## 1. Introduction

Gombos and Hoffman (2013, hereafter Part I) described ensemble-based exigent analysis, a technique to estimate the “exigent” or worst-case scenario (WCS), the forecast that maximizes the damage for a particular weather event and specified risk or confidence level. For each ensemble member, a damage function estimates the potential damage or cost (e.g., the heating demand) from the weather parameters (e.g., the near-surface air temperature). The potential damage is then weighted by what is at risk (e.g., the gridpoint number of inhabitants) to evaluate the actual damage. The exigent damage state (ExDS) is then calculated from the damage ensemble using a Lagrange multiplier optimization. The exigent scenario may be useful to emergency planners, insurers, and the general public because it is the unique [for multivariate Gaussian (mG) ensembles] forecast that maximizes the event-wide damage for a specified risk or confidence level.

This article combines exigent analysis with another ensemble-based technique called ensemble regression (ER; Gombos and Hansen 2008; Gombos 2009; Gombos et al. 2012) to estimate the most probable atmospheric model states expected to precede and coincide with the ExDSs, using the two case studies in Part I as examples. These *pre-exigent* conditions determined by ER are potentially useful to 1) understand the atmospheric dynamics that may lead to the ExDS (Gombos et al. 2012), 2) preemptively update the ExDS probability with incoming observations in advance of relatively slow data assimilations (Gombos 2009), and 3) identify, for purposes including supplementary forecast guidance and adaptive observing, the antecedent atmospheric features to which the ExDS is most sensitive (Gombos et al. 2012).

Ensemble regression is a multivariate linear inverse technique that uses ensemble model output to make inferences about the linear relationships between vector-valued forecast and/or analysis fields, often gridded “maps” of meteorological variables. ER uses the ensemble members of these fields as training samples to estimate a covariance-based matrix that maps an ensemble perturbation in the predictor field(s) to the most probable ensemble perturbation in the predictand field(s).

The multivariate nature of ER distinguishes it from a number of other univariate approaches that use ensemble-based statistics to infer relationships between scalar or vector predictors and scalar predictands, including the ensemble synoptic analysis of Hakim and Torn (2008), the ensemble transform Kalman filter targeting approach of Bishop et al. (2001), and the analysis sensitivity diagnostic of Liu et al. (2009). Whereas univariate techniques use only correlations, ER makes use of the joint distribution between the predictors and predictands to assess sensitivities, thereby ensuring that the predictand perturbation is a statistically feasible member of the predictand distribution. For example, ensemble synoptic analysis might find the statistical relationship between each point in the antecedent geopotential height field and the subsequent surface pressure at the center of a storm, while ER might find the statistical relationship between the *entire* antecedent geopotential height field and the *entire* subsequent surface pressure field.

Previously, using the ensemble covariances of potential vorticity, potential temperature, and geopotential height (*Z*) to approximate an ER operator, Gombos and Hansen (2008) showed that potential vorticity ER and the dynamical piecewise potential vorticity inversion of Davis and Emanuel (1991) yield nearly identical *Z* perturbations. Gombos et al. (2012) used ER to study the sensitivity of Supertyphoon Sepat's (2007) 1000-hPa potential vorticity track to the position and strength of the antecedent 500-hPa *Z* field.

Here, ensemble regression is applied to calculate the pre-exigent conditions for the two case studies of Part I—one on heating demand on 8–9 January 2010 and another on freeze-damage to Florida citrus trees on 11 January 2010. Part I estimated the heating degree days (HDD) 90%-WCS (i.e., the WCS at the 90% confidence level; Fig. 8 in Part I) to yield about 1.26% more heating demand than the ensemble average forecast and the citrus tree 90%-WCS (Fig. 15 in Part I) to damage approximately 14.2 million trees, about 4.3 times more than the ensemble average. For each case study, Part II now uses ER to map the exigent perturbation from Part I backward in time to estimate the corresponding pre-exigent conditions.

This article is organized as follows. Section 2a introduces some notation and sections 2b and 2c briefly review the concepts and mathematics of exigent analysis and of ER, respectively. Section 2d introduces the ER predictand Mahalanobis distance probability *q _{y}*. Section 3 describes two correlation error metrics used to quantify the goodness of the ER. The heating demand and citrus tree case studies are presented in sections 4 and 5, respectively. Section 6 provides a summary and discussion.

