3 METHODS
3.5 Daily POT Joint Probability Methods
3.5.1
Daily POT Bivariate Approach
The previous section showed a bivariate joint probability method which calculated joint return periods for the most extreme combinations of Barcombe Mills flow and Newhaven sea level using return periods estimated from the annual maxima distributions. To test the accuracy of the approach and to further define the full range of interaction of river flow and sea level in the lower River Ouse, the method was developed to calculate daily maxima POT joint probabilities for the primary bivariate variables of Barcombe Mills flow
( )
X and Newhaven sea level( )
Y using the complete observed daily maxima series.Therefore, to calculate the joint probability of exceedance of the threshold u by variables
(
U,V)
with similar marginal distributions and identical probabilities and dependenceχ, equation 2.14 was rewritten and applied directly, thus:(
U u)
P(
U u V u)
P u V u U P( > , > )=1−2 ≤ + ≤ , ≤(
)
[
]
[
(
)]
−χ > − + > − − =1 21 PU u 1 PU u 2(
)
[
1− >]
2 +2[
(
>)]
−1 = PU u −χ PU u (3.3)As with the bivariate extreme joint return period approach, the probabilities were not always identical as the threshold u corresponded to the non-identical threshold levels
(
x*,y*)
for the two observed series(
X,Y)
. Therefore, the probability of exceedance of threshold x* for the variable X may be expressed asP(
U >u)
= P(
X >x*)
and theprobability of exceedance of threshold y* for the variable Y expressed asP
(
V >u)
=P(
Y > y*)
.It was then assumed that the probabilities were not required to be identical (e.g. Hawkes, 2004) for the calculation of the joint probability. For example, a joint probability of 0.5 (i.e. 50%) could be produced by different combinations of probabilities, such as 0.5 & 1.0; 0.707 & 0.707; 1.0 & 0.5 etc. Therefore, equation 3.3 could be transformed to calculate the joint probability P(U >u,V >u) of non-identical probabilities for variables
(
X,Y)
with thresholds(
x*,y*)
and dependence measureχ, thus:= > > , ) (U uV u P
(
) (
)
[
1− P X >x* ⋅PY > y*]
2−χ +2[
P(
X > x*) (
⋅PY > y*)
]
−1 (3.4) Unlike the bivariate extreme joint return period approach, probabilities were instead calculated for the daily exceedance of predetermined threshold levels(
x*,y*)
. For the variable of Barcombe Mills flow, the threshold levels( )
x* were set in increments of 1m3/s, ranging from 1m3/s to 300m3/s to represent the minimum and maximum flow magnitudes from the synthesised series (1981-2006). Similarly, for the second variable of Newhaven sea level, the threshold levels( )
y* were set in increments of 0.02m, ranging from 1.1mAOD to 4.4mAOD to represent the minimum and maximum recorded sea level magnitudes from the observed series (1982-2006). Daily exceedance probabilities were then calculated by counting the number of observations that exceeded each threshold, divided by the total number of observations in the series. The output was a probability curve of exceedance between 0 and 1 for the complete observed tidal range at Newhaven and flow range at Barcombe Mills. Figure 3.2 shows an example probability curve for the daily probability of exceedance of the threshold levels at Newhaven. Appendix G.1 contains the daily joint probability curves.Marginal Probability of Daily Recorded Tide Threshold Exceedance RIVER OUSE: NEWHAVEN
0 1 2 3 4 5 6 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000
Probability of Exceedance / Day
R e c o rd e d N e w h a v e n T id e ( m O D )
Figure 3.2 Daily probability of recorded sea level threshold exceedance at Newhaven (1982-2006)
A probability table was constructed with the bivariate daily probabilities for Barcombe Mills flow and Newhaven sea level on opposing axes, incorporating each increment of sea level and flow. Using equation 3.4, daily joint probabilities were calculated for each pair with the calculated dependence measure ofχ forming a grid containing every
combination of the probabilities. The process was also repeated where full independence
(
χ =0)
was assumed between the sea level and flow variables. A simplified version of the probability matrix is shown in Appendix G.4 for selected magnitudes.The ‘look-up’ algorithm developed for the bivariate joint return period approach was amended to firstly select pairs of flow and sea level which satisfied a desired daily joint probability from the probability table, then to select the corresponding resultant stage from the structure function matrix (Appendix G.6) for that pair. The results were tabulated and the highest stage generated by any pair at the response location (Lewes Corporation Yard or Lewes Gas Works) was then assumed to represent the true joint probability.
To assess the accuracy of the fully-independent and partially-dependent bivariate daily joint probability approaches, daily probabilities were calculated using the daily maxima simulated stage magnitudes at the intermediate locations of interest at Lewes Corporation Yard and Lewes Gas Works. For both Lewes locations, the threshold levels were set in
increments of 0.02mOD, ranging from 1.0mAOD to 5.0mOD* to represent the minimum and maximum observed stage magnitudes from the continuously simulated series (1982- 2006). The daily exceedance probability curves were used as a comparison with the daily joint probabilities calculated for Barcombe Mills flow and Newhaven sea level.
3.5.2
Daily POT Trivariate Approach
As with section 3.4, the daily bivariate joint probability approach was extended to separate the third primary variable of surge at Newhaven from observed sea level. This enabled the exploration of the relationship between river flow and surge and their
combined effects on the joint probability calculations and resultant water levels at Lewes when combined with predicted tide. Daily exceedance probabilities were calculated for Newhaven surge and predicted tide, with the threshold levels set in increments of 0.02m, ranging from -0.3m to 1.3m for surge and 1.0mAOD to 4.0mAOD for predicted tide, to represent the minimum and maximum recorded magnitudes from the observed series (1982-2006).
The daily trivariate joint exceedance approach extended the two probability tables required to calculate the joint probabilities for Barcombe Mills flow, Newhaven predicted tide and Newhaven surge using equation 3.4. Similar to before, the first table produced a grid of joint probabilities for the partially-dependent flow and surge variables using an estimated dependence valueχ, and the second table produced a grid of joint probabilities for the fully-independent predicted tide. The ‘look-up’ algorithm was then extended to select a pair of values of river flow and surge from the first probability table, which then selected a value of predicted tide from the second table which collectively satisfied a desired joint probability when multiplied together. The corresponding river flow and sea level magnitudes (predicted tide plus surge) were then used to select the resultant stage from the structure function matrices. Again, due to their size, simplified daily trivariate probability tables are shown in Appendix G.5, and structure function matrices for the estimation of resultant stage at Lewes Corporation Yard and Lewes Gas Works are shown in Appendix G.6. The performance of the daily trivariate approach was tested against the daily maxima simulated stage magnitudes at the intermediate locations of interest at Lewes Corporation Yard and Lewes Gas Works.
*
Maximum simulated stage at Lewes Corporation Yard was 5.74mAOD which was almost 1.5m above the second highest value of 4.28mOD, therefore daily exceedance probabilities were identical above 4.28mAOD as only one observation exceeded this threshold. The maximum threshold was capped at
The daily trivariate results were tabulated with the highest (worst case) stage generated at the response locations assumed to represent the daily joint probability for the trivariate grouping, which was repeated to for each stage increment at Lewes Corporation Yard and Lewes Gas Works.