A risk model. Content. Distribution of exceedances. Threshold selection. Convergence

Átírás

1 A risk model 2016/2017 Spring semester Content Background: extreme value models Flood simulator Quake simulator Some simulation results Effect of the most important settings The most dangerous areas Summary Distribution of exceedances If X has a max-stable distribution, the conditional distribution function of X-u, under the condition that X>u, can be given as y H( y) 1 1 ~ 1/ if 0, y>0 and ( 1 y / ~ ) 0, where ~ ( u ). ( H(y) is the so called generalized Pareto distribution, GPD). If =0, H(y)=1-e -y, y>0 (the unit exponential distribution). is the same as the shape parameter of the corresponding GEV., Densities of GPD with =1; solid: =0.5, dotted: =-0.1, dots-and-lines: =-0.7, broken: = lim Convergence If F belongs to the MDA of a GEV distribution, then there exists a positive, measurable function a such that 1/ F( u xa( u)) X u (1 x) if 0 P x X u x 1 F( u) a( u) e if u xf 0 This implies that for high enough threshold, the exceedance distribution can be modelled by the GPD. Question: how to choose the threshold in order to ensure that the model is reasonable? Threshold selection Mean excess plot: For any u (threshold), plot the mean of X-u (for those observations for which X>u) against u. If the Pareto model is true, this plot should be nearly linear. The interpretation is made difficult by the great variability near the upper endpoint of the observations. Alternative: Fit the distribution for different values of u and such a threshold should be chosen, where the parameter estimates are stable It is a trade-off between bias (likely to occur if the threshold is too low) and variance (occurs if the threshold is too high, and just a few observations are used for estimation). 1

2 Dependence: a typical 100- day segment of the data Declustering water level One may take just the peak out of a cluster above the threshold. The set of points does not increase monotonically as u decreases. water level x x day day Practical solutions There are physical considerations resulting in a given time period, after which a new high value corresponds to a new flood, thus independence from the previous values is realistic. For water level data there is also a natural choice for the threshold u, as there is a level, when the water fills a wider area around the narrow riverbed. Modelling time-dependence Assumptions: Shape does not change Scale depends linearly on the time: t t Positive coefficient (larger scale) here means larger observations as time advances Maximum likelihood estimator and the (modified) QQ plot for checking the fit are available An example: Budapest The flood simulator Tapasztalati eloszlás Időfüggő QQ-plot Budapest - Eltelt idő: 60 nap Likelihood statisztika értéke: Szignifikancia szintje: Illesztett eloszlás (std. exponenciális) Parameters: time-dependent scale:1.79, scale:23.91, shape:-0.58 relatív gyakoriság Időfüggő hisztogram Budapest - Eltelt idő: 60 nap Az időszak elején érv ény es GPD eloszlás sűrűségf üggv ény e Az időszak v égén érv ény es GPD eloszlás sűrűségf üggv ény e Megfigyelések (cm) The distribution at the end of the time period is much more dangerous Flood frequency estimator By MCMC (result for the last period: climate change ) traditionally Random floods for the stations at upstream river sections Routing of upstream hydrographs by conceptual hydrodynamic models 2

3 Estimating the flood-frequency It is assumed that the intensity is constant over a time period, but jumps are allowed (the data suggested such changes, corresponding to wet or dry periods in the climate) Reversible jump Markov Chain Monte Carlo method is used Apriori distribution for the jumps: uniform for the intensity: Gamma The algorithm for estimating the intensity Data: all flood peaks at the upstream stations (considered as different if the water level is low for at least 20 days) Three type of flood events Flood of the Danube only Flood of the Tisza and/or its tributaries only Flood of both watersheds Steps of the algorithm Change in intensity Change the location of changepoints Introducing a new changepoint Removing an old changepoint Illustration of the results (all floods) The simulated Markov chain can be in different states (different number of change points) The summary of the two change point-models show moderate fluctuation, but an increasing tendency for the last 50 years red: average of the daily intensities, blue: homogenous green/black: density of the first/second changepoint Results for floods of the Danube only changepoints changepoints These models were the most frequent. Weighted average estimator for the last 10 years: Floods of the Tisza and its tributary only Floods affecting both rivers changepoints changepoints Estimator: The fit of the nonhomogenous Poisson process was much better in all cases These models were the most frequent. Weighted average estimator for the last 10 years:

