depicts the magenta AT+ (CT+) contours clustered to the east (west) side of the ensemble mean and the black AT− (CT−) contours clustered to the west (east) side. However, judging by the quick dissipation of L1 as the model evolves (not shown) and the lack of a discernible related signal in Figs. 7b–d, 8b–d, L1 likely is merely correlated to a dynamical steering mechanism rather than being one itself.

Note that the CT ER pattern (Fig. 6a) also shows an elongated positive signal to the southwest of box 2. This signal likely implies that the 72-h CT position is sensitive to the position and breadth of the entire eastern side of the trough. The corresponding CT correlation signal in Fig. 6b (the central and southern parts of which are in white because of their low magnitudes) is similarly shaped but weaker.

Box 3 relates Sepat’s 72-h forecast AT and CT positions to the breadth and western extent of the trough at 6 h. As portrayed by the weak positive (moderate negative) ER signal in box 3 of Fig. 5a (Fig. 6a), anomalously AT (CT) 72-h ensemble members typically have antecedent 500-hPa troughs that extend less far (farther) west than average. The box-3 correlation patterns (Figs. 5b, 6b) are similarly positioned and more extensive than the respective box-3 ER signals (Figs. 5a, 6a). The interpretation that these signals reflect the 72-h AT (CT) sensitivity to the breadth and position of the western side of the trough is consistent with the spaghetti plot in box 3 of Fig. 5b (Fig. 6b), which depicts the magenta AT+ (CT+) contours clustered to the east (west) side of the ensemble mean and the black AT− (CT−) contours clustered to the west (east) side. Note that, although the relevant portions of box 3 pass the significance test at the 10% level, the ER pattern variance in box 3 (Fig. 9) is moderate, which weakens the statistical significance of this signal and its dynamical interpretation.

The positive ER signal in box 3 of Fig. 5a, together with the positive ER signal in the southeastern section of the trough, is likely indicative of a low starting to cut off from the trough—a prominent sensitivity signal in Fig. 7. Box 3 of Fig. 5a reflects that this cutting off occurs anomalously early for ensemble members whose TC is located anomalously AT at 72 h. Figure 7 shows the evolution of this cutoff low at later lead times. The cutting off occurs anomalously south (north) for the ATP+ (ATP−) realization (see Fig. 7b at 24 h) and then the ATP+ (ATP−) cutoff low shifts SSW more (less) quickly than average at later lead times (see Fig. 7c at 42 h). This is consistent with physical expectations, since the ATP+ low is less directly in Sepat’s path than is the ATP− low at 42 (Fig. 7c) and 60 h (Fig. 7d), so that the westerly winds of the ATP+ low would be less likely to impede Sepat’s WNW movement than the westerly winds of the ATP− low. The CT sensitivity evolution in Fig. 8 shows a similar, but considerably weaker, pattern to the AT sensitivity evolution.

Box 4 of Fig. 6 relates Sepat’s 72-h forecast CT position, only, to the southern and southeastern extent of the trough. Box 4 of Fig. 6a and, to a lesser extent, Fig. 6b depicts negative signals, suggesting that a deeper trough at box 4 at 6 h typically precedes an anomalously NNE TC track at 72 h. Consistent with the 582-dm 500-hPa Z spaghetti plots in box 4 of Fig. 6b, CT+ (CT−) members tend to have 6-h 500-hPa troughs that dip farther (less far) southward and southeastward than average. This is consistent with physical expectations, since the stronger (weaker) southwesterly winds at the southeastern section of the trough for CT+ (CT−) members would more (less) strongly steer Sepat to the NNE.

