ERVA IMSC2016 presentation
Extreme value analysis of ocean waves in a changing climate 13th International Meeting on Statistical Climatology (IMSC) Canmore, Canada
Erik Vanem 09 June 2016
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Introduction and background Extreme value analysis of wave climate parameters is important for ocean and coastal engineering – Ships and other marine structures are exposed to environmental loads from wind and waves – Extreme conditions impose extreme loads and need to be accounted for in design and operation – Significant wave height is the dominating parameter in many applications Important question: Will extreme wave heights be affected by climate change?
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Extreme value modelling Typically, return values corresponding to long periods compared to the length of data is required, i.e. 20-year and 100-year extremes – Large uncertainties even without consideration of climate change Different approaches to extreme value modelling used in ocean engineering – Initial distribution approach – Peaks over threshold approach (POT) – Block maxima approach – ACER, (modified) Rice method, etc… How to account for climate change?
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Significant wave height data Three time-series of significant wave height, Hs, are used in this study – Historical period, RCP 4.5 and RCP 8.5 – Same location in the North Atlantic (59.28°N/11.36°W) – 3-hourly data covering 30 years each (1970 – 1999 and 2071 – 2100) – Generated by WAM model – With wind forcings from a global climate model (GFDL-CM3)
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The block maxima approach and the GEV model Extract block maxima from the data (e.g. annual maxima) Under certain assumptions, these block maxima will, asymptotically, follow the Generalized Extreme Value distribution, with cumulative distribution function
𝑥𝑥 − 𝜇𝜇 𝐺𝐺 𝑥𝑥; 𝜇𝜇, 𝜎𝜎, 𝜉𝜉 = 𝑒𝑒𝑒𝑒𝑒𝑒 − 1 + 𝜉𝜉 𝜎𝜎
−1�𝜉𝜉
Three model parameters – 𝜇𝜇 ∈ ℝ – 𝜎𝜎 > 0 – 𝜉𝜉 ∈ ℝ
(location parameter) (scale parameter) (shape parameter)
Three special cases of the GEV model depending on the value of the shape parameter: 𝜉𝜉 → 0 (𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺), 𝜉𝜉 > 0 (𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹) and 𝜉𝜉 < 0 (𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊)
May fit block maxima to this distribution and calculate return levels according to the fitted distribution – Results may be very sensitive to the estimated value of the shape parameter Ungraded
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Annual maximum Hs in each dataset 30 annual maxima and average annual maximum in each dataset
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Stationary extreme value models on separate data subsets Fitted stationary GEV models
Uncertainty estimates based on parametric bootstrap (B = 1000): – Confidence intervals overlap – stationary GEV models are not able to detect statistically significant shifts from historic to future wave climate
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Non-stationary extreme value models Implicit in the GEV modelling are assumptions of stationarity – Contradicts the idea of climate change May model the model parameters as a function of time – Time-dependence on either parameter
𝜇𝜇 𝑡𝑡 = 𝜇𝜇0 + 𝜇𝜇1 𝑡𝑡 ;
Three different approaches – Intra-period trends
– Inter-period linear trends – Inter-period shifts
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𝜎𝜎 𝑡𝑡 = 𝑒𝑒 𝜎𝜎0+𝜎𝜎1 𝑡𝑡 ;
𝜉𝜉 𝑡𝑡 = 𝜉𝜉0 + 𝜉𝜉1 𝑡𝑡
GEV with intra-period trends – location parameter 𝜇𝜇 𝑡𝑡 = 𝜇𝜇0 + 𝜇𝜇1 𝑡𝑡
Mean intra-period trend is decreasing for historical and RCP 4.5, increasing for RCP 8.5 Critical parameter is 𝜇𝜇1 - this is not significantly different from 0 in any dataset
– Likelihood-ratio tests with any reasonable significance levels would reject the alternative hypothesis of 𝜇𝜇1 ≠ 0 – Inclusion of a linear trend would not improve the modelling
Similar results for trends in the scale and shape parameters
Stationary assumption probably OK within each 30-year period of data
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GEV with inter-period linear trends Introduce the covariate t for actual time in year and let it assume values according to the years of the annual maxima
𝑡𝑡 = 1, 2, … , 30, 102, 103, … , 131 1970-1999
2071 - 2100
Fit stationary and non-stationary GEV models to the joint historical and projected data (for RCP 4.5 and RCP 8.5 separately) – Non-stationary location, log-scale and both For RCP 4.5: All non-stationary models rejected, no significant trends – Likelihood ratio tests yield p-values of 0.34, 0.62 and 0.33 For RCP 8.5: Non-stationary model with non-stationary location preferred – P-values of 0.0039, 0.97 and 0.015 – Testing full non-stationary model vs. non-stationary location yields p = 0.93 Ungraded
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GEV with inter-period shifts Combine data from all datasets and introduce the categorical variables Y1 and Y2
Introduce a linear function of these variables as follows
