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Mahsa Mirzargar and Jeffrey L. Anderson

Abstract

Various generalizations of the univariate rank histogram have been proposed to inspect the reliability of an ensemble forecast or analysis in multidimensional spaces. Multivariate rank histograms provide insightful information about the misspecification of genuinely multivariate features such as the correlation between various variables in a multivariate ensemble. However, the interpretation of patterns in a multivariate rank histogram should be handled with care. The purpose of this paper is to focus on multivariate rank histograms designed based on the concept of data depth and outline some important considerations that should be accounted for when using such multivariate rank histograms. To generate correct multivariate rank histograms using the concept of data depth, the datatype of the ensemble should be taken into account to define a proper preranking function. This paper demonstrates how and why some preranking functions might not be suitable for multivariate or vector-valued ensembles and proposes preranking functions based on the concept of simplicial depth that are applicable to both multivariate points and vector-valued ensembles. In addition, there exists an inherent identifiability issue associated with center-outward preranking functions used to generate multivariate rank histograms. This problem can be alleviated by complementing the multivariate rank histogram with other well-known multivariate statistical inference tools based on rank statistics such as the depth-versus-depth (DD) plot. Using a synthetic example, it is shown that the DD plot is less sensitive to sample size compared to multivariate rank histograms.

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Lili Lei and Jeffrey L. Anderson

Abstract

To investigate the impacts of frequently assimilating only surface pressure (PS) observations, the Data Assimilation Research Testbed and the Community Atmosphere Model (DART/CAM) are used for observing system simulation experiments with the ensemble Kalman filter. An empirical localization function (ELF) is used to effectively spread the information from PS in the vertical. The ELF minimizes the root-mean-square difference between the truth and the posterior ensemble mean for state variables. The temporal frequency of the observations is increased from 6 to 3 h, and then 1 h. By observing only PS, the uncertainty throughout the entire depth of the troposphere can be constrained. The analysis error over the entire depth of the troposphere, especially the middle troposphere, is reduced with increased assimilation frequency. The ELF is similar to the vertical localization function used in the Twentieth-Century Reanalysis (20CR); thus, it demonstrates that the current vertical localization in the 20CR is close to the optimal localization function.

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Alicia R. Karspeck and Jeffrey L. Anderson

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The assimilation of sea surface temperature (SST) anomalies into a coupled ocean–atmosphere model of the tropical Pacific is investigated using an ensemble adjustment Kalman filter (EAKF). The intermediate coupled model used here is the operational version of the Zebiak–Cane model, called LDEO5. The assimilation is applied as a means of estimating the true state of the system in the presence of incomplete observations of the state.

In the first part of this study assimilation is performed under the “perfect model” assumption, where SST observations are synthetically derived from a trajectory of the model. The focus is on how and why changes in the filter parameters (ensemble size, covariance localization, and covariance inflation) affect the quality of the analysis. It is shown that isotropic covariance localization does not benefit the analysis even when a small number of ensemble members are used. These results suggest that destruction of the “balance” between variables caused by localization is more detrimental than spurious correlation due to small ensemble size.

In the second part of this study the EAKF is used to assimilate an independent dataset of SST observations. The EAKF/Zebiak–Cane assimilation system is able to correctly estimate the phase and intensity of ENSO, as measured by the average SST anomaly in the eastern equatorial Pacific. A comparison of the analysis herein to independent wind stress and thermocline depth datasets suggests that even with the assimilation of only SST observations it is possible to reproduce over 70% of the interannual variability of thermocline depth in the eastern equatorial Pacific and off the coast of the Philippine Islands. The interannual variability of zonal wind stress in the central and western equatorial Pacific is also well correlated with independent observations (R > 0.75).

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David J. Stensrud and Jeffrey L. Anderson

Abstract

The ability of persistent midlatitude convective regions to influence hemispheric circulation patterns during the Northern Hemisphere summer is investigated. Global rainfall data over a 15-yr period indicate anomalously large July total rainfalls occurred over mesoscale-sized, midlatitude regions of North America and/or Southeast Asia during the years of 1987, 1991, 1992, and 1993. The anomalous 200-hPa vorticity patterns for these same years are suggestive of Rossby wave trains emanating from the regions of anomalous rainfall in the midlatitudes.

Results from an analysis of an 11-yr mean monthly 200-hPa July wind field indicate that, in the climatological mean, Rossby waveguides are present that could assist in developing a large-scale response from mesoscale-sized regions of persistent convection in the midlatitudes. This hypothesis is tested using a barotropic model linearized about the 200-hPa July time-mean flow and forced by the observed divergence anomalies. The model results are in qualitative agreement in the observed July vorticity anomalies for the four years investigated. Model results forced by observed tropical forcings for the same years do not demonstrate any significant influence on the midlatitude circulation. It is argued that persistent midlatitude convective regions may play a role in the development, maintenance, and dissipation of the large-scale circulations that help to support the convective regions.

