Exploring extreme events using joint
probability analyses
CEH Wallingford: Cecilia Svensson (PI), Thomas Kjeldsen, David Jones,
Lisa Stewart
Lancaster University: Jonathan Tawn, Emma Eastoe, Olivia Grigg
Royal Meteorological Society’s National Meeting on
Flood Risk from Extreme Events
Imperial College London, 20 October 2010
Presented by Cecilia Svensson
Outline of presentation
● Objective
● Current techniques for flood estimation
● New techniques
● Summary
Objective
The objective is to undertake an exploratory analysis of observed
moderate and extreme flood events, and of associated environmental
variables related to the causes of flooding, and to identify suitable joint
probability models.
Current techniques for flood estimation
● Flood Estimation Handbook –
statistical approach
● Flood Estimation Handbook –
Statistical approach
Frequency curve
Annual maxima
event approaches
● Continuous simulation
Return period (years)
© Wallingford Hydrosolutions Ltd
New techniques
1) Flood frequency estimates based on
Monte Carlo simulations using a rainfallrunoff model
2) Peak-over-threshold flood frequency
estimates taking into account withincluster dependence
3) Statistical flood frequency estimates
using covariates
1)
Flood frequency estimates based on
Monte Carlo simulations using a
rainfall-runoff model
Monte Carlo simulations
Avoiding bias in event modelling
● Full probability distributions for input variables
● Monte-Carlo simulation of flow events
● Frequency analysis of output peak flows
Frequency curve
Annual maxima
Return period (years)
© Wallingford Hydrosolutions Ltd
Simulation methodology
●Boundary conditions from stochastic models (based on observed events):
− Inter-event arrival time (IEAT) [h]
− rainfall duration (D) [h]
− rainfall intensity (I) [mm/h]
− soil moisture deficit at onset of rainfall event (SMD) [mm]
− initial flow (qs) [m3/s]
● Soil moisture deficit at the end of each flood event from PDM (SMD*) [mm]
rainfall (mm/h)
flow (m3 /s)
time
Event no:
i
i-1
IEATi
IEAT (hours):
SMD (mm):
qs (m3/s):
SMDi1
qs i1
SMD*i 1
SMD i
qs i
SMD *i
Simulate a series of events
(on average 10 per year)
Simulate events taking
into account:
● Seasonality
● Serial dependence
● Conditionality: strong
relationships between
rainfall
Simulate rainfall
(duration, intensity,
temporal sequence)
Simulate flow at start of
event
Rainfall-runoff model (PDM)
Simulate antecedent
wetness (soil moisture
deficit)
Fast runoff
Baseflow
Continuing losses (groundwater
recharge, evapotranspiration)
─ Rainfall duration and
rainfall total
─ Flow at start of event
and soil moisture deficit
flow
peak
with deviation depending on:
2.0
1.5
Typical
SMD
0.0
− time elapsed since the previous event (IEAT)
− the SMD at the end of the previous event
0.5
● SMD at the start of each rainfall event
● Sinusoidal seasonal variation of SMD,
Modelled
SMD
SMD (mm)
1.0
Soil moisture deficit
Typical seasonal variation at time f (fraction of a year):
0.0
0.2
 SMDi

ln 
    f   1   2 sin 2f    3 cos2f 
 S max  SMDi 
0.4
0.6
Fraction of year
SMD at start of event i:
  SMDi*1 

 SMDi 
*
ln
  f    i
    f   exp 4 IEATi  ln
* 
 Smax  SMDi 
  Smax  SMDi 1 

 
0.8
1.0
Untransformed
variables
Rainfall
I
Blyth at Hartford Bridge
D
Red – summer
Blue - winter
● Rainfall duration and intensity show
– dependence
– artificial lower bound (due to selection
of events on total rainfall depth, P)
– duration (gamma): D’= D – 1
– intensity (exp) :I’= (P – Pmin)/(D – 1)
I’ (mm/h)
● Transformed variables
● Two seasons
– May – October, November – April
D’ (h)
Observed annual
maxima (AM)
Results
GEV fitted to AM
from continuous
simulation
AM from
continuous
simulation
AM from MC
simulation
95% CI around
GEV
(bootstrap)
Blyth at Hartford Bridge
2)
Peak-over-threshold flood frequency
estimates taking into account withincluster dependence
Current method for frequency estimation
using peaks-over-threshold
peak
peak
u
● Select a high threshold, u
● Identify flood events
● Select the peak of each flood event
● Model peaks, X, as independent generalised Pareto random variables (GPD)
P( X  x  u | X  u )  (1  x /  ) 1/ 
New Approach
Uses more data
u
● Model all exceedances, Y, above the threshold, u, as marginal generalised
Pareto random variables
● Use recent multivariate extremes models for within-event dependence to give
estimate θ (m linked to event definition):
 (u , m )  P (Y2  u ,..., Ym  u | Y1  u )
● Then model event peaks, X, as
P( X  x  u | X  u) 
 (u  x ,m )
 (u ,m )
(1   x /  ) 1 / 
Describes how within-event dependence changes with threshold
3)
Statistical flood frequency estimates
using covariates
Varying flood frequency estimates
● The values of the frequency distribution parameters
change with the values of the covariates.
●That is, for each day there is a different estimated flood
frequency curve depending on the values of the covariates.
30-year flow
Full line – daily mean flow, with 95% confidence interval in grey (for exceedances of a
threshold only), dotted line – daily max flow (scaled)
Time
Covariates
● The generalised Pareto distribution
is approximated by a censored
Generalised Extreme Value (GEV)
model
– Simplifies threshold choice
●
Covariates used to estimate the
GEV location, scale and shape
parameters:
─ Rainfall
─ Soil moisture deficit
─ Baseflow
● No separate seasonal component,
as covariates have strong seasonality
P-P plot of river flow
exceedances
Summary
● New methods for flood frequency estimation based on joint probability techniques
have been explored
– Monte Carlo simulation using a rainfall-runoff model
– Peak-over-threshold frequency estimates taking into account within-cluster dependence
– Frequency estimation using covariates
References
Kjeldsen, T. R., Svensson, C., Jones, D. A. 2010. A joint probability approach to flood frequency estimation
using Monte Carlo simulation. In Proc. British Hydrological Society Third International Hydrology
Symposium, Newcastle University, Newcastle, UK, 19-23 July 2010.
Grigg, O. Tawn, J. 2010. Threshold models for extremes in the presence of covariates with application to
river flows. Environmetrics (submitted).
Eastoe, E. F., Tawn, J. A. 2010. The distribution for the cluster maxima of exceedances of sub-asymptotic
thresholds. Biometrika (accepted).
Eastoe, E. F., Tawn, J. A. 2010. Statistical models for over-dispersion in the frequency of peaks over
threshold data for a flow series. Water Resources Research, 46, W02510, DOI:10.1029/2009WR007757.