Climate change signal and uncertainty in projections of ocean wave heights

Abstract

In this study, projections of seasonal means and extremes of ocean wave heights were made using projections of sea level pressure fields conducted with three global climate models for three forcing-scenarios. For each forcing-scenario, the three climate models’ projections were combined to estimate the multi-model mean projection of climate change. The relative importance of the variability in the projected wave heights that is due to the forcing prescribed in a forcing-scenario was assessed on the basis of ensemble simulations conducted with the Canadian coupled climate model CGCM2. The uncertainties in the projections of wave heights that are due to differences among the climate models and/or among the forcing-scenarios were characterized. The results show that the multi-model mean projection of climate change has patterns similar to those derived from using the CGCM2 projections alone, but the magnitudes of changes are generally smaller in the boreal oceans but larger in the region nearby the Antarctic coastal zone. The forcing-induced variance (as simulated by CGCM2) was identified to be of substantial magnitude in some areas in all seasons. The uncertainty due to differences among the forcing-scenarios is much smaller than that due to differences among the climate models, although it was identified to be statistically significant in most areas of the oceans (this indicates that different forcing conditions do make notable differences in the wave height climate change projection). The sum of the model and forcing-scenario uncertainties is smaller in the JFM and AMJ seasons than in other seasons, and it is generally small in the mid-high latitudes and large in the tropics. In particular, some areas in the northern oceans were projected to have large changes by all the three climate models.

Appendix: analysis of variance

According to the one-factor ANOVA model (Eq. 4) (see Sect. 3.2), the total sum of squares of variable Y _ts :

$$ SS_{Y} = \sum\limits_{t} \sum\limits_{s} (Y_{ts}-Y_{oo})^2$$

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can be partitioned into two statistically independent variance components (Huitson 1966):

$$SS_{Y} = SS_{\beta} + SS_{\epsilon}, $$

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where

$$\begin{aligned} SS_{\beta} =& S \sum\limits_{t} (Y_{to}-Y_{oo})^{2} = S \sum\limits_{t} \beta^{2}_{t} + (n-1) \hat{\sigma}^{2}_{\epsilon},\\ SS_{\epsilon} =& \sum\limits_{t} \sum\limits_{s} (Y_{ts}-Y_{to})^{2} = n(S-1) \hat{\sigma}^{2}_{\epsilon},\\ \end{aligned}$$

and a “o” is used to replace a subscript when an arithmetic average is taken over that index (this notation is used throughout this section). The null hypothesis

$$ H_{\beta}: \beta_{t} = 0\;\hbox{for}\;t=1,2,\ldots,n $$

can be tested by comparing

$$ F_\beta = \frac{SS_{\beta}/(n-1)}{SS_{\epsilon}/[n(S-1)]}$$

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with the critical value F _p [(n−1), n(S−1)] where F _p [ν₁,ν₂] denotes the p-quantile of the F distribution with ν₁ and ν₂ degrees of freedom. And the proportion of the total variance in Y _ts that is due to the effects of factor β _t is estimated as

$$P_{\beta} = \frac{SS_{\beta} - \frac{n-1}{n(S-1)}SS_{\epsilon}}{SS_{Y}}.$$

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Similarly, according to the two-factor fixed effect ANOVA model (Eq. 5) (see Sect. 3.2), the total sum of squares of variable X _ijt :

$$ SS_{X} = \sum\limits_{i} \sum\limits_{j} \sum\limits_{t} (X_{ijt} - X_{ooo})^{2} $$

