# Calculating travel distance from GPS updates

I am learning about Kalman filters but struggle to apply them for the following problem: tracking the distance traveled from GPS data. The GPS provides position updates every second and an estimate of the current speed. At each update, I can calculate the distance since the last update in two ways:

• distance $$\Delta_1$$ between current and previous coordinates
• distance $$\Delta_2$$ from speed and time: $$\Delta_2 = v * dt$$

(The speed is measured from a Doppler effect and not derived from the change in position.)

Both can be modelled as $$(\mu_i, \sigma_i^2)$$ and combined to $$\Delta = (\mu, \sigma^2)$$ that is used to updated the total distance. The variance $$\sigma^2$$ of $$\Delta$$ is smaller than that of the individual measurements but the variance of the total distance keeps increasing. This makes intuitive sense as we keep adding uncertain values and never have an opportunity to compare this against a measurement of the total distance.

Typical examples for Kalman filters assume that we have an absolute measurement of distance and a relative distance to be added. Hence, they can keep the variance of the total distance small.

Is my understanding correct that just having relative updates is not enough to model this as a Kalman filter? Is there another way to model this? I would assume that this problem comes up in bike computers that track distance from GPS.