In summary, Sepat’s 72-h AT and CT locations are sensitive to the location of L1 and to the position and breadth of the northeastern, western, and southern sections of the 6-h trough. When the JMA model brings Sepat anomalously AT (CT) at 72 h, it also typically has an antecedent 6-h trough characterized by 1) an L1 and northeastern section of the trough that are positioned east (west) of the average (box 2 of Figs. 5 and 6) and 2) a western section of the trough positioned east (west) of the average (box 3 of Figs. 5 and 6). Also, anomalously CT locations at 72 h are typically preceded by anomalously strong south- and southeastward-extending troughs at 6 h (box 4 of Fig. 6). Particularly for the AT sensitivities, the low that cuts off from the trough (near box 3 of Fig. 5) seems most persistent and impactful at later lead times (Fig. 7).

c. 72-h track sensitivities to the 6-h 500-hPa ridge H1

This subsection analyzes the AT sensitivities, only, to the eastern extent of the 500-hPa ridge to the west of Sepat (H1 in Fig. 2). Box 5 of Figs. 5a,b shows a strong negative anomaly that extends from H1 toward the WSW of Sepat, suggesting that an ensemble member with a weaker ridge to the WSW of Sepat at 6 h is often a precursor to an anomalously WNW TC location at 72 h. Box 5 of Fig. 5b supports this notion, as the 585-dm contour (which has been clipped to the east of box 4 for illustrative clarity) of this ridge tends to delineate weaker and more western ridges to the southwest of the trough for AT+ members, whereas it tends to show the ridge farther east for AT− members.

Box 5 of Fig. 5 is consistent with physical expectations since a stronger and farther ENE-positioned H1 would be associated with stronger northwesterlies to the west of Sepat that would impede Sepat’s movement to the WNW. However, this signal does not strongly persist at later lead times, as instead the dominant signal in this region in Figs. 7b–d is associated with the southwestward movement of the cutoff low (see section 7b). Therefore, it is likely that the 6-h signal in box 5 only indirectly influences Sepat at 72 h by not blocking (blocking) the southward movement of the closed low for AT+ (AT−) members.

d. 72-h track sensitivities to the 6-h 500-hPa subtropical high H2

This subsection analyzes the sensitivities of the 72-h 1000-hPa PV AT and CT positions to the 6-h 500-hPa Z subtropical high (H2 in Fig. 2). Box 6 of Figs. 5a,b encloses the strong positive ER and point-correlation signals in AT that mark the sensitivity of the 72-h AT location to the strength and shape of the southeastern edge of H2. As seen by the 587-dm spaghetti contours in box 6 of Fig. 5b, AT+ (AT−) contours uniformly lie south (north) of the ensemble mean and typically do (do not) depict tongues of high pressure that jut southeastward from the high Z center at 6 h. This signal is one of the strongest and most persistent sensitivity features throughout the model integration. The ATP+ realization is characterized by 6-h 500-hPa heights that jut southeastward at 6 h (Fig. 7a), dip southward at 24 (Fig. 7b) and 42 h (Fig. 7c), and then jut to the southeast of Sepat at 60 h (Fig. 7d). Particularly at the later lead times, H2 is located considerably closer to Sepat for the ATP+ realization than for the ATP− realization (see Fig. 7d), causing the strong ESE flow from H2 to steer Sepat more strongly in the AT direction for the ATP+ realization than for the ATP− realization.

The western section of box 6 of CT Fig. 6a (Fig. 6b) encloses a moderately strong negative ER (point correlation) signal that marks the sensitivity of the 72-h CT location to the position of H2 to the north and northeast of Sepat. As seen by the 587-dm spaghetti contours in box 6 of Fig. 6b, CT+ (CT−) contours uniformly lie north (south) of the ensemble mean, indicating that Sepat is likely to be positioned anomalously NNE (SSW) at 72 h for ensemble members with an anomalously northern (southern)-positioned H2 in box 6 at 6 h. This signal is also very strong and persistent at later lead times; Fig. 8 shows that the CTP+ (CTP−) realization is characterized by 6-h 500-hPa heights that shift increasingly north (south) of the average to the north of Sepat with increasing lead times. Note that the relationship between the size and strength of the 500-hPa Sepat signature and the CT position may also contribute to this signal.

Box 7 of Fig. 5 relates Sepat’s 72-h forecast AT position, only, to the 6-h western extent of H2. The positive ER and correlation signals in Figs. 5a,b, respectively, indicate that an anomalously WNW PV perturbation at 72 h is likely preceded by an anomalously strong H2 at 6 h in this region. Again, this is consistent with the spaghetti contours of box 7 of Fig. 5b, where nearly every AT+ (AT−) contour is west (east) of the ensemble mean and farther from (closer to) the center of the high, indicating that AT+ members have uniformly higher heights than AT− members at any given location in this region.

