# Sea Ice Deformation

Introduction

Ice deformation (divergence, shear) and ice vorticity are computed from the RGPS ice motion products. Please refer back to the Introduction to the Data Sets for links to Stern and Moritz [JGR, 2002], which contains a more complete description.

Each ice motion product consists of (essentially) a list of numbers: x,y,u,v where (x,y) are the coordinates of points on a regular 5-km grid, and (u,v) are the displacements of those points over the time interval between the two images. Since the (x,y) lie on a regular 5-km grid, we can think of them as defining a set of square cells. For each cell, we have the displacements of the corners. Using standard finite difference formulas we compute an estimate of ux (partial derivative) at the center of the cell using the corner displacements. Similarly we compute the other partial derivatives uy, vx, vy for the cell. Averaging the ux values over all the cells gives a single large-scale value for ux. It turns out that with this simple averaging procedure, the contributions from interior grid points cancel, and what remains is equivalent to an integration of u (or v) around the boundary of the region, in accord with the divergence theorem. The advantage of this averaging method is that we don't have to identify the outer boundary of grid points explicitly.

Once the large-scale partial derivatives have been computed, the deformation invariants follow easily:
 divergence = ux + vy shear = ( ( ux - vy )2 + ( uy + vx )2 )1/2 vorticity = vx - uy

By restricting our attention to those cells that lie within a certain distance of the SHEBA station, we compute deformation invariants on four different spatial scales: 50 x 50 km, 100 x 100 km, 150 x 150 km, and 200 x 200 km, which is the maximum extent of the ice motion data.

Data Products and File Format

Here are the data files:

The files are in ASCII format. Since there are 184 ice motion products, there are 184 time intervals over which ice deformation is computed. Each file has 736 lines - 4 lines per time interval (4 x 184 = 736). Here are the first 4 lines of the file for the 50 x 50 km region:

```R1000_97305002.LP
1997 305  16  20    75.7611  -143.9476
1997 307  17   2    75.9258  -144.0467
-0.141700   -0.001900    0.023114    2.029175   100
```

The first line is the file name of the RGPS ice motion product from which the deformation was computed, for purposes of cross-referencing.

The second line is the date and time of the first image, and the latitude and longitude of the SHEBA station. The numbers are:

```year, day-of-year, hours, minutes, latitude, longitude
```
where the time is GMT, and latitude and longitude are in degrees. Note that west longitude is negative. In the example above, the date is November 1 (day 305) 1997 and the time is 16:20 GMT.

The third line is the date and time of the second image, and the latitude and longitude of the SHEBA station. Same format as the second line.

NOTE: The position of the SHEBA station is derived from the SAR images and therefore is subject to geolocation error. If accurate positions are needed, please refer to the SHEBA web site (scroll down to "Spline Data ASCII File").

The numbers in the fourth line are:

```vorticity, divergence, shear, delta_t, n_cells
```
The three invariants are dimensionless, delta_t is the time interval in days over which the invariants are computed, and n_cells is the number of 5 x 5 km cells that went into the computation. Thus the area used in the computation is n_cells x 25 square kilometers. One would normally expect n_cells to be 100 for the 50 x 50 km deformation product, since 50 x 50 km = 10 x 10 cells = 100 cells. Similarly one might expect n_cells = 400 for the 100 x 100 km case, n_cells = 900 for the 150 x 150 km case, and n_cells = 1600 for the 200 x 200 km case. However, if the SHEBA station was near the edge of an image, not all the points in the vicinity of the station would be tracked, and therefore some cells would be missing. The n_cells parameter gives an indication of the completeness of the data that went into the computation of deformation.

NOTE: In one case (August 8-9) for the 50 x 50 km region, n_cells = 0. That's because no points could be tracked within the 50 x 50 km region centered at the station, although some points farther away could be tracked. In this one degenerate case, the vorticity and divergence and shear have been set to 999. Note that n_cells dips in August due to the difficulty of summer ice tracking. Interpretation of the ice deformation during this time must take into account the smaller sample sizes.

Graphical Products

Here are the time series plots of the deformation invariants, computed over four spatial scales that nominally encompass areas of 2500 km2, 10000 km2, 22500 km2, and 40000 km2. The fourth plot, Area, shows the actual area that was used in each computation. As previously noted, the actual area is less than the nominal area when the SHEBA station is near the edge of the satellite swath.

The spatial patterns of the magnitude of deformation

|e| = ( (divergence)2 + (shear)2 )1/2

are shown in this SEQUENCE OF 184 PANELS OF |e|. Each panel is 200 x 200 km in extent, centered on the SHEBA station. Each 5-km square within a panel is color-coded according to the size of |e| for that square, from blue (|e| = 0) through the spectrum to red (|e| > 0.25). The location and orientation of each panel is the same as that of the SAR images. Please refer back to the Introduction to the Data Sets for links to Stern and Moritz [JGR, 2002], which contains more details.

The colored panels of |e| are contained in 10 GIF images which are bundled into one tar file here: