Using the distribution of increments in Composite Reference
Using the distribution of increments in Composite Reference difference time series to determine the temperature trend bias caused by inhomogeneities Ralf Lindau
Inhomogeneities Climate records are affected by breaks resulting from relocations or changes in the measuring techniques. For the detection, differences of neighboring stations are considered to reduce the dominating natural variance. Homogenization algorithms identify breaks by searching for the maximum external variance (explained by the jumps). Dipdoc Seminar – 11. November 2019
Composite Reference method Networks of neighboring stations are built (about 10 stations). The average time series is built. (Composite reference). This is subtracted from each candidate station. As result we obtain about 10 difference time series with full break signal from candidate with weaken break signal from the composite without climate signal (assumed to be equal within network, appreciated) without trend bias (because it is just subtracted, not appreciated) Dipdoc Seminar – 11. November 2019
Trend bias Only if the jumps are on average non-zero , they introduce a trend bias, (which is really harmful). Otherwise they introduce only some additional scatter into the data. Thus, we concentrate on the trend bias induced by inhomogeneities. Dipdoc Seminar – 11. November 2019
CR Approach Fails (1/2) a Panel (a) shows the step function of a candidate station and the idealized composite reference b c Panel (b) the saw-shaped difference time series between the two together with the averages of the detected subperiods (thick), Panel (c) the corrected (thick) and the original step function. (explained by the jumps). Some additional steps are inserted but the trend is not corrected. Dipdoc Seminar – 11. November 2019
CR Approach Fails (2/2) In the last example, we assumed only a common network bias and no inter-stational variance of the breaks. Normally, the latter effect is large (compared to the bias) and superimposed. The method finds and corrects these station-specific breaks. As they dominate the variance, the procedure seems to perform well. However, just the bias is missing. Nonetheless, the CR approach can be used to correct the trend bias. Dipdoc Seminar – 11. November 2019
Increment distribution A few large jumps, positive on average A negative trend in between, consisting of many small jumps, negative on average (or vice versa) The mean of all increments within a network is exactly zero, because each break of a station appears with a mirrored sign n times but attenuated by 1/n in the reference. However, the median is negative (when the trend bias is positive) Dipdoc Seminar – 11. November 2019
Main classes and subclasses q Dipdoc Seminar – 11. November 2019
Some statistics (mean) Frequency of main class 1: Frequency of main class 0: Relative frequency of the subclasses k (Binomial distribution): Mean of class 0: Mean of class 1: The overall mean is actually zero Dipdoc Seminar – 11. November 2019
Some statistics (variance) The slightly different means of each subclass impose a small additional variance to the main class. However, as the bias is small, this is negligible. Then: Variance of class 0: Approx. noise variance Variance of class 1: Approx. noise plus break variance The variances s 02 and s 12 are closely connected to the signal-to-noise ratio SNR. Dipdoc Seminar – 11. November 2019
Two main classes Model input: n = 10 p = 0. 05 B =1 K sd = 1 K se = 1 K Model output Main class 0 Number: 10. 450. 296 Mean: - 0. 050 Variance: 1. 091 1 549. 704 0. 950 2. 107 Dipdoc Seminar – 11. November 2019
Median of two Gaussian q Dipdoc Seminar – 11. November 2019
Four terms (1/2) I II IV I: We stand at x 0 and consider class 0: Half of class 0 is reached, but this class contains only a fraction of 1 – p II: We stand at x 0 and consider class 1: Half of class 1 is reached at x 1, but we are –B/s 1 away from x 1. III: We proceed by dx and stand approximately at 0 (class 0): We are p. B/s 0 away from x 0. The normalized increment is dx/s 0. IV: We proceed by dx and stand approximately at 0 (class 1): We are (1 – p)B/s 1 away from x 1. The normalized increment is dx/s 1. Dipdoc Seminar – 11. November 2019
Four terms (2/2) I II IV Neglect IV while replacing 1 -p by 1 in III: Solve for dx: with: q is approx. 1, because we are in both cases very near to 0 in the distribution Dipdoc Seminar – 11. November 2019
Approximation works Formula with q = 1 (thin) Model results (thick) with p = 0. 05 s 0 = 0. 5 K sd = 0. 1 – 0. 9 K The median is a linear function of the product p. B. This is equal to the total temperature change caused by the break bias. The slope depends on the quotient s 0/s 1 Dipdoc Seminar – 11. November 2019
Real data 2°-by-2° grid box in USA Fitting two functions for the inner and the outer distribution provides an estimate for the slope 1 - s 0/s 1 (0. 6) Together with the median (0. 0028 K), we obtain a trend bias of -0. 46 K/cty for this particular grid box. Dipdoc Seminar – 11. November 2019
66 grid boxes A number of 66 2°-by-2° grid boxes in the USA finds a mean trend bias of – 0. 0515 K/cty with a stddev of 0. 6996 K/cty 0. 6996 / 8. 12 > 0. 0515 Not significant Dipdoc Seminar – 11. November 2019
Modelled data, one gridbox Input: Output: n = 10 p = 0. 10 sd = 1. 0 K se = 0. 5 K Remember s 0 = 0. 522 K s 1 = 1. 128 K s 0 = 0. 552 K s 1 = 1. 108 K Dipdoc Seminar – 11. November 2019
Modelled data, 900 grid boxes 900 networks containing 10+1 stations Realistic circumstances s 0 = 0. 522 K s 1 = 1. 128 K p = 10 cty-1 B = 0. 05 K Trend bias = 0. 5 K/cty Finding: Mean deviation from inserted network trend: – 0. 017 K/cty RMS error: 0. 790 K/cty Unbiased method to determine the trend bias. Dipdoc Seminar – 11. November 2019
Conclusion We presented a method to correct the mean trend bias of a network of stations caused by inhomogeneities. The classic Composite Reference method fails to correct it. However, due to the specific characteristics of the data, the trend bias is a linear function of the median of consecutive differences. A rough estimate of the SNR is additionally required to estimate the slope. The trend bias in the USA derived from 66 2°x 2° grid boxes is shown to be not significantly different from zero. Dipdoc Seminar – 11. November 2019
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