Background In deriving basic understanding of atmospheric phenomena, the analysis often revolves around discovering and exploiting relationships between fields, and between locations; e.g. for extratropical cyclones, geostrophic balance relates wind and pressure, and the tropopause has important relations to surface development. Up until now, these relationships have been determined by sampling methods involving long periods of time, such as multiple case studies, composites, and time-series analysis. The advent of probabilistic analyses offers the opportunity to establish these relationships at an instant in time. We call this new analysis technique “ensemble synoptic analysis” because probabilistic samples are derived using an ensemble technique. The Idea Probabilistic analyses provide the best-estimate of the state of the atmosphere (expected value), state uncertainty (e.g., the variance), and the desired relationships between all locations and all variables. An ensemble Kalman filter (EnKF) is used here to generate the probabilistic analyses assuming Gaussian statistics. The ensemble covariance matrix contains the information needed to perform ensemble synoptic analysis. We illustrate the covariance relations for an extratropical cyclone, and show the usefulness of statistically determined Ertel potential vorticity (PV) operators for piecewise PV inversion. We note that in traditional data assimilation (e.g. 3D-VAR), knowledge of established dynamical relationships (e.g. geostrophic balance) is imposed on the covariances and the state is “discovered.” Here we turn that around and use the covariances determined by the EnKF to discover the dynamical relationships. The Method Probabilistic analyses are generated using a limited-area EnKF for the WRF model (in collaboration with C. Snyder, NCAR). The model is run for 100 ensemble members with ~100 km horizontal resolution, 28 vertical levels, and warm-rain microphysics. Further details are given in poster 1.10 in the WAF/NWP conference. Observations consist of 250 randomly spaced surface pressure observations sampled from a truth run using GFS boundary conditions. Here we focus on analyses at 06 UTC 29 March Summary of Ensemble Synoptic Analysis Uses probabilistic analyses to discover kinematic and dynamic relationships. Allows a powerful array of statistical tools to be used on instantaneous fields. Simplifies piecewise Ertel PV inversion. May be useful in mesoscale applications, where conventional balances are intermittent. Potential Vorticity Inversion Typically, PV inversion involves the specification of balance constraints between wind and mass and mass and temperature, boundary conditions, and numerical approximations; i.e. the inversion operator is specified. Here we determine the operator statistically using the analysis ensemble. Let  = L X where  is the ensemble Ertel PV matrix and L is a matrix operator (N x N). X = L -1  defines the PV inversion recovering the state from the PV. X T L T =  T has form “ A X = B.” Undetermined since A = X T ~ M x N; M << N. Solve for L using SVD for the M non-zero singular values of X. Also, since L = cov(X, X) -1 cov(X,  ), observe that L depends on the covariance between the state and PV. This suggests a new definition for balance: the subspace that covaries with PV. Results at 500hPa for a blob of PV near Oklahoma show low geopotential height (dashed lines every 20 m), cyclonic circulation (left panel) and subgeostrophic wind around the low (middle panel). A zonal cross section shows shows warm air above and cold air below the PV anomaly, and maximum winds normal to the section (gold lines every 5 m/s) at the level of the PV anomaly (right panel). invert unapproximated Ertel PV. no balance assumptions. no boundary conditions, map factors, SOR. irregular boundaries & grids are easy. linear model: straightforward superposition. trivial coding (~10 lines of Matlab code) advantages disadvantages results depend on ensemble size (M). M may need to be large. Spurious cov for large distance. L not unique. matrices get very large. This poster may be downloaded from: Ensemble Synoptic Analysis Applied to Ertel PV inversion with no balance constraints, no boundary conditions, and rigorous superposition Gregory J. Hakim and Ryan D. Torn University of Washington Poster P1.17 Sanders Symposium Covariances WRF grid point variables may be expressed as an N x 1 state vector, x, and the ensemble of such states by the N x M matrix, X, where the j th column of X contains the state for the j th ensemble member. The ensemble analysis-error covariance matrix is then C = X’ X’ T where X’ = X – X, and X is the ensemble mean. Note that C ~ N x N (i.e, ~10 6 x10 6 ), but rank (C) = M (i.e., 100). Below are select pieces of C. Results for an Extratropical Cyclone Ensemble-mean surface temperature (colors) and surface pressure (black lines every 4 hPa); ensemble members are shown in gray every 8 hPa (left panel); ensemble-mean 500 hPa height (black lines every 60 m) and Ertel PV (red lines every 0.5 PVU) (middle panel); ensemble-mean tropopause potential temperature (colors, and solid lines every 10 K) (right panel). Covariance between normalized surface pressure at the low center (yellow dot) and surface pressure and wind (left panel), 500 hPa height and wind (middle panel) and tropopause potential temperature and wind (right panel). The sign of the covariances has been reversed to illustrate a deeper surface cyclone: the surface cold front would be displaced eastward, both 500 hPa troughs would be stronger, and the tropopause fronts upstream and downstream would be stronger.