In summary, Sepat’s 72-h AT and CT locations are sensitive to H2’s southeastern tongue and to its western extent. When the JMA model brings Sepat anomalously WNW at 72 h, it also typically has a 6-h H2 characterized by a southwestern-extended tongue (box 6 of Fig. 5) and an anomalously western position (box 7 of Fig. 5). When the JMA model brings Sepat anomalously NNE at 72 h, it also typically has an anomalously northern-positioned antecedent H2 (box 6 of Fig. 6).

e. Differences between ER and point-correlation sensitivities to the 6-h 500-hPa easterly wave T

In general, the ER and corresponding correlation signals in boxes 1–7 of Figs. 5, 6 show very similar characteristics. However, this is not the case for box 8 and the area to its north for the CT sensitivities; Fig. 6a shows strong positive ER signals, whereas Fig. 6b depicts no correlation signals at all. Part of this discrepancy is due to sampling errors; the black dotted region to the north of box 8 in Fig. 9b indicates that the signal in this region is highly variable and very sensitive to the choice of ensemble members. The signal enclosed by box 8, however, is not a sampling artifact, and instead is attributable to the differences between point-correlation sensitivities and the multivariate ER sensitivities developed in this article.

The strong ER patterns in box 8 of Fig. 6a, yet weak point correlations in box 8 of Fig. 6b, can be explained using the notion of the effective perturbation (Gombos and Hansen 2008). As discussed in section 5, although the prescribed PP for this ER application consists only of the CT dipole (approximately equal to the dipole centered on the black dot in Fig. 4c), the ensemble is not able to perfectly resolve this prescribed PP only and therefore the ER actually involves an effective PP equal to the least squares approximation of the prescribed PP (all of Fig. 4c). This effective PP includes several other considerable unprescribed perturbations, including the moderately strong perturbation dipole to the east of the CT dipole that is collocated with the approximate position of T at 72 h. In other words, there exist no ensemble linear combinations that construct the meridional dipole independently of the other features of Fig. 4c, and so these other unprescribed features in the effective PPs affect the resulting ER pattern as well.

If Fig. 6b suggests that the prescribed CT dipole in Fig. 4c is not correlated with the 6-h 500-hPa Z in the region of box 8, then some other unprescribed feature of the effective PP in Fig. 4c must be. The feature that is likely responsible for the strong ER signal in box 8 of Fig. 6a is revealed in Fig. 12, which depicts a point-correlation map of the 6-h 500-hPa Z at the point of the maxima of box 8 in Fig. 6a with the 72-h 1000-hPa PV at all points in the domain. First note that the strong CT dipole in Fig. 4c does not show up in the correlation map in Fig. 12. Instead, there is one dominant positive signal in the eastern central portion of the domain and three nearby negative signals. These four signals are approximately collocated with similar signals in the eastern section of the effective PP in Fig. 4c, suggesting that these 1000-hPa PV signals in the effective PV may be directly correlated to the 500-hPa Z at the location of the ER signal in box 8 of Fig. 6a.

Fig. 12.
Fig. 12.

Filled contours depict the point correlations of the 6-h 500-hPa Z at the point of the maximum ER signal in box 8 of Fig. 6a with the 72-h 1000-hPa PV at all points in the domain.

Citation: Monthly Weather Review 140, 8; 10.1175/MWR-D-11-00002.1

These eastern signals in Fig. 4c are the 72-h ensemble uncertainty signatures of T and of the eastern extent of the subtropical high. As can be seen from Fig. 8a, box 8 at 6 h identifies the sensitivity of the CT location to the eastern extent of the high and to the strength of T along the eastern portion of the high; CTP+ (CTP−) realizations at 72 h tend to be preceded at 6 h by H2s that extend farther (less far) east than average, and by T troughs that are anomalously weak (strong). After 6 h, T continues to ride the perimeter of the ESE portion of H2 and move southwestward from box 8 (not shown); the ensemble sensitivity of this movement to the 72-h CT position can be seen in Figs. 8b–d. By 72 h, the forecast location of T is consistent with the eastern signals in Figs. 4c, 12, suggesting that the uncertainty of T is responsible for the strong signal in box 8 of Fig. 6a.

