You can actually establish (only) the vertical position without measuring acceleration over time. This is accomplished by measuring the angle between the direction to the center of the earth, and the direction to the magnetic north pole.
This only changes (significantly) when the altitude (height) of the phone changes. What you do is use the accelerometer
and magnetometer
to get two float[3] arrays, treat these as vectors, make them unit vectors, and then the angle between any two unit vectors is arccos(AxM).
Note that's dot product ie. math.acos(A[0]*B[0]+A[1]*B[1]+A[2]*B[2])
Any change in this angle corresponds to a change in height. Also note that this will have to be calibrated to real units and the ratio of change in angle to height will be different at various longitudes; But this is a method of getting an absolute value for height; though of course the angle also becomes skewed when undergoing acceleration, or when there are nearby magnets :)