Part 4—Calculate Station Motion
Step 1 – Add a Linear Trendline to the Time Series Plots
- On the sheet that contains each graph, choose Chart > Add Trendline...
- In the Type tab, under Trend/Regression type, choose Linear.
- Click the Options tab. Check the box next to Display equation on chart and click OK.
- On your graph, click the equation to select it and move it to where it is clearly visible. If desired, increase the font size so the equation is readable.
Step 2 – Calculate a Velocity Vector for Each GPS Time Series Plot
- The equation of the trendline generated from each TSP is in the form y = mx + b. In this equation,
For this dataset, m is the velocity of the station in mm/day. Positive slope values indicate motion to the north (or east); negative slope values indicate motion to the south (or west). Velocity directions depend upon which data were used to generate the TSP.
- y represents the North (or East) position in millimeters,
- m represents the slope of the line,
- x represents time in days, and,
- b represents the y intercept. This value is not meaningful for these calculations.
To calculate the average annual velocity of GPS stations in either direction, multiply the daily velocity by 365.25 days/year.
- What is the SEAT station's average annual velocity in the north-south direction?
- What is the SEAT station's average annual velocity in the east-west direction? NOTE: You can find an online calculator by typing an equation into an Internet search box.
Step 3 – Determine the Yearly Motion of the GPS station.
The actual horizontal motion of a GPS station over a year is represented by the vector that results by adding north and east velocity vectors together. Work through examples of both methods used to determine the annual velocity. The examples will use average annual velocities of 3 mm/yr to the north and 6 mm/yr to the east.
Graphical "Head-to-Tail" Method
- Draw an orthogonal set of X and Y axes on a piece of graph paper. Scale each axis from -10 to +10. Be sure to use the same distance per unit on both axes.
- Draw the north-south vector on the Y axis and the east-west vector on the X axis.
- Add the vectors together to graphically get the resultant vector.
- To characterize this vector so you could overlay it on a map, measure its length in the same units you used on the graph paper. Use a protractor to measure the angle from North.
Quantitative values for both the rate of motion and direction (the two components of velocity) of a GPS station can be calculated from the north and east velocities.
- Calculate the length of the vector (the rate of motion) using the Pythagorean Theorem.
a2 + b2 = c2a2 + b2 = c2
32 + 62 = c2
45 = c2
sq. root (45) = c
6.7 mm/yr = c
- Calculate the direction of motion using the relationship that the angle, a, is equal to the inverse tangent of the length of the north vector divided by the length of the east vector. a = tan-1(N/E)
a = tan-1(3/6)
a = tan-1(.5)
a = 26.6 degrees
The solution indicates the direction of motion in degrees from east. Generally, when mapping vectors, it is easier to use degrees from north. Calculate the angle from north by subtracting angle a from 90 degrees.Direction from North = 90 - a
Direction from North = 90 - 26.6
Direction from North - 63.4 degrees