1. A matter of degree, here or there: Changing extreme daily rainfalls, Zarcero, Costa Rica.
Peter Waylen and Marvin Quesada
Universidad de Costa Rica, San Ramón

2. Flooding – Nosara, Costa Rica, October 5, 2018

3. Four Americans and their guide drown during a rafting accident on the Naranjo River, Costa Rica – October 22, 2018.

4. Zarcero

5. Monthly Precipitation Regimes Gainesville and Zarcero

6. Frequency Daily Probability Magnitude Daily Mean
Fifteen-day Moving Averages

7. Definition of Variables
Prendergrass, A.G., 2018
Science, June 8th, vol. 360 p

8. Precipitation Climatology

9. Precipitation Climatology
North Atlantic
Anticyclone H
Northeast
Trades

10. Precipitation Climatology
Cross-
Equatorial
Westerlies 10
Inter-tropical Convergence Zone

11. Dry Season – November-April
N.E.
Trades
ITCZ

12. Pre-Veranillos – May -June
Trades
ITCZ
Cross-
Equatorial
Westerlies

13. Veranillos– July-August
N.E.
Trades
ITCZ
Cross-
Equatorial
Westerlies

14. Post-Veranillos – September-October
N.E.
Trades
ITCZ
Cross-
Equatorial
Westerlies

15. Dry Season – November-April
N.E.
Trades
ITCZ

16. Macro-scale Drivers of Precipitation Variability
HIGH
LOW
Pacific
(ENSO)
Central American
Cordillera
Atlantic
(AMO)

17. (AMO)

18. El Niño - Southern Oscillation (ENSO)
13 Warm Phase (El Niño)
Incomplete/missing
rainfall records
Source: FSU COAPS
Japanese Met . Agency
13 Cold Phase (La Niña)

19. Boring (but essential) Bits
Annual number of extremes (Poisson)
𝑝 𝑥=𝑛 = 𝑒 −Λ Λ n 𝑛!
Λ = Mean number of occurrences per year
Modified for seasonal occurrence (Non-homogeneous
Poisson)
𝑝 𝑡 𝑥=𝑛 = 𝑒 − 𝜆 𝑡 𝜆 𝑡 𝑛 𝑛!
𝜆 𝑡 = Mean number of occurrences per year up to time t
Magnitude above critical level (Exponential)
𝐹 𝑋≤𝑥 =1− (𝑒− 𝛼 𝑥 )
𝛼=Average size of extremes above
selected threshold

20. INCREASING THE THRESHOLD THAT DEFINES EXTREME
Threshold = 60mm 60 80
100
120
140
160
180
200
Pcrit=60mm, # Events = 100, in 50yrs
Λ = 100/50 = 2.0 Events per yr.
Mean = 30mm over threshold (0r 90mm)

21. INCREASING THE THRESHOLD THAT DEFINES EXTREME
Threshold = 60mm
Threshold = 80mm 60 80
100
120
140
160
180
200 60 80
100
120
140
160
180
200
Pcrit=60mm, # Events = 100, in 50yrs
Λ = 100/50 = 2.0 Events per yr.
Mean = 30mm over threshold (0r 90mm)
Pcrit=80mm,

22. INCREASING THE THRESHOLD THAT DEFINES EXTREME
Threshold = 60mm
Threshold = 80mm 60 80
100
120
140
160
180
200 60 80
100
120
140
160
180
200
Pcrit=60mm, # Events = 100, in 50yrs
Λ = 100/50 = 2.0 Events per yr.
Mean = 30mm over threshold (0r 90mm)
Pcrit=80mm, # Events = 50, in 50yrs
Λ = 50/50 = 1.0 Events per yr.

23. INCREASING THE THRESHOLD THAT DEFINES EXTREME
Threshold = 60mm
Threshold = 80mm 60 80
100
120
140
160
180
200 60 80
100
120
140
160
180
200
Pcrit=60mm, # Events = 100, in 50yrs
Λ = 100/50 = 2.0 Events per yr.
Mean = 30mm over threshold (0r 90mm)
Pcrit=80mm, # Events = 50, in 50yrs
Λ = 50/50 = 1.0 Events per yr.
Mean = 30mm over threshold (or 110mm)