Figure 12 and box 8 of Fig. 6 illustrate the nontransitive property of correlation: just because the 72-h CT dipole (strong dipole of Fig. 4c) is correlated with the 72-h T signature (eastern section of Fig. 4c) and the 72-h T signature is correlated with the 6-h 500-hPa Z in box 8 does not mean that the 72-h CT dipole has to be strongly correlated with the 6-h 500-hPa Z in box 8. One of the many possible explanations for this nontransitivity is that T may drive both of these other two processes, but in uncorrelated ways. However, although not strongly directly correlated to Sepat’s 72-h CT positions, the signal shows up in the ER pattern in box 8 of Fig. 6a because it is strongly correlated to the 72-h T signature in the effective PP that is implicitly included in the ER. The implications of this are discussed in section 8.

8. Conclusions, discussion, and future work

The preceding sections presented ER as a means to understand the sensitivities of the track of Supertyphoon Sepat to midtropospheric geopotential heights. Using ensemble data available in real time, the analysis illustrated that statistical patterns computed from ER are consistent with physical expectations and, without the need for an a priori understanding of the system dynamics, ER can identify the atmospheric features most strongly coupled with Sepat’s track at different lead times. In summary, this article showed that Sepat’s 72-h forecast location is sensitive to the

  • forecast positions of Sepat at earlier lead times
  • extent of the southeastern tongue of the 500-hPa subtropical high H2 and the western and southern extents of H2
  • strength and position of the easterly wave T that rides the southern perimeter of H2
  • northeastern, western, and southern positions of the 6-h trough and the position of the low L1 embedded in the trough

This sensitivity guidance might be useful to forecasters attempting to objectively estimate a deterministic forecast from ensemble model output in at least the following ways: first, using the second of the findings itemized above, if the ensembles of models other than the JMA uniformly extend this tongue well to the southeast, a forecaster attempting to issue a single best track forecast could accordingly reduce the likelihood of the JMA ensemble members that position Sepat toward the ESE at 72 h. Second, using a method that combines aspects of data assimilation, preemptive forecasting (e.g., Etherton 2007; Gombos 2009), and adaptive sampling (e.g., Bishop et al. 2001), given recent ER sensitivity information and newly available observations or analyses, a forecaster can estimate the impact of new observations on the forecasts. For example, knowing from Fig. 5a that the development of a tongue of anomalously high 500-hPa Z along the southeastern edge of the 6-h forecast subtropical high is highly correlated with a WNW Sepat track forecast, a forecaster might qualitatively put increased confidence in the WNW Sepat track forecasts if the newly available 500-hPa Z analysis data at that location are consistently greater than those of the ensemble mean used to define the ER sensitivity fields. Moreover, one could potentially make quantitative ER preemptive forecasts (Gombos 2009) using the newly available analyses and a full set of predictors and predictands; given that ER takes seconds to perform, whereas the nonlinear ensemble integration of the operation model takes hours, these preemptive forecasts would be available hours before the operational forecasts.

Section 7e discussed differences between multivariate and point-correlation sensitivities by pointing out that Sepat’s CT location is sensitive to, but not directly correlated to, the 6-h signature of T in box 8 of Fig. 6a because ER includes the effects of unprescribed effective PPs. Depending on the application of this sensitivity information, the fact that ER does not exclude these unprescribed effective PPs should be considered a potentially beneficial characteristic, not a flaw, of the multivariate ER approach. If one is interested in identifying the portions of the 500-hPa Z field that are directly correlated to Sepat’s 72-h CT location, then applying a point-correlation sensitivity technique (e.g., Hakim and Torn 2008) may be appropriate. In that case, new observations of T only might not be useful for forecasting Sepat’s 72-h CT location because T is not directly correlated to Sepat’s 72-h CT location. However, if one is interested in identifying the portions of the 6-h 500-hPa Z field that are correlated to Sepat’s 72-h forecast CT position, conditional on the state of other dynamically related portions of the forecast state vector that may suppress or otherwise affect these point correlations, then multivariate ER sensitivities may be more appropriate. In that case, new observations of T, along with observations of these other dynamically related features, might be useful for forecasting Sepat’s 72-h CT location. Together, this set of observations may improve forecasts of Sepat’s 72-h CT location more than observations only of T or only of these related features.

