Two and a half problems in homogenization of climate series concluding remarks to Daily Stew Ralf Lindau

These are Breaks in German climate station are too small to be detected. But they are large enough to influence the trend significantly (a rather disturbing finding). The pure knowledge of break positions is not sufficient to determine the trend accurately. The correction part of homogenization algorithms is essential and break-aware information is not enough (originally assumed in Daily Stew). If breaks are partly systematic (non-zero overall mean), they induce a spurious mean trend. Is the corresponding break variance large enough to be detectable? Hardly. Dipdoc Seminar – 16. June 2014

Internal and External Variance Consider the differences of one station compared to a neighbor reference. The dominating natural variance is cancelled out, because it is very similar at both stations. Breaks become visible by abrupt changes in the station-reference time series. Internal variance (Noise) within the subperiods External variance (Signal) between the means of different subperiods Break criterion: Maximum external (explained) variance

Explained Variance vs.True Skill X-axis: Normally, we rely on the external or explained variance. Y-axis: For simulated data the true skill is known (measured as RMS 2 difference between true and proposed signal). For SNR of ½ the two measures are only weakly correlated. Dipdoc Seminar – 16. June 2014

RMS Standard vs. arbitrary The skills of standard search and an arbitrary segmentation are comparable. Obviously, the standard search is mainly optimizing the noise, producing completely random results Dipdoc Seminar – 16. June 2014

Which SNR is sufficient? So far we considered SNR = ½. Random segmentation and standard search have comparable skills. RMS skill for different SNRs: 0Random segmentation +Standard search For SNR > 1, the standard search is significantly better. Dipdoc Seminar – 16. June 2014 Random Standard

Conclusion 1a Break search algorithm rely on the explained variance to identify the breakpoints. For signal to noise ratios of ½, the explained variance is not a good measure of the true skill. Consequently, the obtained segmentations do not differ significantly from random. However, for higher SNR the method works.

A priori formula The different reaction of breaks and noise on randomly inserted breaks makes it possible to estimate break variance and break number a priori. If we insert many breaks, almost the entire break variance is explained plus a known fraction of noise. At k = n k half of the break variance is reached (22.8% in total). Dipdoc Seminar – 16. June

Break variance Repeated for all station pairs we find a mean break variance of about 0.2 Thus the ratio of break and noise variance is 0.2 / 0.8 = ¼ The signal to noise ratio SNR = ½ Dipdoc Seminar – 16. June 2014

Conclusions 1b For monthly temperature at German climate stations the SNR can be estimated by an a priori method to just that ominous value of ½. Consequently, breaks are hardly detectable in this data.

Trends Some old-fashion researchers are still interested in the linear trend of climate series. (Modern ones of course in higher-order two-point statistics at the highest resolution possible etc. ;-) Trend errors can easily be estimated by just considering the difference time series between two neighboring stations. Advantage: no need to apply a full complicated homogenization algorithm. Any non-zero trend difference just gives a measure of the uncertainty of trends.

Trend difference (one pair) Difference time series of the monthly temperature anomaly between the stations Aachen and Essen. The thick line denotes the 2-years running mean, the thin line is the linear trend. For neighboring stations the trend difference should be zero. Non-zero trends reflect errors (probably due to inhomogeneities). Here, the trend is K / cty. The pure statistical uncertainty is small with K / cty. Dipdoc Seminar – 16. June 2014

Trend differences (all stations) Trend differences of neighboring stations reflect the true uncertainty of trends (position of crosses). Statistical errors calculated by assuming homogeneous data are much smaller (vertical extend of crosses). We conclude that the data is strongly influenced by breaks. Dipdoc Seminar – 16. June 2014

Conclusions I For signal-to-noise ratios of ½ standard break search algorithms are not superior to random segmentations. They do not work. For monthly temperature at German climate stations the SNR can be estimated by an a priori method to just this ½. Although the relative break variance might be small (½) breaks influence the trend estimates strongly. This is a dilemma: the breaks are too small to be detected, but large enough to influence the trend significantly.

Part II Is a break-aware approach adequate to determine trends? No. Dipdoc Seminar – 16. June 2014

Break-aware idea Breaks are only detected, but not corrected. Calculate the mean trend over all homogeneous subperiods (omitting the known breakpoints). This trend should reflect the true trend. Dipdoc Seminar – 16. June 2014

Internal/External Covariance Dipdoc Seminar – 16. June 2014 Trend is the regression from the data y (depended ) on the time x (independent). Analogous to the variance, also the covariance can be split into an external and an internal part. Cov = C + c Var = V + v

Total trend as weighted average Dipdoc Seminar – 16. June 2014 Total trend External trend Internal trend Total trend is the weighted average of internal and external trend with weights V and v

Total trend as weighted average Dipdoc Seminar – 16. June 2014 The variance of the time  x 2 depends quadratically on the length of the subperiods T. Subperiod length T is reciprocal to subperiod number N The internal trend influences the total trend only marginally: If e.g. N-1 = 5 breaks are contained, only by a factor of 1/36. breaksNvV /43/4 231/98/9

Conclusion II Dipdoc Seminar – 16. June 2014

Part 2 ½ Scenario A certain change in the measurement technique causes in many stations a positive jump hidden by many others, which are random. Only such systematic breaks are critical as they induce a mean spurious trend into the data. What is the relation between the causing jump height (and its corresponding break variance) and the induced trend? Are all relevant jump heights detectable? Dipdoc Seminar – 16. June 2014

Systematic breaks induce trends Dipdoc Seminar – 16. June 2014

Some numbers from data Dipdoc Seminar – 16. June 2014

Conclusion 2 ½ Ok, the effect may vary from station to station. In this way it will be sometimes large enough to be detectable. But this is similar to the undertaking to estimate the mean of an entire distribution by just a few extremes. And what, if the variance is small. So small that we can’t see any extreme. Or so small that we see just one extreme. Dipdoc Seminar – 16. June 2014