4 Illesztett gamma eloszlás paraméterei: alak: 1.86 skála: 2.24 Illesztett gamma eloszlás paraméterei: alak: 1.59 skála: 9.34 Illesztett gamma eloszlás paraméterei: alak: 0.99 skála: 8.56 Illesztett gamma eloszlás paraméterei: alak: 2.58 skála: 1.39 Illesztett gamma eloszlás paraméterei: alak: 0.66 skála: A regressziós egyenes meredeksége: A regressziós egyenes meredeksége: A regressziós egyenes meredeksége: A regressziós egyenes meredeksége: A regressziós egyenes meredeksége: 0.15 The chosen upstream-stations Tisza and its tributaries: Felsőberecki (Bodrog) Tivadar (Tisza) Csenger (Szamos) Gyoma (Kőrös) Makó (Maros) Danube and its tributaries : Sárvár (Rába) Komárom (Duna) Budapest (Duna) Results of the GPD-modelling Scale (1) Scale (2) Shape Frequency Flood/year Sárvár Komárom Budapest , Tivadar Csenger , Felsőberecki Gyoma Makó Duration of the floods Differences among the stations, basically two groups Gamma distribution was used for modelling the length Relatív gyakoriság Relatív gyakoriság Relatív gyakoriság Tivadari szintfeletti árvízhosszak Árvízhossz (nap) Felsőbereckii szintfeletti árvízhosszak Árvízhossz (nap) Makói szintfeletti árvízhosszak Árvízhossz (nap) Relatív gyakoriság Relatív gyakoriság Csengeri szintfeletti árvízhosszak Árvízhossz (nap) Gyomai szintfeletti árvízhosszak Árvízhossz (nap) Szint felett (nap) Szint felett (nap) Szint felett (nap) Dependence between the height and the duration of the floods Meghaladás és az árvízhossz kapcsolata Tivadar Szintmeghaladások (cm) Meghaladás és az árvízhossz kapcsolata Felsőberecki Szintmeghaladások (cm) Meghaladás és az árvízhossz kapcsolata Makó Szintmeghaladások (cm) Szint felett (nap) Szint felett (nap) Meghaladás és az árvízhossz kapcsolata Csenger Szintmeghaladások (cm) Meghaladás és az árvízhossz kapcsolata Gyoma Szintmeghaladások (cm) Linear regression was used, without an intercept and with a heteroscedastic, shifted Gamma distributed error term The flood shape Dependence among the stations 4 different groups of stations was formed Having the height of the flood, 30 such hydrographs were chosen from the given group, where the height was nearest to the current simulated one Having the duration d, such a hydrograph was chosen from the above 30, for which the duration is at least d/2 and at most 2d. The most frequent scenarios out of the possible 31 There were a couple of combinations, which have not occurred among the 164 floods, so a modification was needed Berecki Tivadar Csenger Gyoma Makó Frequency

5 Modified frequencies of joint flood occurrencies Possible conditional dependence between the flood peaks The near-floods were considered as floods with a probability (depending on the difference between the threshold and the observed peak). The chance of multiple flooding was increased. Berecki Tivadar Csenger Gyoma Makó Freq. Modified freq If we know that there were floods at two stations, is there any relation between the height of these floods? The conditional independence can be accepted. Makó Gyoma Makó Összetartozó csúcsok Tivadar Összetartozó csúcsok Csenger Összetartozó csúcsok Makó Gyoma Makó Tapasztalati kopula Tivadar Tapasztalati kopula Csenger Tapasztalati kopula Csenger Csenger Simulations for the stations, where there was no flood Dependence is strong for neighbouring stations (two groups were formed on the Tisza) For the neighbouring stations a height from the conditional distribution of the (non-flood) peaks was chosen, and then a real part of the hydrograph, which had this height as peak and the whole distribution was near enough to the observed one For other stations, only the peak was set as before Summary of the algorithm 1. Number of floods for the given year (from the Poisson distribution) 2. Simulation of the affected upstream stations 3. Simulation of the flood peaks from the timedependent GPD (conditionally independent) 4. Duration of the flood (by regression model) 5. The shape of the flood (random choice from the observed sequences, by transformation) Algorithm/2 The choice of medium and low flow data (from the conditional distribution, as a real, non-transformed sequence) Special case: the data for Budapest is got by Komárom (approx. 100 km upstream from Budapest) (by regression for two groups: were there floods on the catchment between) Hydrological routing By the discrete linear cascade model, for the most important representative floodplain sections: For the Tisza: Vásárosnamény, Záhony, Tokaj, Polgár, Szolnok, Szeged For the Danube: Baja 5