## 2. Methods

### a. Notation

Let **x** be a column vector representing an arbitrary model state or series of such states. Let **p** and **y** define transformations of **x**, which may make a subset of **x** and/or calculate diagnostic quantities from **x** at the analysis time and/or at one or more forecast times. In one example that follows, **p**, the ER predictor, will be a vector of forecast gridpoint HDD, and **y**, the ER predictand, will be a vector of forecast gridpoint 300-hPa geopotential heights. In this case, the time of **p** is after the time of **y**, and the ExDS will be used to predict, in a statistical sense, the antecedent geopotential heights associated with the 90%-WCS heating demand.

Given the ensemble members **p**′ define perturbations with respect to **y** variables be defined in the same way as the **p** variables. Let *t _{p}* and

*t*denote the forecast lead times of the

_{y}**p**and

**y**variables, respectively.

Where subscripts are necessary, the ensemble members are indexed by *n* = 1, … , *N*_{ens}; the elements of **p** are indexed by *i* = 1, … , *I*; and the elements of **y** are indexed by *j* = 1, … , *J*—making *I* × *N*_{ens} matrix and *J* × *N*_{ens} matrix. For example, in section 4, the damage ensemble

### b. Exigent analysis

To define the WCS, Part I introduced the Mahalanobis distance quantile (MDQ) and its associated cumulative density function (CDF) value, the Mahalanobis distance probability (MDP), denoted by *q*, to describe the unusualness of a perturbation **p**′ with respect to the probability distribution function (PDF) defined by the ensemble mean **p**′ whose squared statistical (Mahalanobis) distance (Mahalanobis 1936) to the origin (i.e., the mean of the ensemble perturbations) equals *ν* degrees of freedom, such that

Part I defined the damage functional, *J _{d}* =

**w**

^{T}

**p**, the weighted sum of damage over the

*I*grid points. Here,

**w**is a vector of weights of length

*I*that ascribe a user-determined absolute or relative importance to each grid point. The cumulative probability of the damage functional is referred to as the damage functional probability (DFP) or by the symbol

*β*. The DFP is the probability that

*J*is less than some constant

_{d}*C*and so equals the integral of the damage functional PDF in the half-space bounded by the hyperplane

*J*=

_{d}*C*in the direction −

**w**.

Figures 1 and 2, respectively, show the ExDPs for **w**-weighted versions of the ExDPs in Figs. 1 and 2, respectively. In this article, ER is used to estimate contemporaneous and antecedent conditions associated with these ExDPs.

### c. ER

_{pp}=

_{yp}denotes the cross covariance of

**p**′,

*t*) in the span of

_{y}*t*) in the span of

_{p}**p**′ onto the subspace spanned by

**p**′ resolved by (i.e., not in the null space of)

In realistic applications, *I* and *J* are typically much greater than *N*_{ens}, making *n _{p}* and

*n*PCs, respectively. See Gombos et al. (2012) and Part I for details. Also see the conclusion of Part I for a discussion of how the characteristics of the ensemble may affect the exigent analysis. There it is mentioned that neither the PC filtering approach used here, nor an alternative approach to reduce the impact of small sample size by applying the Gaspari and Cohn (1999) covariance localization, resulted in major changes to the ExDPs.

_{y}For the particular ER application where *t _{p}* precedes

*t*, ER is effectively a linear least squares approximation [subject to sampling and other errors discussed in section 3 and Gombos and Hansen (2008)] to the full nonlinear model used to integrate the ensemble. For these applications,

_{f}For more general applications, such as the ones in this article, for which one (transformed) subset of the state vector at *t _{p}* (e.g., the HDD forecast ensemble) is used to estimate the most probable perturbation (e.g., the ER-estimated preexigent perturbation) of a different subset of the state vector at

*t*(e.g., the 300-hPa

_{y}*Z*forecast ensemble), ER is simply ensemble-based multivariate regression with vector-valued predictors and predictands. For these general applications, ER uses least squares to relate the fields of interest and is not intended to approximate the atmospheric model dynamics.

