# Knowledge Base

# How to convert the horizontal and vertical components to east and north

This article describes how to convert the u (horizontal) and v (vertical) components from the ice motion data set to East and North components on the Earth's surface. Instructions for both the Northern and Southern Hemispheres are described below:

**Northern Hemisphere**

The latitudes and longitudes for the Northern Hemisphere are shown in Figure 1.

The directions in the product are:

- u: toward the right on the grid
- v: upward (toward the top) on the grid
- East is counterclockwise
- North is toward the center of the grid

To compute the East and North components, apply a rotation matrix, which results in:

- E: u * cos L + v * sin L
- N: -u * sin L + v * cos L
- where L is the longitude
- Note the negative sign in the first term of N

This can be checked by verifying that:

- at L=0o, a u-only vector (to the right) corresponds to east (positive E)
- at L=0o, a v-only vector (toward the top) corresponds to north (positive N)
- at L=90o, a u-only vector (to the right) corresponds to south (negative N)
- at L=90o, a v-only vector (toward the top) corresponds to east (positive E)

In Figure 1, both vectors A and B are pointing East.

**Southern Hemisphere**

****The latitudes and longitudes for the Northern Hemisphere are shown in Figure 2.

The directions in the data set are:

- u: toward the right on the grid
- v: upward (toward the top) on the grid
- East is clockwise
- North is outward from the center of the grid

To compute East and North components, apply a rotation matrix, which results in:

- E: u * cos L - v * sin L
- N: u * sin L + v * cos L
- where L is the longitude
- Note the negative sign in the second term of E

This can be checked by verifying that:

- at L=0o, a u-only vector (to the right) corresponds to east (positive E)
- at L=0o, a v-only vector (toward the top) corresponds to north (positive N)
- at L=90o, a u-only vector (to the right) corresponds to north (positive N)
- at L=90o, a v-only vector (toward the top) corresponds to west (negative E)

In Figure 2, both vectors C and D are pointing East.