# Global Sea Surface Temperature and Sea Level Rise Estimation with Optimal Historical Time Lag Data

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Systems Model with Time Lag

_{j,t}, b

_{j,t}and c

_{j}are time lag coefficients which are time lag period dependent. In this model, SST and sea level are the state variables of the system. The discrete solution of this model can be represented as,

_{j,k−i}is the coefficient associated with global SST at previous year(s) i and reflects the impact of global SST in previous year(s) i on current global SST and sea level, b

_{j,k−i}is the coefficient associated with global sea level at previous year(s) i and reflects the impact of the global sea level in previous year(s) i on the current global SST and sea level, c

_{j}is a constant and j = 1, 2. In Equation (2), n prior years is considered to give the equation in a general form. In application, fifteen years are considered and the optimum value of eight years is predicted to give the best results in an optimum sense. We define

**X**(k) is the state vector in year k,

**A**

_{k−i}is a (2 × 2) matrix for i = 1, …, n,

**C**is a (2 × 1) vector. In this model, the system behavior depends on interaction of global SST and sea level in previous n years. This model is an extension of the DSM that was proposed earlier, or the original DSM is a special case of Equation (4) where n = 1 [16].

## 3. Calibration of DSM with Time Lag

**Y**is a 2 × (N − n) matrix,

**B**is a (N − n) × (2n + 1) matrix,

**P**is 2 × (2n + 1) matrix. Applying least squares method, we can identify the parameters of the n time lag matrix coefficients as,

**A**

_{k−i}(k) and

**C**(k) are matrices for i = 1, …, n and vector for year k. It is important to notice that as we conduct this analysis in the year 2013; currently, there is no new data added to the dataset for 2014 and beyond. Thus, to demonstrate this approach, we will use a synthetic data generation process and add the data predicted for year (k + 1) into our dataset to form the new dataset and move on with our computation sequence as described above. We recognize that, in this case, the predicted data we have added to the dataset is not a measured data and will include modeling errors. Thus, the reader needs to recognize that the results obtained for this case will include the propagation of modeling errors over time which will render the one hundred year predictions less reliable. The synthetic data generation process is included here to demonstrate the time-variant modeling approach that is described in this study.

_{p}is the term accounting for the error propagation in dynamic prediction, given by

**X**

_{p}(i) is a (2n + 1) vector used in prediction, given by,

## 4. Determination of Optimal Time Lag

^{2}). We may combine both performance indexes to determine the optimal maximum time lag. The RMSE is a measure of the differences between values predicted by the model and the actual observations [22]. It is defined as,

_{T}and RMSE

_{H}are the root mean square errors for temperature and SLR, and T(k) and H(k) are the historical measurements of temperature and SLR at time step k. The coefficient of determination R

^{2}is a measure of how well future outcomes are likely to be predicted by the model [23]. The R

^{2}is defined as,

^{2}is to one, the better the linear regression fits the data in comparison to the simple average. In model predictions, one wants the proposed model to generate a smaller RMSE but a larger R

^{2}.

^{2}for SST and SLR can be calculated. The RMSE and R

^{2}for each time lag are divided by their maximum values to transfer all RMSE and R

^{2}to an interval [0, 1], and the optimal maximum time lag n should have a minimum value for the performance index given by

^{2}for global SST and SLR, respectively. Equation (16) is a synthetic performance index that reflects the requirements for optimal maximum time lag. In this sense, the model with optimal maximum time lag is identified as the optimal dynamic system time lag model.

## 5. Numerical Results and Discussion

#### 5.1. Time-Invariant DSM (TI-DSM) Application

^{2}are calculated and optimal maximum time lag is determined using Equation (16). This analysis resulted in an optimal time lag of eight years, n* = 8. The coefficients for the corresponding system are given in Table 1 for this case. Since none of these coefficients is zero, the results show that the future global SST and sea levels not only depend on their states in the last year, but also are affected by SST and sea level states in previous 8 years.

^{2}s are 0.71 and 0.98 for global SST and SLR respectively. In comparison to the results obtained in [16], the reconstruction accuracy using this model has been improved. The R

^{2}s obtained for the reconstruction period, in particular, show a significant increase. This implies that the future predictions for global SST and SLR obtained from this model are likely to have higher accuracy.

^{2}was 0.7148 and 0.9773 for SST and SLR, respectively. These values are almost identical to the results obtained from the model identified using the complete dataset from 1880 to 2001.