First, fit a stationary GEV model to the joint data without consideration of which subset they belong to Then, fit non-stationary models with an inter-period shift in – Location parameter only – Scale parameters only – Both location and scale parameters Ungraded
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GEV with inter-period shifts – estimated density functions
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GEV models with inter-period shifts - results Statistically significant positive shift in the location parameter for both future scenarios Shifts in the scale parameter not statistically significant Likelihood ratio tests yield p = 0.011, 0.46 and 0.025 for H0: stationary vs. H1: non-stationary location, scale and both, respectively Models with non-stationary location will be preferred Now, a statistically significant change in the wave climate is detected, which was not detected by stationary GEV models for each dataset separately
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Effect of inter-period linear trend on return values
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Summary and conclusions For these data, assuming a stationary model within each subset of data is defendable The non-stationary model with an inter-period shift in some of the parameters is a good alternative to modelling the data separately for each subset of data – More data available for estimation of common parameters -> less variance – Beneficial for the estimation of the shape parameter,
𝜉𝜉
– Able to detect statistically significant shifts that are not detected by separate stationary models – e.g. caused by climate change Non-stationary models with inter-period trends may be used for pairwise combinations of historical and future data – May estimate “return values” for any year between the data periods by interpolation
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Acknowledgements This work was carried out within the research project ExWaCli, partly funded by the Norwegian Research Council. The data was generated within the project by the Norwegian Meteorological Institute Thanks to Dr. Elzbieta Bitner-Gregersen (DNV-GL) for cooperation and support
Reference Vanem, Erik (2015). Non-stationary extreme value models to account for trends and shifts in the extreme wave climate due to climate change. Applied Ocean Research 52, pp. 201-211
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DNV GL © 2016
09 June 2016
Contact info: [email protected] +47 67 57 99 00
www.dnvgl.com
SAFER, SMARTER, GREENER Ungraded
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DNV GL © 2016
09 June 2016
Erik Vanem 09 June 2016
Ungraded
1
DNV GL © 2016
09 June 2016
SAFER, SMARTER, GREENER
Introduction and background Extreme value analysis of wave climate parameters is important for ocean and coastal engineering – Ships and other marine structures are exposed to environmental loads from wind and waves – Extreme conditions impose extreme loads and need to be accounted for in design and operation – Significant wave height is the dominating parameter in many applications Important question: Will extreme wave heights be affected by climate change?
Ungraded
2
DNV GL © 2016
09 June 2016
Extreme value modelling Typically, return values corresponding to long periods compared to the length of data is required, i.e. 20-year and 100-year extremes – Large uncertainties even without consideration of climate change Different approaches to extreme value modelling used in ocean engineering – Initial distribution approach – Peaks over threshold approach (POT) – Block maxima approach – ACER, (modified) Rice method, etc… How to account for climate change?
Ungraded
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DNV GL © 2016
09 June 2016
Significant wave height data Three time-series of significant wave height, Hs, are used in this study – Historical period, RCP 4.5 and RCP 8.5 – Same location in the North Atlantic (59.28°N/11.36°W) – 3-hourly data covering 30 years each (1970 – 1999 and 2071 – 2100) – Generated by WAM model – With wind forcings from a global climate model (GFDL-CM3)
Ungraded
4
DNV GL © 2016
09 June 2016
The block maxima approach and the GEV model Extract block maxima from the data (e.g. annual maxima) Under certain assumptions, these block maxima will, asymptotically, follow the Generalized Extreme Value distribution, with cumulative distribution function
𝑥𝑥 − 𝜇𝜇 𝐺𝐺 𝑥𝑥; 𝜇𝜇, 𝜎𝜎, 𝜉𝜉 = 𝑒𝑒𝑒𝑒𝑒𝑒 − 1 + 𝜉𝜉 𝜎𝜎
−1�𝜉𝜉
Three model parameters – 𝜇𝜇 ∈ ℝ – 𝜎𝜎 > 0 – 𝜉𝜉 ∈ ℝ
(location parameter) (scale parameter) (shape parameter)
Three special cases of the GEV model depending on the value of the shape parameter: 𝜉𝜉 → 0 (𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺), 𝜉𝜉 > 0 (𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹) and 𝜉𝜉 < 0 (𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊)
May fit block maxima to this distribution and calculate return levels according to the fitted distribution – Results may be very sensitive to the estimated value of the shape parameter Ungraded
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DNV GL © 2016
09 June 2016
