# Filling the Polar Data Gap in Sea Ice Concentration Fields Using Partial Differential Equations

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## Abstract

**:**

## 1. Introduction

## 2. Data

## 3. Method

#### 3.1. Formulation of Data Fill Model

#### 3.2. Stochastic Term Ω

## 4. Results

#### 4.1. Illustrative Examples

#### 4.2. Validation and Performance Assessment

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Yellow outlines indicate the three circular regions used to estimate the parameters of the stochastic term. These three circular regions were also used to validate and assess performance of the solution to Laplace’s equation. Blue circles show polar data gap extent for the three sensors indicated in the legend (see also Table 1). Shading indicates sea ice concentrations for 3 September 2007.

**Figure 2.**Circles indicate empirical estimates of the standard deviation σ of the stochastic term Ω, as a function of day of the year for 1988–2013, where 1988 is selected because it is the first full year after adoption of the moderately-sized pole hole mask (radius 311 km, Table 1). The curve is a fit of Equation (4) with $k=2$. Values exceeding 0.12 (less than 1% of the data) are not shown, but were included in the curve fit.

**Figure 3.**Examples of filling the polar data gap. For 6 January 1985: (

**a**) sea ice concentration; (

**b**) sea ice concentration with polar data gap filled using f (solution to Laplace’s equation plus a stochastic field Ω); and (

**c**) same as (

**b**) but color scheme follows nsidc.org; (

**d**–

**f**) Same as (

**a**–

**c**), but for 30 August 2007.

**Figure 4.**A three-dimensional view of the 30 August 2007 example from Figure 3d–f. Shading and surface elevation indicate concentration, and black curves are provided to help visualize the three dimensionality of the surface. (

**a**) Observed sea ice concentrations around the polar data gap; (

**b**) polar data gap replaced by solution to Laplace’s equation $\psi (x,y)$; (

**c**) polar data gap replaced by ψ plus the stochastic term Ω; (

**d**) same as Figure 3f but rotated to illustrate the geographic orientation of the other panels; (

**e**) polar data gap replaced by bilinear interpolation; and (

**f**) polar data gap replaced by thin plate spline.

**Figure 5.**For each of the regions M1–M3 in Figure 1, color curves indicate values averaged by day of the year: (

**a**) mean observed sea ice concentration; (

**b**) spatial correlation between observed sea ice concentration and the solution to Laplace’s equation ψ; (

**c**) the mean absolute deviation between observed sea ice concentration and ψ; and (

**d**) the spatially averaged bias between observed sea ice concentration and ψ. In each panel, the black curve is the average across the three M regions and the gray curve corresponds to validation via dilation of the polar data gap as described in Section 4.2.

**Table 1.**For the circular “pole hole mask” covering the polar data gap: the satellite platform, the hole radius, the latitude of the hole’s outer edge, and the applicable time periods.

Platform | Radius (km) | Latitude (${}^{\circ}$N) | Time Period |
---|---|---|---|

SMMR | 611 | 84.50 | November 1978–June 1987 |

SSM/I | 311 | 87.20 | July 1987–December 2007 |

SSMIS | 94 | 89.18 | January 2008–present |

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**MDPI and ACS Style**

Strong, C.; Golden, K.M.
Filling the Polar Data Gap in Sea Ice Concentration Fields Using Partial Differential Equations. *Remote Sens.* **2016**, *8*, 442.
https://doi.org/10.3390/rs8060442

**AMA Style**

Strong C, Golden KM.
Filling the Polar Data Gap in Sea Ice Concentration Fields Using Partial Differential Equations. *Remote Sensing*. 2016; 8(6):442.
https://doi.org/10.3390/rs8060442

**Chicago/Turabian Style**

Strong, Courtenay, and Kenneth M. Golden.
2016. "Filling the Polar Data Gap in Sea Ice Concentration Fields Using Partial Differential Equations" *Remote Sensing* 8, no. 6: 442.
https://doi.org/10.3390/rs8060442