For the ER applications in this article, in order to estimate atmospheric conditions antecedent to the ExDP, the prescribed perturbation is *t _{p}*, the ER-estimated predictand perturbation

*t*spanned by

_{y}### d. Ensemble regression Mahalanobis distance probability, d_{y}

*q*is the value of the CDF of the Mahalanobis distance of an ensemble perturbation with respect to a particular ensemble distribution. Here, and in Part I, this ensemble distribution is that of the damage state vector

**p**. A similar quantity is now defined for the ER predictand distribution. Let

*q*denote the value of the CDF of the Mahalonbis distance of an ER predictand perturbation with respect to this distribution. In analogy to Eq. (4) of Part I,

_{y}*q*is defined bywhere

_{y}*ν*denotes the degrees of freedom of

*ν*and the covariance of

## 3. ER error metrics

Throughout this article it is assumed that the ensembles follow mG distributions and that the predictors explain a significant fraction of the total variance of the respective predictands. The validity of the mG assumption is assessed using the Gaussian *Q*–*Q* plot (e.g., Wilks 2006) in Part I. This section outlines two metrics that are used to assess the validity of the covariability assumptions: the composite correlation coefficient (Glahn 1968) and leave-one-out cross validation (LOOCV; e.g., Wahba and Wendelberger 1980; Michaelson 1987; Wilks 2006; Gombos et al. 2012). The

### a. Composite correlation coefficient

*n*predictor PCs are linearly related to the

_{p}*n*predictand PCs. Here,

_{y}*n*predictand PCs accounted for by the

_{y}*n*predictor PCs:where tr denotes the trace of the indicated argument,

_{p}**Σ**is a diagonal matrix of the variances of the

*n*predictand PCs, and

_{y}*j*= 1, 2, … ,

*n*to index the predictand PCs, the

_{y}*j*th entry of the diagonal of

*j*th predictand PC and the

*n*predictor PCs. Note that, because PCs are defined to be orthogonal, the square of the

_{p}*j*th entry of the diagonal of

*j*th predictand PC and each of the

*n*predictor PCs (or, equivalently, the reduction of the variance of the

_{p}*j*th predictand PC by the

*n*predictor PCs) (e.g., Abdi 2007). Note that

_{p}### b. LOOCV

The LOOCV technique discussed in Gombos (2009) and Gombos et al. (2012) is applied to estimate the expected ER accuracy. The technique estimates the error of ER-estimated perturbations by 1) removing a single predictor ensemble perturbation *n _{p}* and

*n*PCs.

_{y}After repeating the LOOCV for each ensemble member, the median of the *n*_{ens} anomaly correlation coefficients (ACCs; e.g., Wilks 2006) between each *t _{p}* precedes

*t*, the median ACC is a measure of how closely

_{y}## 4. HDD case study

This section uses ER to estimate the atmospheric conditions that precede and coincide with the HDD ExDP (Fig. 1). Section 4a describes the HDD data and defines the predictor ensemble, predictor perturbation, and predictand ensembles used for the ER. Section 4b presents

### a. HDD data and synoptic overview

The overall synoptic scenario during the time period of this HDD case study was marked by a strong upper-level trough that swept eastward through the central and eastern United States. The magenta lines in Fig. 3 contour the ensemble mean forecast 300-hPa *Z* at forecast lead times of *t _{f}* = 0, 12, 24, and 36 h. Northerly cold-air advection associated with the strong trough brought anomalously cold temperatures toward the domain of interest. This set the stage for a potentially extreme bout of cold temperatures, for which the

The predictor ensemble used here is the HDD damage ensemble from Part I. That is, the predictor ensemble is the forecast daily average HDD for 8–9 January 2010 computed using the *N*_{ens} = 50 ECMWF ensemble forecast *T* data initialized at 0000 UTC 8 January 2010, retrieved from the THORPEX Interactive Grand Global Ensemble (TIGGE) dataset (Bougeault et al. 2010), and linearly interpolated to 0.25° resolution at points between 31.75° and 40°N and 98° and 80°W. The forecast HDD is computed by applying the HDD damage function [Eq. (9) in Part I] to the average forecast *T* from the four forecast times *t _{p}* centered on 2100 local (central standard) time on 8 January 2010 (i.e., the

*n*= 7 PCs (see appendix C and Fig. 4 in Part I).