#### 5.2. Time-Variant DSM (TV-DSM) Application

**A**

_{k−i}(k) and

**C**(k) at k with years before 2002 are the same as those in the time-invariant DSM as given in Table 1. These matrices are then recalculated year by year from year 2002 forward using the moving data window as described above to form a time-variant dynamic system (TV-DSM). The global SST change and SLR in 21st century using the TV-DSM are shown in Figure 3a,b. The results show that by the end of the 21st century, the SST will reach 2.0 °C with a 90% confidence interval [1.5, 2.5] °C while the sea level will rise to 66.6 cm with a 90% confidence interval [63.2, 69.9] cm. These predictions are much closer to the projections of Scenario B1 [3]. The 90% confidence intervals estimated by Equation (11) are shown as dashed lines in Figure 3. This is similar to the case of the TI-DSM where the widths of the confidence intervals gradually increase as prediction time moves forward. When comparing the results shown in Figure 2 and Figure 3, we may observe two points: (i) the predicted results using the TV-DSM are higher than those obtained using the TI-DSM; and, (ii) the confidence intervals using the TV-DSM are narrower than those obtained using the TI-DSM. These observations indicate that using the TV-DSM and thus new information in system recognition, may improve the prediction reliability although it is synthetic in this application. We must re-emphasize here that the predictions made for this case are based on the synthetic data generation process described earlier.

#### 5.3. Applications Using IPCC Scenarios

_{2}) [2]. In order to assess the impact of greenhouse gas emissions on global warming, IPCC have developed six emissions scenarios based on different patterns of economic development, industrial development, and population growth in the future and modeled physical processes [2,3]. These emission scenarios are labeled A1FI, A1B, A1T, A2, B1 and B2 [2]. Among these scenarios, A1FI and A2 represent the highest emission of greenhouse gases into the atmosphere, A1T and B1 represent the least emissions, and A1B and B2 represent moderate emissions. The IPCC projected the global mean surface temperature dynamic processes from 1990 to 2100 for the six scenarios using the climate models (IPCC, 2007), which are shown in Figure 4. Accordingly, the temperature increase at the end of the 21st century spans from 2 to 4.5 °C depending on which scenario occurs.

## 6. Conclusions

^{2}s of 0.71 and 0.98 for global SST and SLR, respectively, in calibration, which shows an improvement when compared with [16]. The optimal DSM was applied to predict global SST and SLR in the 21st century, and the results show that the global SST will reach 1.9 °C with a 90% confidence interval [0.6, 3.2] °C and the sea level will rise to 56.1 cm with a 90% confidence interval [46.9, 65.2] cm by the end of the 21st century relative to 1990. In order to assess the impact of greenhouse gas emissions on SLR, the second equation of the optimal model was used to project the SLR in the 21st century while the temperature rise of six emission scenarios, simulated by IPCC (2001), is used as the model input. For these cases, the resulting SLR at the end of the 21st century ranges from 61 to 132 cm. The prediction is consistent but higher than the predictions made in previous studies [10,12,13,15,25,26]. The analysis was performed using both TI-DSM and TV-DSM approaches with similar results obtained for each case with the TV-DSM outcome showing higher estimates. Inundation analysis of coastal regions can be evaluated based on the outcome of these predictions as demonstrated in [20].

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Reconstructed numerical results using the TI-DSM with the optimal time lag of 8 years. (

**a**) Global temperature; (

**b**) Sea level rise.

**Figure 2.**Predicted numerical results for the 21st century using the TI-DSM with the optimal time lag of 8 years. (

**a**) Global temperature; (

**b**) Sea level rise.

**Figure 3.**Predicted numerical results for the 21st century using the TV-DSM with the optimal time lag of 8 years. (

**a**) Global temperature; (

**b**) Sea level rise.

**Figure 4.**Temperature change processes simulated by IPCC, based on six greenhouse gas emission scenarios [2].

Time Lag | Matrix |
---|---|

1 | ${A}_{k-1}=\left[\begin{array}{cc}0.4700& 0.0038\\ 0.5701& 0.6372\end{array}\right]$ |

2 | ${A}_{k-2}=\left[\begin{array}{cc}-0.0412& -0.0016\\ -0.0147& 0.0598\end{array}\right]$ |

3 | ${A}_{k-3}=\left[\begin{array}{cc}0.0668& -0.0107\\ -0.59781& 0.1959\end{array}\right]$ |

4 | ${A}_{k-4}=\left[\begin{array}{cc}0.2703& -0.0078\\ 0.8140& 0.0470\end{array}\right]$ |

