daily stew kickoff

Daily Stew Kickoff – 27. January 2011

First Results of the Daily Stew Project

Ralf Lindau


First Steps

Data:

Climate stations of DWD with daily data (or even 7, 14, 21 h)

Although the project is focused on weather indices and extremes, we

consider initially a much easier parameter: Monthly mean temperature

Derive and test methods to:

Create reference time seriesDifferences between a station and its reference are expected be zero

Detect breaksMaximize the external variance by a minimum of breaks


DWD Climate Stations

20001900

1920

1960

1200 Stations in total,but not coexistent.

1900: 251940: 1001960: 5002000: 600


Kriging Approach

• n observations xi at the locations Pi are given.

• Perform a prediction x0 for the location P0 , where no obs is available.

• Construct the prediction by a weighted average of the observations xi.

• Take into account the observation errors xi.

• Determine the weights i.

min1

2

10

m

t

n

iiii xxx


Matrix and Input

Correlations

The spatial autocorrelationis dervided from all availabledata for each of the 12 months.

High correlations for monthly mean temperature.


Potsdam and Reference

A reference for each station is created by kriging of the surrounding 16 stations.

Normalized temperature anomaly in January for station Potsdam.

Station and Reference seems to be nearly identical.


Potsdam and Reference

A reference for each station is created by kriging of the surrounding 16 stations.

Normalized temperature anomaly in January for station Potsdam.

Station and Reference seems to be nearly identical.

However, there is a difference showing a positive trend from 1930 to 2000


Defining breaks

Breaks are defined by abrupt changes in the station-reference time series.

Internal variance

within the subperiods

External variance

between the means of different subperiods

Maximize the external variance by

a minimum number of breaks


Decomposition of Variance

m yearsN subperiodsnk members

The external variance is a weightedmeasure for the variability of thesubperiods‘ means.

The internal variance containsinformation about the error of thesubperiods‘ means.

The seeming external variance hasto be diminished by this errorto obtain the true external variance.


Break Criterion

The true external variance is used as criterion for breaks.


The first break

The difference time series increase from 1930 to 2000 (as already shown)

Between 1965 and 1985 the criterion reaches maximum values.

More than 20% of the total variance can be explained by a break in one of these years.

1970 1968 1969 1979 1967 1978 1980 1971 1972 1981

21.77 21.76 21.67 21.64 21.41 21.33 21.07 20.95 20.87 20.77

criterion

time series


Break Searching Method

Now the first break is not simply fixed where the maximum criterion occured (1970).

But combinations of two breaks are tested which contain one of the 10 best

first-break candidates (10 times 100 permutations).

The 10 best two-breaks combinations are used as seed for the search of

three-breaks combinations.


1970 0.3197 0.3176 0.3150 0.3029 0.2968 0.2941 0.2904 0.2869 0.2857 0.2824 1930 1929 1928 1927 1925 1926 1924 1923 1920 1922

1968 0.3296 0.3270 0.3240 0.3110 0.3039 0.3014 0.2969 0.2931 0.2911 0.2881

1930 1929 1928 1927 1925 1926 1924 1923 1920 1922

1969 0.3232 0.3209 0.3181 0.3056 0.2991 0.2965 0.2924 0.2888 0.2872 0.2840 1930 1929 1928 1927 1925 1926 1924 1923 1920 1922

1979 0.2821 0.2815 0.2804 0.2718 0.2686 0.2656 0.2642 0.2632 0.2621 0.2591 1930 1929 1928 1927 1925 1926 1924 1920 1923 1922

1967 0.3301 0.3273 0.3240 0.3106 0.3032 0.3007 0.2959 0.2919 0.2896 0.2868 1930 1929 1928 1927 1925 1926 1924 1923 1920 1922

1978 0.2818 0.2810 0.2799 0.2710 0.2675 0.2646 0.2630 0.2617 0.2608 0.2577 1930 1929 1928 1927 1925 1926 1924 1920 1923 1922

1980 0.2720 0.2716 0.2707 0.2624 0.2595 0.2565 0.2553 0.2547 0.2534 0.2506 1930 1929 1928 1927 1925 1926 1924 1920 1923 1922

1971 0.3041 0.3023 0.3001 0.2887 0.2831 0.2804 0.2771 0.2739 0.2732 0.2697 1930 1929 1928 1927 1925 1926 1924 1923 1920 1922

1972 0.2987 0.2971 0.2951 0.2841 0.2790 0.2761 0.2732 0.2702 0.2698 0.2662 1930 1929 1928 1927 1925 1926 1924 1923 1920 1922

1981 0.2654 0.2651 0.2643 0.2564 0.2537 0.2508 0.2499 0.2497 0.2497 0.2495 1930 1929 1928 1927 1925 1926 1967 1924 1968 1920

The second break


1 Break


2 Breaks


3 Breaks


4 Breaks

Where to stop?

The searching method is applied to a random time series to define a stop criterion


Random Time Series

2 breaks 30 breaks


Decreasing of internal variance

1 to 400 breakswithin 1000 years

1 to 50 breakswithin 100 years

The remaining internal varianceshrinks rather smoothly for a1000 years time series.

Actually, we are dealing with only a 100 years time series.

Similar behaviour, but lessregular.

Repeat the procedure 500times and consider the change in variance for each added break.


Many Breaks for many random time series

In average 6% of the variance is gained by the first breaks.

The 50th break gains only 0.3%

The 90 and the 95 percentile remain nearly constant at a few percent.

The first step is an exception as here only 100 possibilities are tested, whereas further breaks are searched from 1000 possibilities (10 candidates times 100 years).

Median

90%95%


Observations vs Random

After 4 breaks the gained variance

of the observations is comparable

to that found for random time series.

4 breaks are realistic for the

considered station.

95%Random 90% 50%

Observations


Leaving out one stationJa

nuar

y

Feb

ruar

y

Referencefromnearest 16 stations

ReferencewithoutBerlin-Dahlem


Conclusion

For monthly mean temperatures of DWD climate stations

A method to create reference time series is derived.

A method to detect breaks in difference time series is derived.

daily stew kickoff – 27. january 2011 first results of the daily stew project ralf lindau

Documents

daily data

subperiods external

true external variance

minimum of breaks slide

internal variance

total variance

stationreference time