_{p}Because the goal here is to estimate the antecedent perturbations associated with the HDD ExDP, the predictor perturbation in this case is the

The predictand ensembles use ECMWF ensemble forecasts of 17 fields: 2-m temperature (*T*), and the zonal wind (*u*), meridional wind (*υ*), air temperature (*T _{a}*), and geopotential height (

*Z*) at the 1000-, 850-, 500-, and 300-hPa pressure levels. The ensemble forecasts were initialized at 0000 UTC 8 January 2010 and also linearly interpolated to the same grid as the HDD. For each forecast lead time

*t*= 0, 12, 24, and 36 h, a separate ER is performed to estimate the most probable perturbation. For each lead time, the

_{y}*N*

_{ens}= 50 ensemble members of the 17 fields are concatenated into a single predictand ensemble matrix that is projected onto its leading

*n*PCs. The values for

_{y}*n*for lead times

_{y}*t*= 0, 12, 24, and 36 h are 7, 9, 11, and 11, respectively, based on the respective integer rounded median of the four metrics described in appendix C of Part I (Maaten 2010). See the scree graphs (e.g., Wilks 2006) depicted in Fig. 6. Also note that temperatures are cool enough throughout the domain for all ensemble members so that

_{y}*T*and HDD are linearly related.

### b. Assessing the statistical linear relationship of the HDD predictor and predictand ensembles

The quality of the ER-estimated preexigent perturbations for the HDD case are assessed using the two metrics outlined in section 3. The solid line in Fig. 7 depicts the variation of *t _{y}* = 0, the lead time most distant from the lead times that define the HDD damage ensemble (

*t*= 18, 24, 30, and 36 h)

_{p}*t*= 12 h, and then remains nearly constant through

_{y}*t*= 36 h. Note that the maximum of

_{y}*t*= 12 and 36 h because these lead times are the most contemporaneous with those that define the HDD damage ensemble. An

_{y}*t*= 12 h implies that 67% of the variance of the 300-hPa

_{y}*Z*predictand ensemble PCs is explained by the HDD damage ensemble PCs and suggests that a linear model may be appropriate to model the relationship between perturbations of these two fields.

Figure 7 also displays a boxplot (e.g., Wilks 2006) at each lead time illustrating the variability of the ACC at that lead time. The ensemble median ACC between the ER-predictand perturbation and left-out ensemble member equals 0.40 at *t _{y}* = 0 h, quickly increases to 0.78 at

*t*= 18 h, and then remains nearly constant through

_{y}*t*= 36 h. These ensemble median ACC values approximate the expected ACC values between the ER-estimated antecedent state vector perturbation and the actual antecedent perturbation, implying considerable confidence in the pattern of the ER predictions presented in the following. Note that the choices for

_{y}*n*and

_{p}*n*are based on the four metrics described in appendix C of Part I and are not optimized to maximize the ACC; other choices for

_{y}*n*and

_{p}*n*yield even higher median ACCs (not shown).

_{y}### c. ER-estimated perturbations associated with the HDD ExDP

Using ExDP as the predictor perturbation, ER is employed to investigate the atmospheric conditions that precede and coincide with the *Z*-only portion of the ER predictand perturbations from the four separate ERs with respective lead times *t _{y}* = 0, 12, 24, and 36 h. Magenta contours depict the ensemble mean 300-hPa

*Z*and black contours illustrate the preexigent 300-hPa trough state, the sum of the ensemble mean and the predictand perturbation. The 300-hPa-only portion is displayed here because it illustrates upper-tropospheric dynamics relevant to the anomalously cold temperatures at

*t*.

_{p}In Fig. 3, the ensemble mean 300-hPa *Z* represents a 300-hPa trough that deepens as it progresses eastward between *t _{y}* = 0 h (Fig. 3a) and

*t*= 36 h (Fig. 3d). Compared to the mean trough (magenta), the preexigent 300-hPa trough state (black) is significantly stronger, with a −17-m perturbation at

_{y}*t*= 0 h that rapidly deepens to approximately −37 and −59 m at

_{y}*t*= 12 and 24 h, respectively. The maximum of the

_{y}*t*= 36 h

_{y}*Z*perturbation is approximately 24 m, making the trough weaker than the ensemble mean at that time.

The preexigent 300-hPa *Z* perturbation (Fig. 3) is consistent with the physical expectations of a trough that precedes and coincides with anomalously cold temperatures in the domain of interest between lead times *t _{y}* = 18 and 36 h. A deeper and stronger trough would strengthen northerly cold-air advection on its western side, ushering anomalously cold arctic air southward into the domain throughout this time window. At around

*t*= 36 h, cold air continues to advect southward on the western side of the exigent trough as an upstream ridge progresses eastward toward the center of the domain. See Gombos et al. (2012) for a more detailed example of ER perturbation patterns and their physical interpretations for the case of a tropical cyclone.