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Xiu-Qun Yang and Jeffrey L. Anderson

Abstract

The prognostic tendency (PT) correction method is applied in an attempt to reduce systematic errors in coupled GCM seasonal forecasts. The PT method computes the systematic initial tendency error (SITE) of the coupled model and subtracts it from the discrete prognostic equations. In this study, the PT correction is applied only to the three-dimensional ocean temperature. The SITE is computed by calculating a climatologically averaged difference between coupled model initial conditions and resulting forecasts at very short lead times and removing the observed mean seasonal tendency.

Two sets of coupled GCM forecasts, one using an annual mean SITE correction and the other using a SITE correction that is a function of season, are compared with a control set of uncorrected forecasts. Each set consists of 17 12-month forecasts starting on 1 January from 1980 through 1996. The PT correction is found to be an effective method for maintaining a more realistic forecast climatology by reducing systematic ocean temperature errors that lead to a relaxation of the tropical Pacific thermocline slope and a weak tropical SST annual cycle in the control set. The annual mean PT correction, which allows the model to freely generate its own seasonal cycle, leads to increased prediction skill for tropical Pacific SSTs while the seasonally varying PT correction has no impact on this skill.

Physical mechanisms responsible for improvements in the coupled model’s annual cycle and forecast skill are investigated. The annual mean structure of the tropical Pacific thermocline is found to be essential for producing a realistic SST annual cycle. The annul mean PT correction helps to maintain a realistic thermocline slope that allows surface winds to impact the annual cycle of SST in the eastern Pacific. Forecast skill is increased if the coupled model correctly captures dynamical modes related to ENSO. The annual mean correction leads to a model ENSO that is best characterized as a delayed oscillator mode while the control model appears to have a more stationary ENSO mode; this apparently has a positive impact on ENSO forecast skill in the PT corrected model.

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Yue Ying, Fuqing Zhang, and Jeffrey L. Anderson

Abstract

Covariance localization remedies sampling errors due to limited ensemble size in ensemble data assimilation. Previous studies suggest that the optimal localization radius depends on ensemble size, observation density and accuracy, as well as the correlation length scale determined by model dynamics. A comprehensive localization theory for multiscale dynamical systems with varying observation density remains an active area of research. Using a two-layer quasigeostrophic (QG) model, this study systematically evaluates the sensitivity of the best Gaspari–Cohn localization radius to changes in model resolution, ensemble size, and observing networks. Numerical experiment results show that the best localization radius is smaller for smaller-scale components of a QG flow, indicating its scale dependency. The best localization radius is rather insensitive to changes in model resolution, as long as the key dynamical processes are reasonably well represented by the low-resolution model with inflation methods that account for representation errors. As ensemble size decreases, the best localization radius shifts to smaller values. However, for nonlocal correlations between an observation and state variables that peak at a certain distance, decreasing localization radii further within this distance does not reduce analysis errors. Increasing the density of an observing network has two effects that both reduce the best localization radius. First, the reduced observation error spectral variance further constrains prior ensembles at large scales. Less large-scale contribution results in a shorter overall correlation length, which favors a smaller localization radius. Second, a denser network provides more independent pieces of information, thus a smaller localization radius still allows the same number of observations to constrain each state variable.

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Hisashi Nakamura, Mototaka Nakamura, and Jeffrey L. Anderson

Abstract

Time evolutions of prominent blocking flow configurations over the North Pacific and Europe are compared based upon composites for the 30 strongest events observed during 27 recent winter seasons. Fluctuations associated with synoptic-scale migratory eddies have been filtered out before the compositing. A quasi-stationary wave train across the Atlantic is evident during the blocking amplification over Europe, while no counterpart is found to the west of the amplifying blocking over the North Pacific. Correlation between the tropopause-level potential vorticity (PV) and meridional wind velocity associated with the amplifying blocking is found to be negative over Europe in association with the anticyclonic evolution of the low-PV center, but it is almost zero over the North Pacific. Feedback from the synoptic-scale eddies, as evaluated in the form of 250-mb geopotential height tendency due to the eddy vorticity flux convergence, accounts for more than 75% of the observed amplification for the Pacific blocking and less than 45% for the European blocking. This difference is highlighted in two types of “contour advection with surgery” experiments. In one of them PV contours observed four days before the peak blocking time were advected by composite time series of the low-pass-filtered observational wind, and in the other experiment they were advected by the low-pass-filtered wind from which the transient eddy feedback evaluated as above had been removed at every time step. Hence, the latter data should be dominated by low-frequency dynamics. For the European blocking both experiments can reproduce the anticyclonic evolution of low-PV air within a blocking ridge as observed. For the Pacific blocking, in contrast, the observed intrusion of low-PV air into the higher latitudes cannot be reproduced without the transient feedback. Furthermore, in a barotropic model initialized with the composite 250-mb flow observed three days before the peak time, a simulated blocking development over the North Pacific is more sensitive to the insertion of the observed transient feedback than that over Europe. These results suggest that the incoming wave activity flux associated with a quasi-stationary Rossby wave train is of primary importance in the blocking formation over Europe, whereas the forcing by the synoptic-scale transients is indispensable to that over the North Pacific.

