How to handle a logarithmic term in Kalman filter?
First of all, you're trying to evaluate the derivative of an exponential. If the base of the exponential is positive, the derivative exists. However, if the base is negative, the derivative does not exist. It would be pointless to use a Kalman filter when you have an exponential with a negative base.
Secondly, your model of the echo envelope is
$$ 
A(t) = A_0(\frac{t-\tau}{T})^{\alpha}e(\frac{t-\tau}{T}) 
$$
However, since it is an echo, it does not make sense for the echo envelope to be something else than 0 for $t &lt; \tau$ where $\tau$ is the time of flight
You could add a unit step to the equation to make sure that A(t) = 0 for t &lt; $\tau$
$$ 
A(t) = A_0(\frac{t-\tau}{T})^{\alpha} e(\frac{t-\tau}{T})  u(t-\tau)
$$