_{y}Meanwhile, surface temperatures decrease with lead time and pockets of cold-air anomalies evolve in a manner consistent with the HDD ExDP in Fig. 1. Figure 4 depicts the HDD ExDP from Fig. 1 (magenta lines) overlaid on top of the preexigent 2-m-temperature-only portion of the ER predictand perturbations from the four separate ERs with respective lead times *t _{y}* = 0, 12, 24, and 36 h (filled contours). In the first two panels, the ER predictions are of small amplitude and, for plotting purposes, have been multiplied by 5 and 2 at

*t*= 0 and 12 h, respectively. At

_{y}*t*= 0 h, the preexigent

_{y}*T*perturbation has a cold anomaly in the northern central portion of the domain. By

*t*= 12 h, this pocket begins to spread zonally, and a separate cold pocket forms to the southeast. At

_{y}*t*= 24 h and particularly at

_{y}*t*= 36 h, these cold pockets become collocated with the HDD ExDP (magenta lines). The ACCs between these ER predictions and the HDD ExDP are −0.58, −0.42, −0.82, and −0.84 for

_{y}*t*= 0, 12, 24, and 36 h, respectively, and the ACC between the sum of the 24- and 36-h ER predictions and the ExDP is −0.94. Considering that the weights,

_{y}**w**, are population estimates, Fig. 4 depicts the evolution of a cold outbreak that targets the pattern of the HDD ExDP and is consistent with the ensemble statistics from lead time to lead time.

The title of each panel in Figs. 3 and 4 states the value of *q _{y}* [Eq. (5)] for the respective predictand perturbations. Note that

*q*for “perfect” ER applications for which

_{y}## 5. Citrus tree case study

In contrast to the HDD case study, the statistical relationship between the citrus tree damage ensemble and the relevant upstream antecedent state vector predictand ensemble is weak for reasons explained below in section 5b.

### a. Citrus tree data

The predictor ensemble for ER applications in this section is the citrus tree damage ensemble used in Part I. That is, the citrus freeze damage function [Eq. (10) from Part I] is applied to the *N*_{ens} = 50 ECMWF ensemble forecast *T* data initialized at 1200 UTC 10 January 2010 and retrieved from the TIGGE dataset (Bougeault et al. 2010) to estimate the citrus freeze damage ensemble. This damage ensemble is used to approximate the forecast covariance of the fraction of trees damaged at *t _{p}* = 24 h (near dawn local time). Note that a single forecast time

*t*is used to define the citrus damage ensemble, whereas four forecast times are used to define the HDD damage ensemble. The notation in gray on the timeline in Fig. 5 depicts the sequence of events for the citrus tree case study. These ECMWF

_{p}*T*ensemble data are linearly interpolated to 0.25° resolution at points between 25.75° and 29.5°N and 82.75° and 80°W. The citrus tree damage ensemble is projected onto its leading

*n*=

_{p}*ν*= 5 PCs (see appendix C and Fig. 4 in Part I).

The predictor perturbation for this section equals the *t _{p}*. Again, note that Fig. 15 in Part I is the corresponding 90%-WCS anomaly damage map, the citrus-tree-density-weighted version of the

The predictand ensembles used to assess the predictor–predictand relationship in the following subsection are the same as those described in section 4a, except that the ensemble data are initialized at 1200 UTC 10 January 2010 and interpolated to 0.25° resolution between 22° and 40°N and 103° and 80°W, which is a somewhat larger area than the area used for the HDD case. Predictand ensembles are projected onto their leading *n _{y}* = 7, 10, 11, 10, or 11 PCs for

*t*= 0, 6, 12, 18, or 24 h, respectively, based on the four metrics (Maaten 2010) described in appendix C of Part I.

_{y}### b. Citrus predictor–predictand relationship and expected errors

The *t _{y}* = 0 h and remains nearly constant through

*t*= 24 h. Correspondingly, the median cross-validated ACCs (dashed line in Fig. 8) are also poor for the citrus tree case study; the ensemble median ACC between the ER-predictand perturbation and left-out ensemble member equals 0.12 at

_{y}*t*= 0 h and remains nearly constant through

_{y}*t*= 24 h.

_{y}The poor expected quality of the estimate of the preexigent conditions implied by the low *I* × *N*_{ens} = 195 × 50 = 9600) where the temperature is above −4.2°C, and to equal one over a small part of the domain for a few ensemble members (14 times out of 9600) where the temperature is below −6.7°C. However, as discussed below, the citrus damage ensemble is at least very strongly linearly related to the contemporaneous *T* ensemble.

