Displacements at Onsala inferred from the Superconducting Gravimeter for the Maule Earthquake of 2010-02-27

The first two figures show the surface wave displacements inferred from the gravimeter record of the morning of Feb. 27, 2010.
The analysis suggests: 4 mm peak-to-peak

Figure 1  - Resulting displacements derived according to the description on this page.

Figure 2 - Zooming on the maximum surface wave action ±6 minutes around 07:30:00 UT.

Deriving these charts was no simple task. The SCG record clips at ±10 V  (0.776 mGal), and the accelerations were much larger thant this during much of the surface wave action. Figure 3 shows the clipped and the restored signals. See "Signal restoration" below; the process will be desginated with symbol CSR for clipped signal restoration.

In principle, displacements can be computed by double-integration over time with carefully constraining the constants of integration. In the present case, the signal was high-pass filtered (HPF) (suppressing a narrow range near zero frequency (0 - 10 mHz) and a sharp transition to the pass-band (121 + 121 filter coefficients using a Kaiser-Bessel Window design with shape parameter 2.1. Spurious DC-offsets were deleted (DCD).

Integration was carried out with the simple scheme
   INT: y(i+1) = y(i) + x(i) dt
which advances series y by 1/2 time step, dt = 1s

   u = DCD ( HPF ( INT ( DCD ( INT ( HPF ( DCD ( CSR ( DCD ( x )))))))))

Figure 3 - Signal restoration. The original gravimeter data with over-range values deleted (red curve), and the restored signal (in cyan).

Figure 4 - Stages of signal restoration and displacement computation. The time series are not synchronous; SCG and ClipRest are unaffected by the 121-s truncation of the high-pass filter, and F4 and Displacement are affected two times by the truncation. F1 (dark blue) is after the first hi-pass, F2 (petrol) after the first integration,  F3 (purple)  after the second integration, and F4 (green) afterthe second filtering. 

Signal restoration.
This might be a matter of debate. The method employed was to
(1) cut the signal at ±10V and identify the gaps
(2) before and after each gap, find the times where maximum signal change (x(i+1) - x(i)) occurs.
(3) a sinus half-wave is clamped with its zero-crossings at these two points, from which we get the frequency
(4) a linear ramp is added to run through these two points.
(5) the sine is given a linearly changing amplitude since the slope in the branches before and after the breaks may not be mirror-symmetric.

So there are four parameters to solve for the interpolation formula: Sine amplitude: a1 and a2; ramp start and end ordinate: x1 and x2. The clamping points before and after the gap are named mb and mf (m-backward, m-forward), respectively.

See the Mathematica notebook print-out (pdf).

The fit is done with a nonlinear minimum finder (Numerical Recipes dfpmin); the reason for this is that we anticipated (erroneously) that we could adjust the frequency. All too often, the frequency parameter runs away, so that many more than half a cycle is filled into the gap.

You may ask for the routines unclip.f (F77) and ask the author of this page for Ddfpmin.f Dlnsrch.f (Num.Recipes slightly adapted for the present purpose).

The procedure which carried out the process is tslist / tsfedit (=> UNCLIP) (=> IIR) (=> FILTER D:WD), see the resulting protocol of the process.  

The relations of gravity to displacement are more complicated when periods are long (mHz or below) - see a pdf memo.

Göteborg, 2010-07-12
Hans-Georg Scherneck