6 Megfigyelt Szimulált (exp. súly) Szimulált (kvadr. súly) Súlyozatlan The most dangerous area Data base for floods Tapasztalati eloszlás Időfüggő QQ-plot Szolnok - Eltelt idő: 60 nap Likelihood statisztika értéke: Szignifikancia szintje: Illesztett eloszlás (std. exponenciális) Parameters: time-dependent scale:1.22, scale:133, shape:-0.63 relatív gyakoriság Időfüggő hisztogram Szolnok - Eltelt idő: 60 nap Az időszak elején érv ény es GPD eloszlás sűrűségf üggv ény e Az időszak v égén érv ény es GPD eloszlás sűrűségf üggv ény e Megfigyelések (cm) The simulated floods were even more dangerous As the simulation of floods for the upstream stations and the hydrological routing are computer intensive, the simulation uses a data base of floods generated as described above floods were generated from the described model another from a bootstrap version, where the error of the GPD fit was also taken into account 1000 floods were chosen so that the fit to the observed data should be acceptable, but the sample should also include exceptionally dangerous ones (900 from the original sample, 100 from the bootstrap). Weights for the simulator We had dangerous floods in the data base, but the simulator should comply with the reality. We set the chosen risk by weights. The chosen characteristic was the duration (time length) of the flood. Exponential weights: by the likelihood-ratio of the Gamma distribution, fitted to the observed and simulated flood length at the given station. Quadratic weights: the extreme long floods were downweighted by a quadratic function (minimising the discrepancy between the observed and the simulated regression model for the exceedance/duration pair). suruség suruség Examples Szolnok Tivadar Megfigyelt Szimulált (exp.súly) Szimulált (kvadr.súly) Megfigyelt Szimulált (exp.súly) Szimulált (kvadr.súly) Szeged Budapest cm Megfigyelt Szimulált (exp.súly) Szimulált (kvadr.súly) sűrűség cm Density N = 67 Bandwidth = Choosing from the weights The floodplain sections The quadratic weights put more weight on the longer floods ( wet years ), The exponential weights provide floods more similar to the observed ones ( dry years ). Both cases are more risky than the observed data. The are more than 100 such sections, bordered by natural or artificial dykes. The dykes are not everywhere in good shape, the risks are determined by the weakest part in the floodplain sections. Thus a just safe (critical) level was determined for all floodplain sections, and it was supposed that a 140 cm higher flood automatically causes a dyke breachment. 6

