Statistical Weather Forecasting Independent Study Daria Kluver From Statistical Methods in the Atmospheric Sciences by Daniel Wilks

Perfect Prog and MOS Classical statistical forecasts for projections over a few days are not used. Current dynamical NWP models are more accurate. 2 types of classical statistical wx forecasting are used to improve aspects of NWP forecasts. (by post-processing the NWP data) Both methods used large multiple regression equations.

3 reasons why statistical reinterpretation of dynamical NWP output is useful for practical weather forecasting: 1. NWP models simplify and homogenize sfc conditions. Statistical relationships can be developed btwn NWP output and desired forecast quantities. 1. NWP models simplify and homogenize sfc conditions. Statistical relationships can be developed btwn NWP output and desired forecast quantities. 2.NWP model forecasts are subject to error. To the extent that these errors are systematic, statistical forecasts based on NWP info can correct forecast biases. 3. NWP models are deterministic. Using NWP info in conjunction with statistical methods allows quantification of the uncertainty associated with different forecast situations.

1 st Statistical approach for dealing with forecasts from NWP Perfect Prog (Klein et al. 1959) ◦ Takes NWP model forecasts for future atmosphere assuming them to be perfect ◦ Perfect prog regression equations are similar to classical regression equations except they do not incorporate any time lag. Example: equations specifying tomorrows predictand are developed using tomorrow’s predictor values. If the NWP forecasts for tomorrow’s predictors really are perfect, the perfect-prog regression equations should provide very good forecasts. Example: equations specifying tomorrows predictand are developed using tomorrow’s predictor values. If the NWP forecasts for tomorrow’s predictors really are perfect, the perfect-prog regression equations should provide very good forecasts.

2 nd Statistical approach Model Output Statistics (MOS) ◦ Preferred because it can include directly in the regression eqns the influences of specific characteristics of different NWP models at different projections into the future. ◦ To get MOS forecast eqns you need a developmental data set with historical records of predictand, and records of the forecasts by NWP model. ◦ Separate MOS forecast equations must by made for different forecast projections. Example: predictand is tomorrows mb thickness as forecast today by a certain NWP model.

Advantages and Disadvantages of Perfect-Prog and MOS:

Perfect Prog Advantages: Large developmental sample (fit using historical climate data) Equations developed without NWP info, so changes to NWP models don’t require changes in regression equations Improving NWP models will improve forecasts Same equations can be used with any NWP models Disadvantages: ◦ Potential predictors must be well forecast by the NWP model

MOS Advantages: ◦ Model-calculated, but un-observed quantities can be predictors ◦ Systematic errors in the NWP model are accounted for ◦ Different MOS equations required for different projection times ◦ Method of choice when practical Disadvantages: ◦ Requires archived records (several years) of forecast from NWP model to develop, and models regularly undergo changes. ◦ Different MOS equations required for different NWP models

Operational MOS Forecasts Example: FOUS14 MOS equations underlying the FOUS14 forecasts are seasonally stratified: warm season (Apr- Sep) and cool season (Oct-Mar). ◦ A finer stratification is preferable with sufficient developmental data Forecast equations (except t, t d and winds) are regionalized. Some MOS equations contain predictors representing local climatological values Some equations are developed simultaneously to enhance consistency (example: a t d higher than t doesn’t make sense)

Ensemble Forecasting

First, lets talk about the birth of Chaos… Lorenz Royal McBee Sensitive dependence on initial conditions Lorenz’s data

What does Chaos have to do with NWP? The atmosphere can never be completely observed A NWP model will always begin calculating forecasts from a state slightly different from the real atmosphere. These models have the property of sensitive dependence on initial conditions.

Stochastic Dynamical Systems in Phase Space Stochastic dynamic prediction- physical laws are deterministic, but the equations that describe these laws must operate on initial values not known with certainty ◦ can be described by a joint probability distribution. This process yields, as forecasts, probability distributions describing uncertainty about the future state of the atmos.

Phase space Phase space is used to visualize the initial and forecast probability distributions Phase space – geometrical representation of the hypothetically possible states of a dynamical system, where each of the coordinate axes defining this geometry pertains to one of the forecast variables of the system.

Pendulum Animation Lorenz attractor and the butterflybutterfly

Phase space of a atmospheric model has many more dimensions. ◦ Simple model has 8 dimensions! Operational NWP models have a million dimensions More Complications: ◦ The trajectory is not attracted to a single point like the pendulum ◦ Pendulum did not have sensitive dependence to initial conditions.

Uncertainty about initial state of the atmosphere can be conceived of as a probability distribution in phase space. The shape of the initial distribution is stretched and distorted at longer forecast projections. ◦ Also, remember there is no single attractor. A single point in phase space is a unique weather situation. The collection of possible points that equal the attractor can be interpreted as the climate of the NWP model.