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can be partitioned into four statistically independent variance components (Huitson 1966):

$$ SS_{X} = SS_{\gamma} + SS_{\theta} + SS_{\delta} + SS_{\varepsilon}, $$

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where

$$\begin{aligned} SS_{\gamma} =& nq \sum\limits_{i} (X_{ioo}-X_{ooo})^{2} = nq\sum\limits_{i} \hat{\gamma}_{i}^{2} + (m-1) \hat{\sigma}^{2}_{\varepsilon},\\ SS_{\theta} =& n m \sum\limits_{j} (X_{ojo}-X_{ooo})^{2} = nm \sum\limits_{j} \hat{\theta}_{j}^{2} + (q-1) \hat{\sigma}^{2}_{\varepsilon},\\ SS_{\delta} =& n \sum\limits_{i} \sum\limits_{j} (X_{ijo} - X_{ojo} - X_{ioo} + X_{ooo})^{2}= n \sum\limits_{i} \sum\limits_{j} \hat{\delta}_{ij}^{2} + (m-1)(q-1) \hat{\sigma}^{2}_{\varepsilon},\\ SS_{\varepsilon} =& \sum\limits_{i} \sum\limits_{j} \sum\limits_{t}( X_{ijt}-X_{ijo})^{2} = mq(n-1)\hat{\sigma}^{2}_{\varepsilon}. \\ \end{aligned}$$

Therefore, the null hypothesis

$$ H_{\gamma}: \gamma_{i} = 0\;\hbox{for}\;i=1,2,\ldots,m $$

can be tested by comparing

$$ F_{\gamma} = \frac{SS_{\gamma}/(m-1)}{SS_{\varepsilon}/[mq(n-1)]}$$

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with the critical value F _p [(m−1), mq(n−1)]. And an unbiased estimator of the proportion of the total variance in X _ijt that is due to the effects of factor γ _i is

$$ P_{\gamma} = \frac{SS_{\gamma}-\frac{m-1}{mq(n-1)} SS_{\varepsilon}}{SS_{X}}. $$

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Similarly, the null hypothesis

$$ H_{\theta}: \theta_{j} = 0\;\hbox{for}\;j=1,2,\ldots,q $$

can be tested by comparing

$$ F_{\theta} = \frac{SS_{\theta}/(q-1)}{SS_{\varepsilon}/[mq(n-1)]}$$

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with the critical value F _p [(q−1), mq(n−1)]. And the proportion of the total variance in X _ijt that is due to the effects of factor θ _j is estimated as

$$ P_{\theta} = \frac{SS_{\theta} - \frac{q-1}{mq(n-1)} SS_{\varepsilon}}{SS_{X}}. $$

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The null hypothesis

$$ H_{\delta}: \delta_{ij} = 0\;\hbox{for}\;i=1,2,\ldots,m\;\hbox{and}\;j=1,2,\ldots,q $$

can be tested by comparing

$$ F_{\delta} = \frac{SS_{\delta}/[(m-1)(q-1)]}{SS_{\varepsilon}/[mq(n-1)]}$$

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with the critical value F _p [(m−1)(q−1), mq(n−1)]. And the proportion of the total variance in X _ijt that is due to the interaction effects δ _ij is estimated as

$$ P_{\delta} = \frac{SS_{\delta} - \frac{(m-1)(q-1)}{mq(n-1)} SS_{\varepsilon}}{SS_{X}}. $$

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Also, the null hypothesis

$$ H_{{\gamma + \theta + \delta}}: \gamma_{i} = \theta_{j} = \delta_{ij} = 0\;\hbox{for}\;i = 1,2,\ldots,m\;\hbox{and}\; j=1,2,\ldots,q $$

can be tested by comparing

$$ F_{{\gamma + \theta + \delta}} = \frac{(SS_{\gamma} + SS_{\theta} + SS_{\delta})/(mq-1)}{SS_{\varepsilon}/[mq(n-1)]} $$

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with the critical value F _p [(mq−1), mq(n−1)]. And the proportion of the total variance in quantity X _ijt that is due to the total effects of all factors (i.e., γ _i , θ _j , and δ _ij ) is estimated as

$$ P_{{\gamma + \theta + \delta}} = \frac{(SS_{\gamma} + SS_{\theta} + SS_{\delta}) - \frac{(mq-1)}{mq(n-1)} SS_{\varepsilon}}{SS_{X}}.$$

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