Despite its novelty, ER is not a drastic departure from traditional means of estimating sensitivities. Analogous to the countless numerical experiments that draw physical conclusions from model sensitivities to parameter tunings, ER draws inferences from the sensitivities of models to changes in initial (or forecast) conditions. The primary difference is that traditional sensitivity experiments typically tune a single parameter or variable, whereas ensemble techniques necessitate the joint tunings of all initial condition fields in accordance with physical requirements. Although one may consider the inability to isolate the cause of model changes as a drawback of ER, its accordance with the laws of physics (i.e., balance considerations) and probabilistic information (i.e., tuning one parameter or variable implies tuning all correlated parameters) results in ER offering more realistic and consistent sensitivity information than that yielded by single-parameter tuning techniques. These two types of sensitivity experiments should be considered complementary. Moreover, ER can potentially be extended to handle disjoint multiparameter tuning by assuming that each parameter has a persistence evolution equation and by forming an initial ensemble in which only the parameters being tuned are perturbed.

Another potential area of future ER research stems from the fact that ER sensitivities using European Centre for Medium-Range Weather Forecasts (ECMWF) ensemble data of Sepat tracks to 500-hPa Z (not shown) are nonnegligibly different from those from the JMA ER, even though ECMWF and JMA generate ensemble members in similar ways. Model sensitivity comparisons can potentially be used to identify fundamental model differences and model errors, to compute model-specific sensitivities of dynamical features for specific forecast decisions, and to suggest that optimal targeting observing sites are highly model dependent, necessitating model-specific targets for observation.

Rather than just treating models independently, ER can also potentially combine aspects of different models. An ER operator formed using ensemble analyses and/or forecasts from multiple models, each with different physical specifications and parameterizations, will sample various regions of the model parameter probability density function and effectually combine characteristics of multiple models into a single model. Future work might assess the value of multimodel ER forecasting and identify how its forecast errors compare to those from the constituent-model ERs.

It is crucial to note in this regard that all ER results are model dependent and that ER results depict model-specific sensitivities, not real-world sensitivities. Inherent in any use of ER to make dynamical inferences about the real atmosphere is the assumption that the ensembles accurately represent the real atmosphere.

It is the hope of the authors that the demonstrated accuracy and versatility of ensemble regression will not only improve the understanding, modeling, and forecasting of the atmosphere in the years to come, but will also expose to both forecasters and researchers the wealth of untapped information contained in ensemble model output.

Acknowledgments

This work is based upon a portion of the first author’s Ph.D. dissertation at the Massachusetts Institute of Technology (MIT). The authors thank Dr. Jason Sippel (NASA GSFC), Dr. Brian Tang (National Center for Atmospheric Research), Prof. Kerry Emanuel (MIT), Dr. Cecile Penland (NOAA/Earth Systems Research Laboratory), and two anonymous reviewers for their helpful comments and suggestions. The authors gratefully acknowledge funding provided by National Science Foundation Grant 0838196.

APPENDIX A

ER Regularization

In realistic applications, I and J are typically much greater than Nens, making and rank-deficient and not strictly invertible. Therefore, to maximize numerical stability and to ensure computational tractability, the predictor and predictand ensembles are projected onto their leading np and ny PCs, and , respectively, before computing the ER operator [viz., Eq. (2)]. Here,
ea1
and
ea2
where and are the eigenvectors of the covariances of and , respectively (e.g., Barnett and Preisendorfer 1987; Wilks 2006). The method for optimally choosing np and ny is described in appendix B. After also projecting the prescribed PP, p′, onto this subspace via
ea3
the ER proceeds as usual [viz., Eqs. (2) and (3)], except , , and are substituted for , , and p′, respectively. At the conclusion of the ER, the estimated predictand perturbation is projected back into physical space from the PC subspace via
ea4
for plotting and analysis purposes. This regularization procedure reduces the dimensionality of the predictor ensemble, predictand ensemble, and ER operator to np × Nens, ny × Nens, and np × ny, respectively, thereby stabilizing the ER by reducing the variance of the error of the estimated ER parameters in (e.g., Wilks 2006; Penland 1989). [See GH08 and Gombos (2009, chapter 3) for more information.]

