# Sensitivity Analysis of Arctic Sea Ice Extent Trends and Statistical Projections Using Satellite Data

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

_{i}, i = 1, … 5 were optimized by minimizing the sum of squared errors between the model and data via a simplex-based search approach. The Simplex method is a widely used procedure for linear programming, first proposed by Dantzig in 1947 [16]. It can be used to solve efficiently the optimization of a multivariate cost function by employing a sequence of simplices. Nelder and Mead [17] presented a Simplex algorithm for nonlinear unconstrained optimization. The built-in Nelder–Mead Simplex Method Matlab function from [18] has been utilized in this analysis.

## 3. Results

#### 3.1. Sensitivity to Averaging Methods and Averaging Intervals

#### 3.2. Sensitivity of Arctic Ice-Free Projection to Time Domain of Linear Regression

^{6}km

^{2}/decade) (Table 2 and Figure 3). Assuming this trend is persistent, the Arctic summer is projected to be ice-free after 2058 (zero-crossing at year 2069). The linear trend from 1996 to 2015 (the last 20 years of the available time series) is about −1.47 (10

^{6}km

^{2}/decade), which projects the Arctic summer to be ice-free after 2036, in less than 20 years (zero-crossing at year 2043). In comparison, the trend for the current climate normal period (1981–2010) is −0.82 (10

^{6}km

^{2}/decade), which would result in Arctic ice-free summer beginning at year 2062 (zero-crossing at 2074). The trend of the climate normal period is slightly slower than that based on the trend from the whole record period (1979–2015) as the time period for the climate normal does not include the record low in 2012. On the other hand, the linear trend from the first 20 years of the time series (1979–1998) projects 2147 for ice-free and 2174 for zero-crossing. Therefore, it seems that the accelerated reduction of sea ice extent in the last 20 years (1996–2015) has put the prospect of an ice-free Arctic summer well into our near future. Although all of the above linear trends for the annual SIE minimums are significant at the 99% confidence level, the question is whether any of these linear trends will persist in the future. There is a drastic jump in decadal trends between the first 20 and last 20 years (−0.38 vs. −1.47 10

^{6}km

^{2}/decade). The acceleration, even in the linear sense, is very distinct and is reflected in the ice-free projections shown in Figure 3. If this trend-acceleration continues, the FIASY could continue to move closer to the present day.

#### 3.3. Sensitivity of Different Statistical Curve Fitting Functions

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Time series (solid) of annual Arctic sea ice extent (SIE) maximums (left panels) and minimums (right panels) and their linear trends (dashed) for moving (red) and subsetting (blue) averaging methods. Plots for different time intervals (5-, 7-, 10-, and 30-day) are shown, along with the percentage change of decadal trends. The five-day moving average is a widely adopted method in the cryosphere community, and the 30-day subsetting average is similar to a monthly average.

**Figure 2.**Time series (solid) of annual Arctic SIE maximums (left panels) and minimums (right panels) and their linear trends (dashed) for moving average (top two panels) and subsetting average (bottom two panels) methods for different averaging intervals (5-, 10-, and 30-day). The percentage changes of decadal trends from the 10- and 30-day averages relative to that from the five-day averages are displayed.

**Figure 3.**(

**a**) Linear regression trend lines based on different subsets of the annual Arctic SIE minimum time series; and (

**b**) Arctic ice-free projections based on these linear trends. The annual Arctic SIE minimum values from five-day moving average based on daily sea ice concentrations are denoted by black circles in (

**a**) and black solid line in (

**b**). Different color and line styles in (

**a**,

**b**) denote the linear regression trend lines and the FIASY projections based on the linear trends for the period of first 20 years (1979–1998; green dotted), first 30 years (1979–2008; cyan dash-dotted), climate normal period (1981–2010; thick blue dashed), all 37 years (1979–2015; black dashed), last 30 years (1986–2015; magenta dash-dotted), and last 20 years (1996–2015; red dotted), respectively. Open circles in (

**b**) denote the end of data fitting and the start of the projection. The horizontal black dotted line in (

**b**) denotes the ice-free threshold of one million square kilometers. The zero-crossing of Arctic SIE minimums is denoted by “0+”, which represents a state in this analysis that all data grid cells in the Arctic Ocean have less than 15% sea ice area fraction.

