Let’s work through a visualization of how velocities from three PBO GPS stations can be used to determine the local rate of infinitesimal strain.

We pick three PBO GPS sites in the red box, along the Oregon coastline just northwest of Salem. These sites are on continental crust of the North American plate, and are located above the Cascadia subduction zone along which the Juan de Fuca plate is subducting under the North American plate. The thick black arrows indicate the average direction in which the Juan de Fuca plate is moving relative to the stable interior of North America.

The PBO website provides a Google Map plot of the three GPS stations we have chosen to use in our example.

We superimpose a plotting grid on top of each of the GPS sites We superimpose a plotting grid on top of each of the GPS sites. The units here are in millimeters per year.

We superimpose a plotting grid on top of each of the GPS sites We superimpose a plotting grid on top of each of the GPS sites. The units here are in millimeters per year.

We superimpose a plotting grid on top of each of the GPS sites We superimpose a plotting grid on top of each of the GPS sites. The units here are in millimeters per year.

We plot the north-south and east-west components of the GPS site velocities, which are provided on the PBO web resources associated with each of the sites.

We add the two component vectors together to define the instantaneous horizontal velocity vector for each site. The magnitude of each of the three blue horizontal-velocity vectors is expressed in millimeters per year. This image has become quite cluttered, so let’s simplify it a bit.

There, that’s better. The three GPS sites are all moving toward the northeast, relative to the stable interior of North America.

The three sites form the apices of a triangle.

We find the center of the triangle using simple geometry

And inscribe a circle that is concentric with the centroid of the triangle.

This figure is the initial state or datum from which we will determine the strain using GPS velocities.

We now add the instantaneous horizontal velocity vectors we computed earlier…

We now add the instantaneous horizontal velocity vectors we computed earlier…

…and imagine how the original triangle and circle will change consistent with those velocities. Clearly, this illustration shows finite strain rather than infinitesimal strain. This illustration is meant to give you a feel for how the triangle and circle would change shape in accordance with the horizontal velocities measured at the GPS sites. The actual analysis performed by the students is a true infinitesimal-strain analysis, and not a graphic analog like this picture. You can perhaps see that the lengths and interior angles of the red triangle are different than those of the black triangle, and that the black circle has changed to form the red ellipse.

…and imagine how the original triangle and circle will change consistent with those velocities. Clearly, this illustration shows finite strain rather than infinitesimal strain. This illustration is meant to give you a feel for how the triangle and circle would change shape in accordance with the horizontal velocities measured at the GPS sites. The actual analysis performed by the students is a true infinitesimal-strain analysis, and not a graphic analog like this picture. You can perhaps see that the lengths and interior angles of the red triangle are different than those of the black triangle, and that the black circle has changed to form the red ellipse.

…and imagine how the original triangle and circle will change consistent with those velocities. Clearly, this illustration shows finite strain rather than infinitesimal strain. This illustration is meant to give you a feel for how the triangle and circle would change shape in accordance with the horizontal velocities measured at the GPS sites. The actual analysis performed by the students is a true infinitesimal-strain analysis, and not a graphic analog like this picture. You can perhaps see that the lengths and interior angles of the red triangle are different than those of the black triangle, and that the black circle has changed to form the red ellipse.

The average of the three blue horizontal-velocity vectors is the green translation vector, which extends from the centroid of the original (black) triangle to the centroid of the displaced (red) triangle.

We can subtract the translation vector from the displaced red triangle…

…and bring the two triangles back together at their centroids. Time constraints prevent me from showing how we find the rotational component, but the displaced triangle is rotated slightly clockwise from the original triangle. You can perhaps see that the red ellipse is inside of the black circle to the lower left and upper right, and is slightly beyond the black circle at the top left and bottom right. That red ellipse represents the strain ellipse.

The minor axis of the strain ellipse is the blue line and is associated with a negative extension, while the major axis is red and is associated with a small positive extension. Finding the strain ellipse is the end of a typical strain analysis in a structural geology course, but we can go further because we know the sign and magnitude of the horizontal principal extensions.

The usual map symbol for this sort of horizontal strain ellipse has black-filled arrows indicating shortening and white-filled arrows indicating extension. The size of the arrow indicates the relative absolute magnitudes of the extensions, so in this case the absolute value of the extension along shortening axis is greater than that along the extension axis.

Placing that symbol in the middle of our GPS site triangle gives the visual cue that this area is experiencing shortening toward the east-northeast and extension toward the north-northwest.

Backing-up to view the regional context, we can have wonderfully fruitful discussions with students about the possible cause of deformation along the leading edge of the North American Plate over the Cascadia Subduction Zone. This discussion can expand in many directions such as the consideration of earthquakes, volcanoes, uplift and subsidence, episodic tremor and slip, geologic hazards and risk assessment.

What is the reason for this state of crustal strain between these three GPS stations? We can start to address this question by considering the direction the subducting plate is moving relative to North America, give or take 10 degrees or so.