APPENDIX B

ER Error Analysis

Meaningful inference from ER requires that the chosen predictor explains a considerable part of the variance of the predictand, and that the method is relatively insensitive to the exact choice of the domain and to the size of the ensemble. This appendix employs LOOCV (e.g., Wilks 2006) to assess the likely quality of the rank-deficient (Nens = 50, I = J = 22 444) Sepat track ERs in section 6 and to choose the optimal values for np and ny. LOOCV is a tool that enables ER model parameter fitting and forecast quality evaluation by comparing the ER-computed predictand perturbation to the actual predictand perturbation that was left out. That is, for each of the ensemble members, LOOCV removes a unique ensemble member from the ensembles that form the ER operator, applies the left-out predictor ensemble member prescribed PP to the ER using the operator defined by the remaining Nens −1 ensemble members, and compares the ER prediction of the predictand state to the actual left-out predictand ensemble member. The method produces Nens sample ER forecasts that together indicate the likely correspondence of the predictand perturbation from the ER of any arbitrary prescribed PP to the actual predictand state associated with that prescribed PP.

As in section 6, the Nens = 50 member predictor ensemble for the LOOCV is the 72-h forecast JMA 1000-hPa PV initialized at 1200 UTC 14 August 2007 and the Nens = 50 member predictand ensemble is the forecast JMA 500-hPa Z initialized at 1200 UTC 14 August 2007 (i.e., the 500-hPa Z prior to Sepat’s landfall). To assess the ER prediction skill sensitivity to lead time, np, and ny, the LOOCV is repeated separately using 500-hPa Z forecast lead times of 0–72 h in 6-h intervals, and with ny and np equal to 5 through 45 PCs in intervals of 5.

The quality of the predictions is assessed via the median over the 50 LOOCV calculations of the ensemble anomaly correlation coefficient (e.g., Wilks 2006) of the ER-computed predictand field and the actual ensemble predictand field, with respect to the (Nens = 49) ensemble mean rather than the traditionally used climatological mean. The ACC is typically used to evaluate the skill of extended-range forecasts and is designed to be more sensitive to the pattern rather than to the magnitude of the observed field (Wilks 2006). This makes the ACC suitable for the assessment of ER forecasts, since dynamical inferences from ER are derived primarily from perturbation anomaly patterns, not magnitudes.

Because overfit regressions yield highly uncertain (i.e., large variance) results (e.g., Wilks 2006), it is advantageous to define ER operators using the smallest possible values for np and ny without sacrificing ER accuracy. For each of the lead times considered in the LOOCV, the median ACC increased with increasing numbers of PCs until a threshold number of PCs was reached, beyond which the median ACC remained nearly constant or decreased (not shown) with increasing numbers of PCs. Therefore, the optimal values of ny and np at each lead time are those that correspond to this ACC threshold; using values of ny and np greater than this threshold contributes little to the expected ER accuracy but contributes considerably to the variance of the ER.

Figure B1 shows boxplots (e.g., Wilks 2006) of ensemble ACCs as a function of predictand lead time for the optimal values of ny and np at each lead time (in parentheses). Figure B1 indicates that the median ACC increases and the ACC variance decreases as the difference between the predictor and predictand lead times decreases; ER predictions improve and become less variable as the predictand and predictor ensembles become more contemporaneous. Also, in general, np < ny is optimal for this ER application, where 15 ≤ ny ≤ 20 and 5 ≤ np ≤ 15.

Fig. B1.
Fig. B1.

Boxplots of ensemble ACCs as a function of predictand lead time for the optimal values of ny and np at each lead time (in parentheses, ordered as ny and then np). The plot presents the median ACC (denoted by the large dot) and the variability of the ACCs (illustrated by the extent of the boxplot) at each lead time when the ER is performed using the optimal number of PCs. See text for more details.