**Figure 4.**Model predictions of annual SIE minimums using different sub-periods of the dataset for training. Circles denote data values used for model training and the predictions over the training set are denoted by solid lines. Plus-signs denote data points excluded from the fitting and dotted lines are the extrapolated predictions based on the resultant fitting. (

**a**) First 20 years (1979–1998); (

**b**) first 30 years (1979–2008); (

**c**) climate normal (1981–2010); (

**d**) the whole record period (1979–2015); (

**e**) last 30 years (1986–2015); and (

**f**) the last 20 years (1996–2015).

**Figure 5.**W–Akaike weights: (

**a**) for each data period grouped by statistical models; and (

**b**) for each statistical model grouped by data periods.

**Figure 6.**The projected FIASY by six statistical models grouped by time domain of calibration: the first 30 years (1979–2008), the climate normal period (1980–2010), the whole record period (1979–2015), and the last 30 years (1986–2015), respectively. The long-dashed lines denote the minimum and maximum of the projections for each record period. The short-dashed lines denote the mean of the projections for each record period. The values bounded by solid arrow lines denote the spread of the projections. The means and standard deviations of the projected FIASY for each period are noted at the top of the graph.

**Table 1.**SIE decadal linear trends (10

^{6}km

^{2}/decade) from different averaging methods and intervals for the period of 1979–2015, along with their percentage changes relative to those of the original daily SIE time series.

Original | Moving Average Interval (in Days) | Subsetting Average Intervals (in Days) | |||||||
---|---|---|---|---|---|---|---|---|---|

5 | 7 | 10 | 30 | 5 | 7 | 10 | 30 | ||

SIE Decadal Trends in Annual Maximum (10 ^{6} km^{2}/decade) | −0.346 | −0.353 | −0.35 | −0.346 | −0.332 | −0.348 | −0.342 | −0.345 | −0.325 |

Percentage Change to Original (%) | 0 | −2.02 | −1.16 | 0.00 | 4.05 | −0.58 | 1.16 | 0.29 | 6.07 |

SIE Decadal Trends in Annual Minimum (10 ^{6} km^{2}/decade) | −0.868 | −0.866 | −0.868 | −0.869 | −0.861 | −0.867 | −0.87 | −0.876 | −0.867 |

Percentage Change to Original (%) | 0 | 0.23 | 0.00 | −0.12 | 0.81 | 0.12 | −0.23 | −0.92 | 0.12 |

Case ID | Data Period | Trend (10 ^{6} km^{2}/Decade) | Margin of Error (10 ^{6} km^{2}/Decade) * | First Ice-Free Summer (Year) | Zero-Crossing (Year) |
---|---|---|---|---|---|

First 20 years | 1979–1998 | −0.38 | 0.19 | 2147 | 2174 |

First 30 years | 1979–2008 | −0.74 | 0.29 | 2069 | 2083 |

Climate Normal | 1981–2010 | −0.82 | 0.32 | 2062 | 2074 |

All years | 1979–2015 | −0.87 | 0.30 | 2058 | 2069 |

Last 30 years | 1986–2015 | −1.06 | 0.41 | 2048 | 2057 |

Last 20 years | 1996–2015 | −1.47 | 0.73 | 2036 | 2043 |

*****The margin of error in a confidence interval denotes the maximum expected difference between the true and a sample estimate of the statistic. It is proportional to the standard error of the statistic. The margin of error for the linear trend at the 95% confidence level is computed as: ~2 * StdErr, where StdErr denotes the standard error of the linear regression slope.

**Table 3.**Metrics from model optimization. Entries in bold indicate the best fit among the six models for the time period.

Model | Exponential | Gompertz | Log | Quadratic | Linear | Linear w Lag | |
---|---|---|---|---|---|---|---|

Period | |||||||

1979–1998 (first 20 years) | RMSE (in) (10 ^{6} km^{2}) | 0.32 | 0.32 | 0.32 | 0.32 | 0.32 | 0.32 |

RMSE (out) (10 ^{6} km^{2}) | 0.83 | 0.85 | 0.84 | 0.98 | 0.83 | 0.82 | |

AICc [(10 ^{6} km^{2})^{2}] | −78.61 | −78.62 | −78.46 | −78.69 | −80.99 | −73.39 | |

W [unitless] | 0.14 | 0.14 | 0.13 | 0.14 | 0.45 | 0.010 | |

1979–2008 (first 30 years) | RMSE (in) (10 ^{6} km^{2}) | 0.39 | 0.39 | 0.46 | 0.41 | 0.47 | 0.41 |

RMSE (out) (10 ^{6} km^{2}) | 1.23 | 0.85 | 0.32 | 0.39 | 0.35 | 0.38 | |

AICc [(10 ^{6} km^{2})^{2}] | −62.99 | −62.54 | −50.30 | −58.57 | −51.06 | −53.18 | |

W [unitless] | 0.52 | 0.42 | 0.00091 | 0.057 | 0.0013 | 0.0039 | |

1981–2010 (climate normal) | RMSE (in) (10 ^{6} km^{2}) | 0.40 | 0.40 | 0.45 | 0.40 | 0.46 | 0.40 |

RMSE (out) (10 ^{6} km^{2}) | 0.52 | 0.46 | 0.32 | 0.37 | 0.34 | 0.36 | |

AICc [(10 ^{6} km^{2})^{2}] | −61.25 | −61.43 | −52.25 | −60.40 | −52.79 | −55.10 | |