5 | ${A}_{k-5}=\left[\begin{array}{cc}-0.1380& 0.0042\\ -0.9916& -0.0347\end{array}\right]$ |

6 | ${A}_{k-6}=\left[\begin{array}{cc}0.1918& 0.0060\\ 0.8953& 0.0033\end{array}\right]$ |

7 | ${A}_{k-7}=\left[\begin{array}{cc}0.0061& 0.0144\\ 0.1940& -0.0347\end{array}\right]$ |

8 | ${A}_{k-8}=\left[\begin{array}{cc}0.0466& -0.0016\\ 0.2314& 0.1115\end{array}\right]$ |

$C=\left[\begin{array}{c}0.0217\\ 0.7023\end{array}\right]$ |

**Table 2.**System matrices and vector obtained from 10-fold validation for the optimal time lag ${n}^{*}$ = 8.

Time Lag | Matrix |
---|---|

1 | ${A}_{k-1}=\left[\begin{array}{cc}0.4667& 0.0053\\ 0.5543& 0.6342\end{array}\right]$ |

2 | ${A}_{k-2}=\left[\begin{array}{cc}-0.0446& -0.0023\\ 0.0009& 0.0591\end{array}\right]$ |

3 | ${A}_{k-3}=\left[\begin{array}{cc}0.0769& -0.0110\\ -0.6084& 0.1977\end{array}\right]$ |

4 | ${A}_{k-4}=\left[\begin{array}{cc}0.2616& -0.0077\\ 0.8255& 0.0473\end{array}\right]$ |

5 | ${A}_{k-5}=\left[\begin{array}{cc}-0.1344& 0.0032\\ -0.9993& -0.0320\end{array}\right]$ |

6 | ${A}_{k-6}=\left[\begin{array}{cc}0.1888& 0.0074\\ 0.9011& -0.0002\end{array}\right]$ |

7 | ${A}_{k-7}=\left[\begin{array}{cc}0.0096& 0.0133\\ 0.1861& -0.0331\end{array}\right]$ |

8 | ${A}_{k-8}=\left[\begin{array}{cc}0.0485& -0.0015\\ 0.2163& 0.1132\end{array}\right]$ |

$C=\left[\begin{array}{c}0.0219\\ 0.6979\end{array}\right]$ |

Scenario | Sea Level Rise (cm in 2100 Relative to 1990) | |||
---|---|---|---|---|

Best Estimate | 90% Confidence Interval | Rahmstorf’s Projections [10] | IPCC Projections [2,3] | |

A1FI | 110.2 | [88.7, 131.7] | 102.1 | [26, 59] |

A1B | 89.9 | [73.9, 106.0] | 84.4 | [21, 48] |

A1T | 90.1 | [73.8, 106.4] | 84.7 | [20, 45] |

A2 | 92.6 | [75.9, 109.3] | 87.2 | [23, 51] |

B1 | 73.2 | [60.6, 85.8] | 70.0 | [18, 38] |

B2 | 83.7 | [69.2, 98.1] | 79.5 | [20, 43] |

Scenario | Sea Level Rise (cm in 2100 Relative to 1990) | |||
---|---|---|---|---|

Best Estimate | 90% Confidence Interval | Rahmstorf’s Projections [10] | IPCC Projections [2,3] | |

A1FI | 136.4 | [119.9, 152.9] | 102.1 | [26, 59] |

A1B | 102.4 | [93.0, 111.7] | 84.4 | [21, 48] |

A1T | 98.1 | [87.0, 109.1] | 84.7 | [20, 45] |

A2 | 112.7 | [101.5, 123.7] | 87.2 | [23, 51] |

B1 | 77.7 | [69.6, 85.7] | 70.0 | [18, 38] |

B2 | 93.3 | [85.7, 100.9] | 79.5 | [20, 43] |

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Aral, M.M.; Guan, J.
Global Sea Surface Temperature and Sea Level Rise Estimation with Optimal Historical Time Lag Data. *Water* **2016**, *8*, 519.
https://doi.org/10.3390/w8110519

**AMA Style**

Aral MM, Guan J.
Global Sea Surface Temperature and Sea Level Rise Estimation with Optimal Historical Time Lag Data. *Water*. 2016; 8(11):519.
https://doi.org/10.3390/w8110519

**Chicago/Turabian Style**

Aral, Mustafa M., and Jiabao Guan.
2016. "Global Sea Surface Temperature and Sea Level Rise Estimation with Optimal Historical Time Lag Data" *Water* 8, no. 11: 519.
https://doi.org/10.3390/w8110519