Citation: Monthly Weather Review 140, 8; 10.1175/MWR-D-11-00002.1

The most important finding portrayed by Fig. B1 is that the ensemble median ACCs using the optimal ny and np combinations range between 0.44 and 0.71. These ACCs suggest that, although some subtle features of the predictand fields may be inaccurately estimated (particularly at the earlier lead times when confidence in the exact nature of the predicted smaller-scale features should be somewhat low), ER is highly capable of capturing the majority of the large-scale features. Therefore, the chosen predictor and predictand ensembles are certainly viable for a statistically significant synoptic-scale analysis of Sepat track sensitivities. Moreover, since some of the error is attributable to poor resolvability of the prescribed PP, and given that the actual prescribed PPs chosen for the ER in the Sepat track sensitivity ER in section 6 are nearly perfectly resolvable by the full ensemble, the ER predictions for the Sepat track sensitivity are likely to be good relative to the cross-validated ER predictions. Note that ER prediction accuracy would likely improve with the addition of qualified predictors, but this analysis has been confined to just 1000-hPa PV and 500-hPa Z for the reasons stated in section 5.

To test the sensitivities of both the ER patterns and the values of ny and np to the size of the domain, the exact same ER experiments performed in section 6 were repeated using a domain approximately half the size (I = J = 11 440 rather than I = J = 22 444 as in section 6). Within the smaller domain, the differences between Figs. 4, 5a, 6a and the equivalent figures for the smaller domain (not shown) are negligible. Moreover, the effective PP resolved by the predictor ensemble (not shown) remained nearly identical in the smaller domain. In this case, the overall ER signals seem insensitive to considerable changes in the domain size.

The LOOCV experiment was also repeated for this smaller domain. On average, over the 13 separate forecast leads, the median ACCs increased by approximately 0.05 for the smaller domain, indicating a small increase in predictability. The optimal values for np and ny were sensitive to the change in the domain size; the average value of ny was smaller for the smaller domain by approximately five PCs and the average value of np was larger for the smaller domain by approximately five PCs. It is highly recommended that these parameters be optimized for every ER experiment.

REFERENCES

  • Ancell, B., , and G. Hakim, 2007: Comparing adjoint- and ensemble-sensitivity analysis with application to observation targeting. Mon. Wea. Rev., 135, 41174134.

    • Search Google Scholar
    • Export Citation
  • Anderson, J., 2007: Exploring the need for localization in ensemble data assimilation using a hierarchical ensemble filter. Physica D, 230, 99111.

    • Search Google Scholar
    • Export Citation
  • Barnett, T., , and R. Preisendorfer, 1987: Origins and levels of monthly and seasonal forecast skill for United States surface air temperatures determined by canonical correlation analysis. Mon. Wea. Rev., 115, 18251850.

    • Search Google Scholar
    • Export Citation
  • Bishop, C., , B. Etherton, , and S. Majumdar, 2001: Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129, 420436.

    • Search Google Scholar
    • Export Citation
  • Bougeault, P., and Coauthors, 2010: The THORPEX Interactive Grand Global Ensemble. Bull. Amer. Meteor. Soc., 91, 10591072.

  • Davis, C., , and K. Emanuel, 1991: Potential vorticity diagnostics of cyclogenesis. Mon. Wea. Rev., 119, 19291953.

  • Ertel, H., 1942: Ein neuer hydrodynamischer wirbelsatz. Meteor. Z., 59, 271281.

  • Etherton, B., 2007: Preemptive forecasts using an ensemble Kalman filter. Mon. Wea. Rev., 135, 34843495.

  • Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99, 10 14310 162.

    • Search Google Scholar
    • Export Citation
  • Gombos, D., 2009: Ensemble regression: Using ensemble model output for atmospheric dynamics and prediction. Ph.D. thesis, Massachusetts Institute of Technology, 190 pp.

  • Gombos, D., , and J. Hansen, 2008: Potential vorticity regression and its relationship to dynamical piecewise inversion. Mon. Wea. Rev., 136, 26682682.

