A unified statistical framework for detecting trends in multi-timescale precipitation extremes: application to non-stationary intensity-duration-frequency curves

Abstract

There is a large agreement that global warming induces changes of precipitation regimes of different nature and amplitude depending on the timescale considered. This question is of special concern regarding extreme rainfall that might have critical socio-environmental consequences. A unified framework is proposed here for detecting trends in extreme rainfall. It is based on the GEV distribution, whose parameters depend both on a simple scaling formulation to account for multiple time durations of rainfall and on time to account for the non-stationarity deriving from climatic trends. The implementation of the model is illustrated in the Sahel region by analyzing 30 in situ rainfall series of 28 years measured at time-steps from 2 to 24 h. While the separate analysis of the point series proves inconclusive for detecting trends at any of the time-steps considered, the inclusion of all the series and time-steps into the proposed unified model allows trends to be detected at a high level of confidence (p-value < 1%). This trend essentially appears in the scale parameter of the regional GEV distribution, involving a 15 to 20% increase of the 10-year rainfall in 28 years, and a 23 to 30% increase of the 100-year rainfall. The main advantages of the proposed framework are (i) its parsimony, allowing for reducing the uncertainty associated with the model inference; (ii) its capacity for detecting trends either in the mean and/or in the variability of the extreme events; and (iii) its ability for producing non-stationary Intensity-Duration-Frequency curves that are coherent over a range of durations of accumulation.

Appendices

Appendix 1: Check of the Simple Scaling model validity for the stations and durations considered in the study

The suitability of a simple scaling modeling of the temporal scale invariance of annual maxima is checked over the 30 stations of the ACN dataset in two steps:

• First, we check the linearity of the log(E^(q)[I(D)]) vs log(D) (5) for q = {1, 2, 3, 4}. The results are shown in Fig. 11, where each marker represents a station. Figure 12 shows the Pearson coefficient of the linear regression obtained at each station, for each moment order; they are all very close to − 1.

• Second, we check the shape of the relationship between the slopes of the linear regressions of step 1 for each moment order (referred to as k(q)) and the moment orders q. For the SS model to be considered a valid approximation, we need the linearity of k(q) vs q. Otherwise, a multi-scaling model should be preferred. As shown in Fig. 13 (left), the k(q) vs q linear fit is very satisfying for all the stations. Figure 13 (right) shows the slope (corresponding to the η parameter of the SS model) and the Pearson coefficient of these regressions.

Fig. 11

Log of the moments of order q = {1,2,3,4} as a function of the log of durations D = {2,4,6,12,18,24h} for the 30 stations of the ACN dataset (markers)

Fig. 12

Distribution among the 30 stations of the ACN dataset of the log(E^(q)[I(D)]) vs log(D) linear regression Pearson coefficients (r-values)

Fig. 13

(left) k(q) as a function of q = {1,2,3,4} and (right) slopes and Pearson coefficient (r-values) of the k(q) vs q linear regressions for the 30 stations of the case study. Markers are as in Fig.11

Fig. 14

Pearson coefficients of correlation (r) between the time series of (a) annual maximum of rainfall intensities for D= 2h (lower left) and D= 24h (upper right) and (b) annual totals at the 30 stations of the ACN dataset. Correlations significant at the 5% level are outlined in black. The medians across the stations are indicated in the corners

2: Cross-correlation matrices of annual maxima for D= 2h and D= 24h and annual totals

Figure 14

3: Mean and variance of the GEV distribution

Equations 20 and 21 give the relationship between the mean (M) and variance (V), respectively, of the GEV distribution and its parameters.

$$ M = \mu + \frac{\sigma}{\xi} (g_{1} - 1) ~ \text{if} ~ \xi \neq 0 ~ \text{and} ~ \xi < 1 $$

(20)

$$ V = \frac{\sigma^{2}}{\xi^{2}} (g_{2} - {g_{1}^{2}}) ~ \text{if} ~ \xi \neq 0 ~ \text{and} ~ \xi < 1/2 $$

(21)

with g_k = Γ(1 − kξ) where Γ is the gamma function.