**LECTURE 3: **

**Probabilistic downscaling: Uncertainty of the forecast and its assessment; multimodel peculiarities**

**Vladimir Kryjov**

RULES of CALCULATION of ERRORS

Let’s assume that

and all the parameters x,y,z…. are measured with some errors:

*d *

*d *x, *d *y, *d *z…..(all the d ’s are non-negative!)

If *d *x, *d *y, *d *z… are random and independent

Error in *q *will be

if we cannot assume that errors are independent

From these equations it follows that (for independent x, y, z…)

for

for where n is a fixed known number

for

for where n is fixed known number

ERRORS (UNCERTAINTIES) IN FORECAST BASED ON REGRESSION *Y=f(X)*

**Notations:**

*y _{t}* – series of predictand - a target variable anomaly at the target station (T2m, T850, precipitation, etc.) - observations

x_{t} – series of predictor – single model ensemble mean – series of anomaly of some quantity (grid-point value, area mean, PC, etc.) of Z500, wind component, etc.

*t = 1…T*

*T *– training period length (years)

Time (year) *f *– time (year) of forecast.

*y _{f}* – target variable forecast value

*x _{f }*– model forecast value (ensemble mean)

*xe _{f}* – model ensemble members in year

*f*; N – ensemble size.

*M* – number of models (*m = 1….M*)

**General scheme: estimation of forecast by each individual model separately, then combination of the individual model forecasts **

SINGLE MODEL

*Forecast of predictand value using ensemble mean (deterministic forecast) *

Assumption for each single model participated in MME:

*a* and *b* are estimated by Least Squares Estimation using observations and forecasts (ensemble means) from the training period. Due to anomalies, *a*=0 (we write it just to not forget). Forecast value of *y *by *a* single model:

Estimation of uncertainty

Regression error variance

Errors in estimates of regression parameters *a* and* b* (although we deal with anomalies and *a*=0, error in *a* estimate is not zero):

Uncertainty of forecast model ensemble mean

Error (uncertainty) in the forecast by a single model (under assumption that all the errors are independent upon each other):

Deterministic forecast with uncertainty by individual models

Forecast of mean value with uncertainty by a single m-model:

Where

- it is a deterministic forecast by *m*-model

*?* and *?* are Gaussian PDF parameters.

This gives us the way to probabilistic interpretation.

MULTIMODEL COMBINATION

There may be at least three options with several sub-options:

1.1. Assumption: ?m’s are comparable, meanwhile difference in forecast *?m‘s* may be quite significant. (Actually, it’s our ordinary approach in MME1 or regression corrected composite (when regression applied) – simply, we never care about all those *?m’s*).

Multimodel forecast *?* and uncertainty* ?* are:

( - it is like in ordinary deterministic forecast )

(Under assumption of no dependency between mean_?m and ?? , otherwise - without squares).

Applied above Gaussian PDF for ?m‘s is quite questionable and depends upon a model set and regressions – though, usually, we accept it as a default assumption.

Also, for a combination of already regression corrected forecasts with small difference in ?m‘s (as I hope) it may make more sense to use method 2.1 (see the next page)).

This is a **deterministic forecast with estimated uncertainty**.

We can also treat it in probabilistic sense. These *? and ?* are parameters of predicted Gaussian PDF.

We construct predicted PDF and estimate tercile probabilities by comparing it with climatological PDF.

The QUESTION to be solved: what to take as climatological PDF – observations? – incomparable variance. Or to construct multimodel climatological PDF based on hindcast data? – looks better.

1.2. The same assumption as in 1.1. “Pure” probabilistic forecast. We estimate tercile probabilities using *?m and ?m* for each model separately and then combine them using TPF with all *P(mdlm) = 1/M.* Climatological PDFs are those of individual models.

2.1. Assumption: difference between individual model forecasts *?m‘s* is not significant, meanwhile a difference between *?m’s *may be large. Formally it can be expressed as there is no statistically significant difference between *?i and ?j* , meanwhile there are no restrictions on *?m’s*. Multimodel forecast *?* and uncertainty *?* are:

where

This is a **deterministic forecast with estimated uncertainty**. We can also treat it in probabilistic sense. These ? and ? are parameters of predicted Gaussian PDF. We construct predicted PDF and estimate tercile probabilities by comparing it with climatological PDF.

The QUESTION to be solved: what to take as climatological PDF – observations? – incomparable variance. Or to construct multimodel climatological PDF based on hindcast data? – looks better.

2.2. The same assumption as in 2.1. “Pure” probabilistic forecast. We estimate tercile probabilities using ?m and ?m for each model separately and then combine them using TPF with P(mdlm) proportional to uncertainty in forecast probability associated with wm. Climatological PDFs are those of individual models.

3. Assumption: there may be significant difference in both ?m‘s and ?m’s – we don’t know and can’t check. In this case it makes sense to estimate forecast probabilities for each model separately using ?m and ?m and then to combine them by using TPF (like we do it in APCC PMME)

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