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 ID: 136208
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Slide1
Statistical Weather Forecasting
Independent Study
Daria
Kluver
From Statistical Methods in the Atmospheric Sciences by Daniel
WilksSlide2
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.Slide3
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.
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. Slide4
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 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.Slide5
2nd 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.Slide6
Advantages and Disadvantages of Perfect-
Prog
and MOS:Slide7
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 modelSlide8
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 modelsSlide9
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 valuesSome equations are developed simultaneously to enhance consistency (example: a t
d higher than t doesn’t make sense)Slide10
Ensemble ForecastingSlide11
First, lets talk about the birth of Chaos…
Lorenz
Royal
McBee
Sensitive dependence on initial conditions
Lorenz’s dataSlide12
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.Slide13
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.Slide14
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.Slide15
Pendulum Animation
Lorenz attractor and the
butterflySlide16
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.Slide17
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.Slide18
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.Slide19
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 deviationSlide20
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.Slide21Slide22Slide23Slide24Slide25
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)Slide26
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.Slide27
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.Slide28
Assessing Discrete Probabilities
Tricks forecasters can use, like spinning wheels or playing betting games.Slide29
Chapter 7: Forecast VerificationSlide30
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.Slide31
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.Slide32
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 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.Slide33
Scalar 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.
ResolutionThe degree to which the forecasts sort the observed events into groups that are different from each other.DiscriminationConverse 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.Slide34
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.Slide35
Next time
Continue to talk about forecast verification
Looking at some forecast data
NWS
vs
weather.com
vs climatology
2x2 contingency tablesConversion from probabilistic to nonprobabilisticQuantile plots
Probability forecasts of discrete predicandsProbability forecasts for continuous predictandsAccuracy measures, Skill scores, Brier Score, MSE
Multi-category events