24. INCREASING THE THRESHOLD THAT DEFINES EXTREME
Threshold = 60mm
Threshold = 80mm 60 80
100
120
140
160
180
200
Threshold = 150mm 60 80
100
120
140
160
180
200 60 80
100
120
140
160
180
200
Pcrit=60mm, # Events = 100, in 50yrs
Λ = 100/50 = 2.0 Events per yr.
Mean = 30mm over threshold (0r 90mm)
Pcrit=80mm, # Events = 50, in 50yrs
Λ = 50/50 = 1.0 Events per yr.
Mean = 30mm over threshold (or 110mm)
Pcrit=150mm,

25. INCREASING THE THRESHOLD THAT DEFINES EXTREME
Threshold = 60mm
Threshold = 80mm 60 80
100
120
140
160
180
200
Threshold = 150mm 60 80
100
120
140
160
180
200 60 80
100
120
140
160
180
200
Pcrit=60mm, # Events = 100, in 50yrs
Λ = 100/50 = 2.0 Events per yr.
Mean = 30mm over threshold (0r 90mm)
Pcrit=80mm, # Events = 50, in 50yrs
Λ = 50/50 = 1.0 Events per yr.
Mean = 30mm over threshold (or 110mm)
Pcrit=150mm, # Events = 5, in 50yrs
Λ = 5/50 = 0.1 Events per yr.

26. INCREASING THE THRESHOLD THAT DEFINES EXTREME
Threshold = 60mm
Threshold = 80mm 60 80
100
120
140
160
180
200
Threshold = 150mm 60 80
100
120
140
160
180
200 60 80
100
120
140
160
180
200
Pcrit=60mm, # Events = 100, in 50yrs
Λ = 100/50 = 2.0 Events per yr.
Mean = 30mm over threshold (0r 90mm)
Pcrit=80mm, # Events = 50, in 50yrs
Λ = 50/50 = 1.0 Events per yr.
Mean = 30mm over threshold (or 110mm)
Pcrit=150mm, # Events = 5, in 50yrs
Λ = 5/50 = 0.1 Events per yr.
Mean = 30mm over threshold (or 180mm)

27. For Modeling Purposes, What is the Initial Critical Level?
In absence of “applied” answer, let the data speak for themselves.
About 99% of all historic (56 years), non-zero, daily rainfall totals are less than 70mm.

28. Annual Number of days with Extreme Rainfalls
AMO
Phase

29. Annual Average Number of Events – 1.41
AMO
ENSO
1.33
1.08
Warm
1.05
1.40
Neutral
1.81
1.77
Cold

30. Number and Timing of Events AMO ENSO
Between Drivers

31. Number and
Timing
AMO
ENSO
Ext
reme
Phases

32. Changes in Mean Number of Events.
Independence of Ocean Phases.
ENSO Phase
Period
Warm
Cold
Neutral
Sum
Rel. Frequ. 4 7 15
0.268 12 20
0.357 5 11 21
0.375 13 30 56
1.000
0.232
0.536
Chi-squared contingency test:
Insufficient evidence to reject null hypothesis of independence between frequency of ENSO phases and the phase of the AMO at the 0.05 level. (i.e. they seem to be independent).

33. Thanks Caroline!
Cai, W., Wang, G., Dewitte, B., Wu, L., Santoso, A., Takahashi, K., Yang, Y., Carréric, A. and McPhaden, M.J., Nature, 564(7735), p.201.
Cheng, L., Abraham, J., Hausfather, Z. and Trenberth, K.E., How fast are the oceans warming?. Science, 363(6423), pp

34. Changing the Critical Level AMO
Daily rainfall totals exhibit exponential-like distributions in their upper tail.
ENSO

35. Really Pushing It To The Extreme !
Closed Circles – data used in calibration
Open Circles – data used for validation
Exponential does not appear to model Extreme Tail Appropriately. Alternatives:
Generalized Pareto
Gamma
Mixed Exponential

36. Exponential ?

37. Conlusions: Searching for simple linear-type trends is at best naïve.
Although still the largest known cause of interannual variability of climate, ENSO is not the only cause and its effect may be moderated or amplified by other variables.
Such extant causes of variability, unless understood, can obscure any trends, yet once enumerated and accounted for, may actually highlight lower frequency changes.
The empirical results that emerge concur with the behaviors observed in a variety of simulation studies.
The exponential model of magnitudes is inadequate to represent the upper end of the distribution of daily rainfall totals. Particularly interested in pursuing the option of mixed exponentials with Prof. Patra.