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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.

Thanks in advance,

PS: This question will be downvoted for closing (XD??!). If you have something to add, please use the comments or message me through any other question.

A new version of this question, can be found here.

I've been requiring this answer from a lot of time, and some huge project implications are involved. Thank you for your time.

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closed as unclear what you're asking by Andy aka, Daniel Grillo, Voltage Spike, Dave Tweed Sep 4 '16 at 11:48

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  • \$\begingroup\$ Unfortunately, some national organisms and most if not all structural specialist on my country still think on that as a valid method.... I am just trying to obtain a clear and solid theoretical argument against that. \$\endgroup\$ – Brethlosze Sep 1 '16 at 4:01
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    \$\begingroup\$ I disagree the process becomes invalid, the result is simply a tolerance error due to noise, drift and resolution. But if the geological interest is seismic, one can reset the velocity between events due to drift error or offset in acceleration. By "structural", are you referring to fragility boundary tests of v(t) vs a(t) \$\endgroup\$ – Sunnyskyguy EE75 Sep 1 '16 at 5:04
  • \$\begingroup\$ Civil and Structural Specialists from the 80' firmly believe that a "very precise" instrument will nullify the "baseline". There is actually invalid after a number of samples, as calculated below. This is traditionally applied, but incorrect. The tendency of integrating \$a(t)\$ is indeed \$v(t)\$, plus a wiener process, which can lead you anywhere inside the \$(\sigma\sqrt(t),-\sigma\sqrt(t))\$ cone iwth 1-sigma confidence. That is my concern of the method. \$\endgroup\$ – Brethlosze Sep 1 '16 at 5:34
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    \$\begingroup\$ The integral of e(tau) should approach 0 for bigger t. I think the problem are non-symmetric aberrations which keep increasing absolute error with growing t. \$\endgroup\$ – JimmyB Sep 1 '16 at 16:14
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    \$\begingroup\$ So, the current best practice way of estimating \$v_0(t)\$ is the Kalman filter (at least as far as I am aware). Is there a particular reason you aren't looking at that? \$\endgroup\$ – Andrew Spott Sep 1 '16 at 18:59