Annual maximum Hs in each dataset 30 annual maxima and average annual maximum in each dataset
Ungraded
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DNV GL © 2016
09 June 2016
Stationary extreme value models on separate data subsets Fitted stationary GEV models
Uncertainty estimates based on parametric bootstrap (B = 1000): – Confidence intervals overlap – stationary GEV models are not able to detect statistically significant shifts from historic to future wave climate
Ungraded
7
DNV GL © 2016
09 June 2016
Non-stationary extreme value models Implicit in the GEV modelling are assumptions of stationarity – Contradicts the idea of climate change May model the model parameters as a function of time – Time-dependence on either parameter
𝜇𝜇 𝑡𝑡 = 𝜇𝜇0 + 𝜇𝜇1 𝑡𝑡 ;
Three different approaches – Intra-period trends
– Inter-period linear trends – Inter-period shifts
Ungraded
8
DNV GL © 2016
09 June 2016
𝜎𝜎 𝑡𝑡 = 𝑒𝑒 𝜎𝜎0+𝜎𝜎1 𝑡𝑡 ;
𝜉𝜉 𝑡𝑡 = 𝜉𝜉0 + 𝜉𝜉1 𝑡𝑡
GEV with intra-period trends – location parameter 𝜇𝜇 𝑡𝑡 = 𝜇𝜇0 + 𝜇𝜇1 𝑡𝑡
Mean intra-period trend is decreasing for historical and RCP 4.5, increasing for RCP 8.5 Critical parameter is 𝜇𝜇1 - this is not significantly different from 0 in any dataset
– Likelihood-ratio tests with any reasonable significance levels would reject the alternative hypothesis of 𝜇𝜇1 ≠ 0 – Inclusion of a linear trend would not improve the modelling
Similar results for trends in the scale and shape parameters
Stationary assumption probably OK within each 30-year period of data
Ungraded
9
DNV GL © 2016
09 June 2016
GEV with inter-period linear trends Introduce the covariate t for actual time in year and let it assume values according to the years of the annual maxima
𝑡𝑡 = 1, 2, … , 30, 102, 103, … , 131 1970-1999
2071 - 2100
Fit stationary and non-stationary GEV models to the joint historical and projected data (for RCP 4.5 and RCP 8.5 separately) – Non-stationary location, log-scale and both For RCP 4.5: All non-stationary models rejected, no significant trends – Likelihood ratio tests yield p-values of 0.34, 0.62 and 0.33 For RCP 8.5: Non-stationary model with non-stationary location preferred – P-values of 0.0039, 0.97 and 0.015 – Testing full non-stationary model vs. non-stationary location yields p = 0.93 Ungraded
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DNV GL © 2016
09 June 2016
GEV with inter-period shifts Combine data from all datasets and introduce the categorical variables Y1 and Y2
Introduce a linear function of these variables as follows
First, fit a stationary GEV model to the joint data without consideration of which subset they belong to Then, fit non-stationary models with an inter-period shift in – Location parameter only – Scale parameters only – Both location and scale parameters Ungraded
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DNV GL © 2016
09 June 2016
GEV with inter-period shifts – estimated density functions
Ungraded
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DNV GL © 2016
09 June 2016
GEV models with inter-period shifts - results Statistically significant positive shift in the location parameter for both future scenarios Shifts in the scale parameter not statistically significant Likelihood ratio tests yield p = 0.011, 0.46 and 0.025 for H0: stationary vs. H1: non-stationary location, scale and both, respectively Models with non-stationary location will be preferred Now, a statistically significant change in the wave climate is detected, which was not detected by stationary GEV models for each dataset separately
Ungraded
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DNV GL © 2016
09 June 2016
Effect of inter-period linear trend on return values
Ungraded
14
DNV GL © 2016
09 June 2016
Summary and conclusions For these data, assuming a stationary model within each subset of data is defendable The non-stationary model with an inter-period shift in some of the parameters is a good alternative to modelling the data separately for each subset of data – More data available for estimation of common parameters -> less variance – Beneficial for the estimation of the shape parameter,
𝜉𝜉
– Able to detect statistically significant shifts that are not detected by separate stationary models – e.g. caused by climate change Non-stationary models with inter-period trends may be used for pairwise combinations of historical and future data – May estimate “return values” for any year between the data periods by interpolation
Ungraded
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DNV GL © 2016
09 June 2016
Acknowledgements This work was carried out within the research project ExWaCli, partly funded by the Norwegian Research Council. The data was generated within the project by the Norwegian Meteorological Institute Thanks to Dr. Elzbieta Bitner-Gregersen (DNV-GL) for cooperation and support
Reference Vanem, Erik (2015). Non-stationary extreme value models to account for trends and shifts in the extreme wave climate due to climate change. Applied Ocean Research 52, pp. 201-211
Ungraded
16
DNV GL © 2016
09 June 2016
Contact info: [email protected] +47 67 57 99 00
www.dnvgl.com
SAFER, SMARTER, GREENER Ungraded
17
DNV GL © 2016
09 June 2016
