# Statistical Weather Forecasting

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Statistical Weather Forecasting

Independent Study

Daria

Kluver

From Statistical Methods in the Atmospheric Sciences by Daniel

Wilks

Slide2Perfect 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.

Slide33 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.

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.

Slide41st 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 perfectPerfect 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.

Slide52nd 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 1000-800mb thickness as forecast today by a certain NWP model.

Slide6Advantages and Disadvantages of Perfect-

Prog

and MOS:

Slide7Perfect 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

Slide8MOS

Advantages:Model-calculated, but un-observed quantities can be predictorsSystematic errors in the NWP model are accounted forDifferent MOS equations required for different projection timesMethod of choice when practicalDisadvantages: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

Slide9Operational MOS Forecasts

Example: FOUS14MOS 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 dataForecast equations (except t, td and winds) are regionalized. Some MOS equations contain predictors representing local climatological valuesSome equations are developed simultaneously to enhance consistency (example: a td higher than t doesn’t make sense)

Slide10Ensemble Forecasting

Slide11First, lets talk about the birth of Chaos…

LorenzRoyal McBeeSensitive dependence on initial conditions

Lorenz’s data

Slide12What 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.

Slide13Stochastic 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.

Slide14Phase 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.

Slide15Pendulum Animation

Lorenz attractor and the

butterfly

Slide16Phase 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.

Slide17Uncertainty 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.

Slide18Ensemble 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 randomlyThese are the ensemble of initial conditionsThe movement of the initial-state probability distribution through phase space is approximated by this sample’s trajectoriesEach point provides the initial conditions for a separate run of the NWP model.

Slide19Ensemble 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

Slide20Graphical 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.

Slide21Slide22

Slide23

Slide24

Slide25

Effects of Model Errors

2 types of model errors:1. models operate at a lower resolution than reality2. 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)

Slide26Statistical 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.

Slide27Subjective 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.

Slide28Assessing Discrete Probabilities

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

Slide29Chapter 7: Forecast Verification

Slide30Purposes 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.

Slide31The Joint Distribution of Forecasts and Observations

Forecast = Observation = The joint distribution of the forecasts and observations is denotedThis is a discrete bivariate probability distribution function associating a probability with each of the IxJ possible combinations of forecast and observation.

Slide32The joint distribution can be factored in two ways, the one used in a forecasting setting is:

Called calibration-refinement factorizationThe 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 yi 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 yi sometimes called the refinement of a forecast.

Slide33Scalar Attributes of Forecast Performance

Partial list of scalar aspects, or attributes, of forecast quality

Accuracy

Average correspondence between individual forecasts and the events they predict.

Bias

The correspondence between the average forecast and the average observed value of the

predictand

.

Reliability

Pertains to the relationship of the forecast to the average observation,

for specific values of the forecast.

Resolution

The degree to which the forecasts sort the observed events into groups that are different from each other.

Discrimination

Converse of resolution, pertains to differences between the conditional averages of the forecasts for different values of the observation.

Sharpness

Characterize

the unconditional distribution (relative frequencies of use) of the forecasts.

Slide34Forecast 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.

Slide35Next time

Continue to talk about forecast verificationLooking at some forecast dataNWS vs weather.com vs climatology2x2 contingency tablesConversion from probabilistic to nonprobabilisticQuantile plotsProbability forecasts of discrete predicandsProbability forecasts for continuous predictandsAccuracy measures, Skill scores, Brier Score, MSEMulti-category events

## Statistical Weather Forecasting

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