    • Search Google Scholar
    • Export Citation
  • Hakim, G., , and R. Torn, 2008: Ensemble synoptic analysis. Synoptic-Dynamic Meteorology and Weather Analysis and Forecasting: A Tribute to Fred Sanders, Meteor. Monogr., No. 55, Amer. Meteor. Soc., 147–162.

  • Hawblitzel, D., , F. Zhang, , Z. Meng, , and C. Davis, 2007: Probabilistic evaluation of the dynamics and predictability of the mesoscale convective vortex of 10–13 June 2003. Mon. Wea. Rev., 135, 15441563.

    • Search Google Scholar
    • Export Citation
  • Holton, J., 1992: An Introduction to Dynamic Meteorology. Academic Press, 511 pp.

  • Hoskins, B., , M. McIntyre, , and A. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877946.

    • Search Google Scholar
    • Export Citation
  • Jazwinski, A., 1970: Stochastic Processes and Filtering Theory. Academic Press, 276 pp.

  • Molteni, F., , R. Buizza, , T. Palmer, , and T. Petroliagis, 1996: The new ECMWF Ensemble Prediction System: Methodology and validation. Quart. J. Roy. Meteor. Soc., 122, 73119.

    • Search Google Scholar
    • Export Citation
  • Nakagawa, M., 2009: Outline of the High Resolution Global Model at the Japanese Meteorological Agency. RSMC Tokyo-Typhoon Center Tech. Rev. 11, 13 pp.

  • Nicholls, N., 1987: The use of canonical correlation to study teleconnections. Mon. Wea. Rev., 115, 393399.

  • Penland, C., 1989: Random forcing and forecasting using principal oscillation pattern analysis. Mon. Wea. Rev., 117, 21652185.

  • Penland, C., , and P. Sardeshmukh, 1995: The optimal growth of sea surface temperature anomalies. J. Climate, 8, 19992024.

  • Penland, C., , and L. Matrosova, 1998: Prediction of tropical Atlantic sea surface temperatures using linear inverse modeling. J. Climate, 11, 483496.

    • Search Google Scholar
    • Export Citation
  • Rabier, E., , H. Järvinen, , E. Klinker, , J. Mahfouf, , and A. Simmons, 2000: The ECMWF operational implementation of four-dimensional variational assimilation. I: Experimental results with simplified physics. Quart. J. Roy. Meteor. Soc., 126, 11431170.

    • Search Google Scholar
    • Export Citation
  • Sampson, C., , and A. Schrader, 2000: The Automated Tropical Cyclone Forecasting System (version 3.2). Bull. Amer. Meteor. Soc., 81, 12311240.

    • Search Google Scholar
    • Export Citation
  • Sippel, J., , and F. Zhang, 2008: A probabilistic analysis of the dynamics and predictability of tropical cyclogenesis. J. Atmos. Sci., 65, 34403459.

    • Search Google Scholar
    • Export Citation
  • Torn, R., , and G. Hakim, 2008: Ensemble-based sensitivity analysis. Mon. Wea. Rev., 136, 663677.

  • Velden, C., , and L. Leslie, 1991: The basic relationship between tropical cyclone intensity and the depth of the environmental steering layer in the Australian region. Wea. Forecasting, 6, 244253.

    • Search Google Scholar
    • Export Citation
  • Wilks, D., 2006: Statistical Methods in the Atmospheric Sciences. 2nd ed. Elsevier, 672 pp.

  • Zhang, F., 2005: Dynamics and structure of mesoscale error covariance of a winter cyclone estimated through short-range ensemble forecasts. Mon. Wea. Rev., 133, 28762893.

    • Search Google Scholar
    • Export Citation
1

The singular value decomposition represents as , where Σ is a diagonal matrix of singular values and and are matrices that define the orthonormal bases of ensembles and , respectively. The factor illustrates that the first action of is to project the prescribed PP, p′, onto this orthonormal basis of , so that all vectors with equal projections onto this basis will have equal resulting predictand perturbations . Therefore, .

2

This can be explained using the analogy that the reading ability among youths is correlated to shoe size only because shoe size is correlated to age; the ER operator may have predictive power only because the two fields that define the operator are both correlated to an exogenous field or mechanism.

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