7 Risk model Elöntési valószínûségek Localisation Probability of breaching the dikes was given by hydrologists, a distinction between partial and full inundation of the floodplain section was done. Valószínûség teljes részleges kritikus szint Árvíz magassága - a kritikus szint Even for larger dike failures it is usually possible to localise the flooding There are official contingency plans to confine inundation for the two largest floodplain sections Budapest-Baja Fegyvernek-Mesterszállás where secondary protection lines are marked by the plans For the others: a decrease of the probability for complete flooding in accordance with the area of the floodplain section was included into the model. The cities of Szolnok, Szeged, Győr are expected to get exceptional care, so their flooding is less likely (it is supposed to happen only if the peak level is higher by more than 140 cm than the level of the surrounding flood embankments/levees) Effect of dike breach The water level is decreased by a couple of centimetres in the neighbouring downstream stations. This effect depends on the size and topology of the inundated floodplain section. We used a simple model: the effect is 10cm/100km 2, which is halved every 50 km. Experiences In the last decades there were just a few breaching of dykes Bereg, 2001 being the most recent (where we had claim data from) These corresponded to extreme high floods (with estimated return period over 100 years) There were problems with the dykes around Szolnok during the recent floods. This validates our observation of this area being the most dangerous. Claim ratio It is possible to choose from different distributions (beta, or normal/t for the logit transform). The expected value and variance of the claim can be parametrised. The default value is the estimator we got by the available data (it depends on the wall type, floor etc.) The dependency between the building and content-losses is also included into the model. Here we used the copulas for finding suitable structure to a given correlation. Some parameters of the losses partial flooding total flooding building content building content stone adobe other These losses are supposed to be halved for 2 nd floor flats and further halved for higher levels. 7

8 Intensity It is widely used for modelling the destruction capability of an earthquake. It is convenient to use it on a continuous scale, but its definition is based on a discrete scale. III. Many people indoors feel movement. Hanging objects swing back and forth. IV. Most people indoors feel movement. Hanging objects swing. Dishes, windows, and doors rattle. The earthquake feels like a heavy truck hitting the walls. V. Almost everyone feels movement. Sleeping people are awakened. Doors swing open or close. Dishes are broken. Pictures on the wall move. VI. Everyone feels movement. People have trouble walking. Objects fall from shelves. Plaster in walls might crack. Damage is slight in poorly built buildings. VII. People have difficulty standing. Loose bricks fall from buildings. Damage is slight to moderate in well-built buildings; considerable in poorly built buildings. VIII. Drivers have trouble steering. Houses that are not bolted down might shift on their foundations. Tall structures such as towers and chimneys might twist and fall. Well-built buildings suffer slight damage. Poorly built structures suffer severe damage. Magnitude It is directly related to the energy of the quake. It is calculated on a logarithmic scale, so a magnitudeincrease by 1 means an energy-increase by , approximately a factor of We have used the following formula for the epicentral intensity i 0 : m=a i 0 +blog10(h)+c where m is the epicentral magnitude and h is the depth of the quake. The estimated value of the constants for Hungary are a=0.68, b=0.96, c= The quake-attenuation The following formula by Kövesligethy was used: i 3log( r h / h) 3 i r log( e)( r 0 h i r is the intensity at the distance of r from the epicentre h is the depth α is the absorption coefficient h ) The coefficient α α depends on the epicentral depth as follows: h (km) Absorption coefficient α ± ± ± ± ± The effect of the estimation error Effect of depth and distance As we saw, there is a substantial vagueness in the estimation of α Its effect is also important However, our simulation showed that we get at least the observed losses, so we do not suggest to use a more pessimistic version intenzitás Intenzitás-csökkenés a távolság függvényében h=4 km, becs h=16 km becs h=4 km alsó h=16 km alsó h=4 km felső h=16 km felső intenzitás Intenzitás-csökkenés a távolság függvényében h=0.5 km h=4 km h=16 km h=40 km távolság az epicentrumtól (km) távolság az epicentrumtól (km) 8