Ensemble Forecasts The ensemble forecast procedure begins by drawing a finite sample from the probability distribution describing the uncertainty of the initial state of the atmos. Members of the point cloud surrounding the mean estimated atmospheric state are picked randomly These are the ensemble of initial conditions The movement of the initial-state probability distribution through phase space is approximated by this sample’s trajectories Each point provides the initial conditions for a separate run of the NWP model.

Ensemble Average and Ensemble Dispersion To obtain a forecast more accurate than 1 model run with the best estimate of the initial state of the atmosphere, members of the ensemble are averaged. The atmospheric state corresponding to the center of the ensemble in phase space will approximate the center of the stochastic dynamic probability distribution at the future time. Doing this with weather models averages out elements of disagreement and emphasizes shared features. Over long time periods, the average smoothes out and looks like climatology. We get an idea of the uncertainty ◦ More confidence if the dispersion is small ◦ Formally calculated by ensemble standard deviation

Graphical Display of Ensemble Forecast Information Current practice includes 3 general types of graphics: ◦ displays of raw ensemble output, ◦ displays of statistics summarizing the ensemble distribution, and ◦ displays of ensemble relative frequencies for selected predictands.

Effects of Model Errors 2 types of model errors: ◦ 1. models operate at a lower resolution than reality ◦ 2. certain physical processes- predominantly those operating at scales smaller than the model resolution- are represented incorrectly. To represent the residuals of fig 6.31, random numbers can be added to the parameterization function. Called “stochastic physics” and used at ECMRF The parameterization (smooth curve) does not fully capture the range of behaviors for the parameterized process that are actually possible (scatter of points)

Statistical Postprocessing: Ensemble MOS You can do MOS post processing on the ensemble mean: ◦ There is still research on how best to do this. ◦ Multiple ways, which involve probability distributions, which will not be discussed here.

Subjective Probability Forecasts The nature of subjective forecasts: ◦ Subjective integration and interpretation of objective forecast info forecast guidance. ◦ Includes deterministic forecast info from NWP, MOS, current obs, radar, sat, persistence info, climate data, individual previous experiences. ◦ Subjective forecasting – the distillation by a human forecaster of disparate and sometimes conflicting info.  A subjective forecast- one formulated on the basis of the judgment of 1 or more individuals. Good one will have some measure of uncertainty.

Assessing Discrete Probabilities Tricks forecasters can use, like spinning wheels or playing betting games.

Chapter 7: Forecast Verification

Purposes of Forecast Verification ◦ Forecast verification- the process of assessing the quality of forecasts. ◦ Any given verification data set consists of a collection of forecast/observation pairs whose joint behavior can be characterized in terms of the relative frequencies of the possible combinations of forecast/observation outcomes. ◦ This is an empirical joint distribution ◦ It is important to do verification to improve methods, evaluate forecasters, estimate error characteristics.

The Joint Distribution of Forecasts and Observations Forecast = Observation = The joint distribution of the forecasts and observations is denoted This is a discrete bivariate probability distribution function associating a probability with each of the IxJ possible combinations of forecast and observation.

The joint distribution can be factored in two ways, the one used in a forecasting setting is: Called calibration-refinement factorization The refinement of a set of forecasts refers to the dispersion of the distribution p(yi) If y i has occurred, this is the probability of o j happening. Specifies how often each possible weather event occurred on those occasions when the single forecast y i was issued, or how well each forecast is calibrated. If y i has occurred, this is the probability of o j happening. Specifies how often each possible weather event occurred on those occasions when the single forecast y i was issued, or how well each forecast is calibrated. The unconditional distribution, which specifies the relative frequencies of use of each of the forecast values y i sometimes called the refinement of a forecast.

Scalar Attributes of Forecast Performance Partial list of scalar aspects, or attributes, of forecast quality

Forecast Skill Forecast skill- the relative accuracy of a set of forecasts, wrt some set of standard control, or reference, forecast (like climatological average, persistence forecasts, random forecasts based on climatological relative frequencies) Skill score- a percentage improvement over reference forecast. accuracy Accuracy of reference Accuracy that would be achieved by a perfect forecast.

Next time Continue to talk about forecast verification ◦ Looking at some forecast data  NWS vs weather.com vs climatology ◦ 2x2 contingency tables ◦ Conversion from probabilistic to nonprobabilistic ◦ Quantile plots ◦ Probability forecasts of discrete predicands ◦ Probability forecasts for continuous predictands ◦ Accuracy measures, Skill scores, Brier Score, MSE ◦ Multi-category events