Another factor contributing to the low

### c. ER-estimated perturbations associated with the citrus tree ExDP

Although the entire upstream state vector for this citrus tree case study cannot be accurately estimated, ER nevertheless can accurately estimate certain portions of this state vector. This section displays the ER-estimated contemporaneous *T* perturbation associated with the citrus tree ExDP and, given the low

The leading *n _{p}* = 5 PCs of the citrus damage ensemble explains a high fraction (90%) of the variance of the

*t*= 24 h forecast

_{y}*T*over Florida (i.e., for the same domain and time as the damage ensemble) with

*T*ensemble and that the

*T*values over a portion of the domain (approximately 6% of the grid points) are in the linear interval between −6.7° and −4.2°C.

The filled contours of Fig. 9 display the most probable contemporaneous *t _{p}* =

*t*= 24 h

_{y}*T*perturbation associated with the citrus tree ExDP. As is to be expected, given

*q*= 0.88 approximately equals

_{y}Perhaps counterintuitively, the minimum of this contemporaneous *T* perturbation (Fig. 9) is located well to the northeast of the maximum citrus tree density (Fig. 12 in Part I). As portrayed by the ensemble variance map of the forecast *T* ensemble (magenta line contours in Fig. 9), this location mismatch is attributable to the significantly greater variability at the location of the temperature minimum than at the location of high citrus tree density. At the location of the variance maximum, temperatures are free to vary significantly without being extreme relative to the ensemble PDF, allowing for anomalously cold, yet plausible, temperatures at that location. On the other hand, the ensemble temperature variations are significantly smaller at the location of the high citrus tree density (Fig. 12 in Part I), so temperatures any lower at that location would correspond to the MDP exceeding 0.9. The positioning of the temperature perturbation in Fig. 9 is a result of having sufficiently cold temperatures at the locations of the citrus trees to maximize damage and strong positive temperature correlations at neighboring locations with high ensemble variability.

## 6. Summary and discussion

This is the second part of a two-part series on ensemble-based exigent analysis. In Part I, Gombos and Hoffman (2013) used a Lagrange multiplier technique to derive the equation for the exigent damage perturbation (ExDP) and then estimated the ExDP for two case studies from a cold outbreak in January 2010: one of a HDD 90%-WCS and another of a citrus tree 90%-WCS. This article combines exigent analysis with ensemble regression (ER) to predict the most probable perturbations expected to precede and/or coincide with the ExDPs from Part I. For the HDD case study, the ER results are consistent with physical expectations; the trough that precedes and coincides with the anomalously cold temperatures during the HDD case study associated with the ExDP (i.e., the ER-estimated preexigent 300-hPa geopotential height trough) is approximately 59 and 17 m deeper than the ensemble mean at around the time of the ExDP and 24 h earlier, respectively.

For the HDD case study, leave-one-out cross-validation (LOOCV) statistics suggest that the anomaly correlation coefficient (ACC) between the ER-estimated preexigent perturbation and the true (assuming perfect forecasts) preexigent perturbation varies between 0.40 and 0.78 depending on the lead time. For the citrus tree case study,

Estimating preexigent conditions using ER has many potential applications. For example, the methods described in this article can be used to 1) gain insights into the atmospheric dynamics associated with a particular *minimizes* a damage functional, at an MDP level that reduces the damage to a reasonable degree dictated by resources and other requirements, and apply ER to find the associated preexigent perturbation that will minimize the damage. These and other possible ER applications have potential uses for forecasting and planning in future extreme weather situations that pose risks to life and property.

## Acknowledgments

The authors gratefully acknowledge funding provided by National Science Foundation Grant 0838196.

## APPENDIX

### Embedding ER into Exigent Analysis

An equivalent alternative to the two-step algorithm of approximating the preexigent perturbation (by first estimating the ExDP and then using it as the predictor perturbation for ER) is extended exigent analysis, a one-step procedure that embeds ER directly into exigent analysis.

*I*+

*J*) ×

*n*

_{ens}matrix

**w**is replaced by the (

*I*+

*J*) × 1 vector

**w**

_{p}equals the original weighting matrix

**w**and

**w**

_{y}is a

*J*× 1 vector of zeros. Then, if

*q*and

*ν*are chosen as in the original exigent analysis, the values of

*Q*and

_{p}*Q*are unchanged and the solution of the extended problem, obtained from Eq. (1), may be written aswhere

_{w}_{pp}=

_{yy}is the covariance of

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