9 Depth distribution Modification h (km) prob h (km) prob relatív gyakoriság Mélységeloszlás illesztése Gamma eloszlással Historikus gyakoriság Illesztett eloszlás gyakorisága Illesztett eloszlás sűrűségfüggvénye Mélység (km) Better fit was achieved through the mixture of two Gamma distributions: relatív gyakoriság Mélységeloszlás illesztése keverék Gamma eloszlással Historikus gyakoriság Illesztett eloszlás gyakorisága Illesztett eloszlás sűrűségfüggvénye For the simulation, we needed a continuous distribution. First try: Gamma Mélység (km) The seismically active zones The parameters of the zones The figure shows the zones and the quakes from the last 100 years, with m>4. EOV(y) A megfigyelt nagy földrengések és a zónák EOV(x) number of zone max. magnitude freq (1/1000 y) number of zone max. magnitude freq (1/1000 y) Zone 2 is the most dangerous, as Budapest belongs to it. We investigate the magnitude distribution in it in some detail. The other zones behave similarly, so there is no need to repeat the procedure for all zones. Magnitude distribution The original distribution is shown on the figure. It is uniform over the intervals and the probability of the largest (half) interval looks to large. Modification is done by the extreme value magnitúdó theory (GPD). Magnitude distribution for zone 2, given by the seismologists gyakoriság A 2. zóna magnitúdó-gyakoriságai Estimation method We assume that there is a single shape parameter for all zones Usual maximum likelihood is very unstable (too few large observations) The possible maximum of magnitudes (given by the seismologists) can be considered as fixed, thus only the scale parameter has to be estimated. ksi Threshold selection is crucial A GPD alakparaméter változása a küszöb függvényében magnitúdó The change of the GPD shape parameter, as a function of the threshold 9

10 Possible shape parameters The frequencies for zone 2 The threshold of 4.4 A megfigyelt magnitúdó-eloszlás és az illesztett GPD magnitude looks reasonable (in this case the frequency of 4 full intervals can be ksi=-0.5 ksi=-0.4 ksi=-0.31 estimated by the GPD) Different, possible distributions are shown on the figure ξ=-0.4 was chosen magnitúdó Observed magnitude distribution and different GPD densities sűrűség The end point and the shape uniquely determines the distribution ξ=-0.4 gave a good fit interval original estimate ξ = E ξ = E ξ = observed length of the observed time period Checking the results Historical earthquakes We chose the largest intensity quakes from the cathalogue Year Month Day Latitude Longitude Depth Magn. Intens. Error code X E E E C X B C We simulated quakes in zone 2 by the model above, and compared it to the seismologist s calculations B B C B Historical simulations: methodology The magnitude, depth and location were random, within the given error. The code gives the possible location error, ranging from 2 km (X) to over 50 km (E). The magnitude has an error margin of 1, or even more for the older ones. 100 simulations were carried out for each quake Loss was generated by the algorithm above Historical simulations: results We see the results of the simulations (losses in thousands of HUF) in the figure. The quakes, possibly affecting Budapest, gave the highest 0e+00 2e+08 4e+08 6e+08 Historikus károk values

11 Highest simulated losses at the 1561 earthquake The losses grouped by settlements Quantitative results for the 3 highest simulations: Settlement Budapest_XI Budapest_VII Budapest_XIV Budapest_XIII By settlement Tatabánya Nyergesújfalu Alcsútdoboz Vértesacsa Esztergom Szentendre Pomáz Budapest_I. Budapest_I. Budapest_XI. Biatorbágy Budapest_XI. Ráckeresztúr Ercsi Nagymaros Érd Szigetszentmiklós Kiskunlacháza Ráckeve Building losses (Thousands of Millions of HUF) Dunakeszi Dömsöd Budapest_XVI. Budapest_XVI. Budapest_XV I. Content losses (Thousands of Millions of HUF) Gyál Total Vác Göd Bugyi Galgamácsa Gödölő Isaszeg Sülysáp Monor Csévharaszt Dabas Újlengyel Tura Kóka Tápióság Albertirsa Hatvan NA Quantile calculations by the historical simulations The difference in the observed period should be taken into account: having T years for a given quake-size, we chose 2000*n/T such quakes into the bootstrap sample. The results are in Thousands of Millions HUF. These simulations resulted in slightly smaller values that the seismic models. Mean 2.9 Variance % quantile (median) % quantile % quantile % quantile % quantile % quantile % quantile % quantile % quantile 376 Maximum 735 Convolution of the flood and the quake losses The simplest, and also the most accurate method was the simulation. The flood results are calculated for the wet years, including the effect of the climate changes. visszatérési szintek, Md Ft Az árvíz-és földrengéskárok összege szimuláltak összege földrengés árvíz év vissz.szint (Md Ft) év year return level (thousands of M HUF) 11