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Jeffrey L. Anderson and William F. Stern

Abstract

A method is presented for determining when an ensemble of model forecasts has the potential to provide some useful information. An ensemble forecast of a particular scale quantity is said to have potential predictive utility when the ensemble forecast distribution is significantly different from an appropriate climatological distribution. Here, the potential predictive utility is measured using Kuiper's statistical test for comparing two discrete distributions. More traditional measures of the potential usefulness of an ensemble forecast based on ensemble mean or variance discard possibly valuable information by making implicit assumptions about the distributions being compared.

Application of the potential predictive utility to long integrations of an atmospheric general circulation model in a boundary value problem (an ensemble of Atmospheric Model Intercomparison Project integrations) reveals a number of features about the response of a GCM to observed sea surface temperatures. In particular, the ensemble of forecasts is found to have potential predictive utility over large geographic areas for a number of atmospheric fields during strong El Niño-Southern Oscillation anomalous events. Unfortunately, there are only limited areas of potential predictive utility for near-surface fields and precipitation outside the regions of the tropical oceans. Nevertheless, the method presented here can identify all areas where the GCM ensemble may provide useful information, whereas methods that make assumptions about the distribution of the ensemble forecast variables may not be able to do so.

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Jeffrey L. Anderson and Huug M. van den Dool

Abstract

The skill of a set of extended-range dynamical forecasts made with a modern numerical forecast model is examined. A forecast is said to be skillful if it produces a high quality forecast by correctly modeling some aspects of the dynamics of the real atmosphere; high quality forecasts may also occur by chance. The dangers of making a conclusion about model skill by verifying a single long-range forecast are pointed out by examples of apparently high “skill” verifications between extended-range forecasts and observed fields from entirely different years.

To avoid these problems, the entire distribution of forecast quality for a large set of forecasts as a function of lead time is examined. A set of control forecasts that clearly have no skill is presented. The quality distribution for the extended-range forecasts is compared to the distributions of quality for the no-skill control forecast set.

The extended-range forecast quality distributions are found to be essentially indistinguishable from those for the no-skill control at leads somewhat greater than 12 days. A search for individual forecasts with a “return of skill” at extended ranges is also made. Although it is possible to find individual forecasts that have a return of quality, a comparison to the no-skill controls demonstrates that these return of skill forecasts occur only as often as is expected by chance.

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Jeffrey L. Anderson, Bruce Wyman, Shaoqing Zhang, and Timothy Hoar

Abstract

An ensemble filter data assimilation system is tested in a perfect model setting using a low resolution Held–Suarez configuration of an atmospheric GCM. The assimilation system is able to reconstruct details of the model’s state at all levels when only observations of surface pressure (PS) are available. The impacts of varying the spatial density and temporal frequency of PS observations are examined. The error of the ensemble mean assimilation prior estimate appears to saturate at some point as the number of PS observations available once every 24 h is increased. However, increasing the frequency with which PS observations are available from a fixed network of 1800 randomly located stations results in an apparently unbounded decrease in the assimilation’s prior error for both PS and all other model state variables. The error reduces smoothly as a function of observation frequency except for a band with observation periods around 4 h. Assimilated states are found to display enhanced amplitude high-frequency gravity wave oscillations when observations are taken once every few hours, and this adversely impacts the assimilation quality. Assimilations of only surface temperature and only surface wind components are also examined.

The results indicate that, in a perfect model context, ensemble filters are able to extract surprising amounts of information from observations of only a small portion of a model’s spatial domain. This suggests that most of the remaining challenges for ensemble filter assimilation are confined to problems such as model error, observation representativeness error, and unknown instrument error characteristics that are outside the scope of perfect model experiments. While it is dangerous to extrapolate from these simple experiments to operational atmospheric assimilation, the results also suggest that exploring the frequency with which observations are used for assimilation may lead to significant enhancements to assimilated state estimates.

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