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# Velocity from Acceleration

## Velocity from Acceleration

(OP)
In geological instrumentation, it is frequent to find the "integration of the acceleration" measurement, as the "empirical estimate" of the velocity, namely, the Particle Point Velocity (PPV).

Assuming:

- An real physical acceleration and velocity magnitudes -$a_0(t)$ and v_0(t)- for a steadily oscillating system, unknown. This is standard on sensors placed on ground. This is not the case of aeronautics and navigation necessarily,
- An electrical additive noise e(t)~N(0,\sigma), unknown,
- An acceleration measurement, a(t), known,
- A sampling time \tau over time, functions and integrals (just for treating the process discretely, not required though).

An accelerometer register can be simply modeled as:

$$a(t)=a_0(t)+e(t)$$

Thus the velocity estimate would be:

$$v(t)=\int_0^ta_0(\tau)d\tau+\int_0^te(\tau)d\tau=v_0(t)+\int_0^te(\tau)d\tau$$

For cancelling the "noisy" tendency in v(t), a "baseline correction" procedure is often stated as the "correct" solution. Which is nothing more than fitting a customary function to the wiener process.

This is a "Dead Reckoning" process.

**Which should be the optimal way for estimating v_0(t)? from a mathematical point of view**.

PS: The estimate at the instant t, of a Wiener Process with conditional deviation \sigma is:

\$w(t)\$~\$N(0,\sigma \sqrt t)\$

Hence, if the estimated resolution -from DAQ resolution- of the velocity measurement is \nu, after the time (\frac{\nu}{\sigma})^2 the integrated noise will surpass the sensor resolution with "1-sigma" confidence, invalidating the procedure.
Actually, after the time \$t\$ the wiener process itself will be anywhere in 1-sigma confidence interval (-\sigma\sqrt t,\sigma\sqrt t).

IS there an standard method, taking in place the inherent gaussian integrated error into account, available?.
I am trying to solve -still just trying- this problem by imposing:

- The velocity has no real "baseline" - i.e. a "real" velocity measurement will behave like an accelerometer measure, centered at zero.
- The velocity at every smaller interval should have no "baseline". This is incorrect at very small intervals, but a potential solution.
- An easy solution is to take any smoother filter, with a look&feel filtering power. This way a trend is estimated, removing the random walk noise and leaving a gaussian noise. This is the solution i have at this moment.
- Standard solutions like butterworth lowpass filters as baseline correction, though proper, are not correct, because the butterworth are meant for removing specific low frequencies, and the Wiener process is a ramp in PSD.
- A potential solution could be mimic the Wiener PSD, and apply that filtering.

### RE: Velocity from Acceleration

"In geological instrumentation... ...not the case of aeronautics and navigation..."

Inertial Navigation in an aircraft, without other corrections, might drift of about a km per hour.

I assume that geological instrumentation measurements would require errors many orders of magnitude less than that.

The real world practical solution would be to add GPS and a Kalman filter. Won't work if you're underground.

### RE: Velocity from Acceleration

The gaussian integrated error is a random process itself caused by integrating the noise. The optimal solution to minimize this is, I think is a wiener filter as you mentioned.

Once you have the filter designed the error estimate of the integration will need the variance of the filtered result.

The first answer here seems like it might help.

http://dsp.stackexchange.com/questions/8629/varian...

From the variance of the filtered noise the random walk ( error ) statistics can be generated.

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