# Accelerometer first-order Markov chain model

I am trying to implement the algorithm for altitude estimation presented here. In the beginning of section 2.1 the authors present the mathematical model they will be using for the accelerometer. The accelerometer signal is modeled as follows (equation 1.b):

$$s_A = {}^{s}g + {}^{s}a + n_A$$

Where, as far as I understand, it says that the accelerometer signal is the sum of the gravity plus the vehicle acceleration plus noise and that all of these values are expressed in the sensor's frame (left superscript). I think this makes perfect sense

However, just after it they present the actual acceleration model, for which they use a first order Markov chain (equation 2):

$${}^{s}a_t = c_a{}^{s}a_{t-1} + \epsilon_{a, t}$$

I had never before seen the acceleration modeled in this way and my question is: How can I determine the value of the constant $c_a$? Is there a standard procedure to do so?

P.S: I don't really know if this is the proper place to ask this question. If it is not I will gladly move it to where it corresponds.

That is a discrete-time differential equation. It models the progress of $^sa_t$ through time: it has a contribution from the previous time instant (weighed by $c_a$) and is deviated by an exogenous factor $\epsilon_{a,t}$.
Disconsidering exogenous influence, one can affirm that, if $|c_a| < 0$, the signal will decay in amplitude with time. If $|c_a| > 0$, the signal will gain magnitude until it becomes unbounded. We can then call $c_a$ a "natural decay" constant.
To obtain this constant, one could perform some sort of experiment in which exogenous influences are nullified and the signal has an initial value different from zero. By sampling the signal $^sa_t$ along time, the rate of decay of the signal can be estimated. Sorry, but I can't read korean, so these generic concepts are as good as I can provide.