2016 Schleiss IMSC Canmore
A new Framework for Analyzing The Scaling Properties of Intermittent Precipitation Marc Schleiss Civil and Environmental Engineering, Princeton University, NJ, USA
13th International Meeting on Statistical Climatology 6-10 June 2016, Canmore, Alberta
Most studies focus on rainfall amounts or intensities However, the way water is distributed in space and time also matters! Ithaca, NY
Monahans, TX
1065 mm/y ; 185 dry days 1.2 mm/h when rainy
336 mm/y ; 322 dry days 2.3 mm/h when rainy
→ Need to consider both the occurrence and the intensity!
Rainfall statistics strongly depend on scale Occurrence and intensity do not scale the same way!
Why is this relevant? A good understanding of rainfall intermittency is necessary to: • Study extremes at different scales → floods and droughts • Detect and quantify changes in precipitation patterns • Design efficient statistical downscaling algorithms • Diagnostic tools to assess the quality of model outputs
Goal of this talk “Propose a new mathematical framework to analyze rainfall intermittency based on the concept of inter-amount times”
Precipitation inter-amount times Doing the same thing, just a little differently
Precipitation inter-amount times Measuring the times between successive amounts Classical Sampling
Inter-amount times
Variable amounts, fixed durations ∆t:
Variable durations, fixed amounts ∆r :
rn (∆t) = R(n∆t) − R(n∆t − ∆t)
τn (∆r ) = T (n∆r ) − T (n∆r − ∆r ) T (x) = inf{t | R(t) ≥ x}
A good example of IATs Tipping bucket rain gauge:
Choice of inter-amount ∆r depends on application: ∆r → 0: dry period durations ∆r → ∞: climatology
Main advantages of IATs • Single random process • Strictly positive values (no zeros) • Good interpretation in terms of multiplicative random cascade
Main research questions in this talk Question 1: What can we learn from inter-amount times? Question 2: What is the distribution of IATs at a given scale? Question 3: Can we use IATs to downscale precipitation?
Data used in this study: USCRN Network 10 years of data, 139 stations, 5 min resolution, 0.2 mm amounts
1. What can we learn from inter-amount times? IATs give insight into local rainfall climatologies
Reference: Schleiss, M. and J. A. Smith, 2016: Two simple metrics for quantifying rainfall intermittency: the burstiness and memory of inter-amount times. J. Hydrometeorol., 17 (1), 421–436.
2. What is the distribution of IATs at a given scale? -Normal at large scales (> 200 mm) -Weibull and Gamma at intermediate scales (10-200 mm) -Lognormal and Lognormal-Pareto at small scales (< 10 mm)
3. Can we use IATs to downscale precipitation? Yes! Using discrete multiplicative cascades. (1)
(2)
2
2
τ∆r → {τ ∆r , τ ∆r } (1)
τ ∆r = W τ∆r 2
(1)
(2)
2
2
τ ∆r + τ ∆r = τ∆r
W ∈ (0, 1)
Problem: The law of W depends on the scale ∆r and τ∆r .
Logit-normal Model Z = ln
σ = σ0 · ∆r
W 1−W −a b
·τ
∼ N(0, σ 2 ) σ0 , a, b > 0
→ Complete downscaling algorithm with only 3 parameters!
Thank you for your attention! Any questions? For more information: Marc Schleiss [email protected] Reference: M. Schleiss and J. A. Smith, 2016: Two simple metrics for quantifying rainfall intermittency: the burstiness and memory of inter-amount times. J. Hydrometeorol., 17 (1), 421–436.
How does the autocorrelation of IATs change with scale? Stretched exponential model: ρ∆r (n) ≈ exp(−A · [n∆r ]B ) Satisfying scaling relation: ρλ∆r (n) = ρ∆r (nλ)
Multifractal Analysis
13th International Meeting on Statistical Climatology 6-10 June 2016, Canmore, Alberta
Most studies focus on rainfall amounts or intensities However, the way water is distributed in space and time also matters! Ithaca, NY
Monahans, TX
1065 mm/y ; 185 dry days 1.2 mm/h when rainy
336 mm/y ; 322 dry days 2.3 mm/h when rainy
→ Need to consider both the occurrence and the intensity!
Rainfall statistics strongly depend on scale Occurrence and intensity do not scale the same way!
Why is this relevant? A good understanding of rainfall intermittency is necessary to: • Study extremes at different scales → floods and droughts • Detect and quantify changes in precipitation patterns • Design efficient statistical downscaling algorithms • Diagnostic tools to assess the quality of model outputs
Goal of this talk “Propose a new mathematical framework to analyze rainfall intermittency based on the concept of inter-amount times”
Precipitation inter-amount times Doing the same thing, just a little differently
Precipitation inter-amount times Measuring the times between successive amounts Classical Sampling
Inter-amount times
Variable amounts, fixed durations ∆t:
Variable durations, fixed amounts ∆r :
rn (∆t) = R(n∆t) − R(n∆t − ∆t)
τn (∆r ) = T (n∆r ) − T (n∆r − ∆r ) T (x) = inf{t | R(t) ≥ x}
A good example of IATs Tipping bucket rain gauge:
Choice of inter-amount ∆r depends on application: ∆r → 0: dry period durations ∆r → ∞: climatology
Main advantages of IATs • Single random process • Strictly positive values (no zeros) • Good interpretation in terms of multiplicative random cascade
Main research questions in this talk Question 1: What can we learn from inter-amount times? Question 2: What is the distribution of IATs at a given scale? Question 3: Can we use IATs to downscale precipitation?
Data used in this study: USCRN Network 10 years of data, 139 stations, 5 min resolution, 0.2 mm amounts
1. What can we learn from inter-amount times? IATs give insight into local rainfall climatologies
Reference: Schleiss, M. and J. A. Smith, 2016: Two simple metrics for quantifying rainfall intermittency: the burstiness and memory of inter-amount times. J. Hydrometeorol., 17 (1), 421–436.
2. What is the distribution of IATs at a given scale? -Normal at large scales (> 200 mm) -Weibull and Gamma at intermediate scales (10-200 mm) -Lognormal and Lognormal-Pareto at small scales (< 10 mm)
3. Can we use IATs to downscale precipitation? Yes! Using discrete multiplicative cascades. (1)
(2)
2
2
τ∆r → {τ ∆r , τ ∆r } (1)
τ ∆r = W τ∆r 2
(1)
(2)
2
2
τ ∆r + τ ∆r = τ∆r
W ∈ (0, 1)
Problem: The law of W depends on the scale ∆r and τ∆r .
Logit-normal Model Z = ln
σ = σ0 · ∆r
W 1−W −a b
·τ
∼ N(0, σ 2 ) σ0 , a, b > 0
→ Complete downscaling algorithm with only 3 parameters!
Thank you for your attention! Any questions? For more information: Marc Schleiss [email protected] Reference: M. Schleiss and J. A. Smith, 2016: Two simple metrics for quantifying rainfall intermittency: the burstiness and memory of inter-amount times. J. Hydrometeorol., 17 (1), 421–436.
How does the autocorrelation of IATs change with scale? Stretched exponential model: ρ∆r (n) ≈ exp(−A · [n∆r ]B ) Satisfying scaling relation: ρλ∆r (n) = ρ∆r (nλ)
Multifractal Analysis