W [unitless] | 0.36 | 0.39 | 0.0039 | 0.23 | 0.0052 | 0.016 | |

1979–2015 (all years) | RMSE (in) (10 ^{6} km^{2}) | 0.52 | 0.52 | 0.55 | 0.52 | 0.57 | 0.51 |

RMSE (out) (10 ^{6} km^{2}) | N/A | N/A | N/A | N/A | N/A | N/A | |

AICc [(10 ^{6} km^{2})^{2}] | −41.26 | −41.83 | −38.16 | −42.15 | −37.69 | −37.22 | |

W [unitless] | 0.23 | 0.31 | 0.034 | 0.36 | 0.039 | 0.031 | |

1986–2015 (last 30 years) | RMSE (in) (10 ^{6} km^{2}) | 0.50 | 0.49 | 0.50 | 0.49 | 0.50 | 0.49 |

RMSE (out) (10 ^{6} km^{2}) | 0.22 | 0.20 | 0.31 | 0.18 | 0.36 | 0.18 | |

AICc [(10 ^{6} km^{2})^{2}] | −45.31 | −45.64 | −44.66 | −45.58 | −46.23 | −40.76 | |

W [unitless] | 0.17 | 0.21 | 0.13 | 0.20 | 0.28 | 0.018 | |

1996–2015 (last 20 years) | RMSE (in) (10 ^{6} km^{2}) | 0.41 | 0.40 | 0.41 | 0.38 | 0.41 | 0.41 |

RMSE (out) (10 ^{6} km^{2}) | 0.96 | 1.57 | 0.95 | 2.62 | 0.96 | 0.93 | |

AICc [(10 ^{6} km^{2})^{2}] | −59.51 | −61.75 | −59.48 | −64.35 | −61.88 | −54.15 | |

W [unitless] | 0.051 | 0.16 | 0.050 | 0.57 | 0.17 | 0.0035 |

**Table 4.**Projected FIASYs from optimized statistical models. Entries in bold indicate projections from models with the best fit (see AICc and W information from Table 3).

Case ID | Data Period | Exponential | Gompertz | Log | Quadratic | Linear | Linear w Lag |
---|---|---|---|---|---|---|---|

First 20 years | 1979–1998 | >2100 | >2100 | 2077 | N/A | >2100 | >2100 |

First 30 years | 1979–2008 | 2014 | 2017 | 2047 | 2024 | 2069 | 2025 |

Climate Normal | 1981–2010 | 2018 | 2022 | 2044 | 2025 | 2062 | 2025 |

All years | 1979–2015 | 2027 | 2033 | 2043 | 2030 | 2058 | 2035 |

Last 30 years | 1986–2015 | 2031 | 2037 | 2038 | 2032 | 2048 | 2037 |

Last 20 years | 1996–2015 | 2036 | 2067 | 2036 | N/A | 2036 | 2036 |

**Table 5.**Projected first Arctic zero-crossing summer years from optimized statistical models. Entries in bold indicate projections from models with the best fit (see AICc and W information from Table 3).

Case ID | Data Period | Exponential | Gompertz | Log | Quadratic | Linear | Linear w Lag |
---|---|---|---|---|---|---|---|

First 20 years | 1979–1998 | >2100 | N/A | 2083 | N/A | >2100 | >2100 |

First 30 years | 1979–2008 | 2015 | N/A | 2053 | 2027 | 2083 | 2028 |

Climate Normal | 1981–2010 | 2020 | N/A | 2050 | 2028 | 2074 | 2029 |

All years | 1979–2015 | 2030 | N/A | 2049 | 2034 | 2069 | 2041 |

Last 30 years | 1986–2015 | 2035 | N/A | 2044 | 2036 | 2057 | 2044 |

Last 20 years | 1996–2015 | 2043 | N/A | 2043 | N/A | 2043 | 2042 |

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**MDPI and ACS Style**

Peng, G.; Matthews, J.L.; Yu, J.T.
Sensitivity Analysis of Arctic Sea Ice Extent Trends and Statistical Projections Using Satellite Data. *Remote Sens.* **2018**, *10*, 230.
https://doi.org/10.3390/rs10020230

**AMA Style**

Peng G, Matthews JL, Yu JT.
Sensitivity Analysis of Arctic Sea Ice Extent Trends and Statistical Projections Using Satellite Data. *Remote Sensing*. 2018; 10(2):230.
https://doi.org/10.3390/rs10020230

**Chicago/Turabian Style**

Peng, Ge, Jessica L. Matthews, and Jason T. Yu.
2018. "Sensitivity Analysis of Arctic Sea Ice Extent Trends and Statistical Projections Using Satellite Data" *Remote Sensing* 10, no. 2: 230.
https://doi.org/10.3390/rs10020230