Modeling data acquisition drift

To model data drift we imagine a vector $\bold q$ of random numbers that will be passed thru a low-pass filter (like a leaky integrator) $\bold L$.The modeled data drift is $\bold L\bold q$.We will solve for $\bold q$.A price we pay is an increase of the number of unknowns. Augmenting earlier fitting goals (23) we have:  

$$\begin{array} {lllll} \bold 0 &\approx& \bold r_d &=& \bold... ...bold 0 &\approx& \bold r_q &=& \epsilon_2 \bold q, \end{array}$$ (26)

where ${\bf h}={\bf H^{-1}p}$ estimates the interpolated map of the lake, and where ${\bf L}$ is a drift modeling operator (leaky integration), ${\bf q}$ is an additional variable to be estimated, and $\lambda$ is a balancing constant to be discussed. We then minimize the misfit function,

$$g_2(\bold p,\bold q) = \Vert\bold r_d\Vert^2+\epsilon_1^2\Vert\bold r_p\Vert^2+\epsilon_2^2\Vert\bold r_q\Vert^2,$$ (27)

Now the data $\bold d$ is being modeled in two ways by two parts which add, a geological map part $\bold B \bold {H^{-1}} \bold p$and a recording system drift part $\lambda\bold L \bold q$.Clearly, by adjusting the balance of $\epsilon_1$ to $\epsilon_2$ we are forcing the data to go one way or the other. There is nothing in the data itself that says which part of the theory should claim it.

 

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Figure 20 Top left: Estimated ${\bf p}$ without attenuation of the tracks, i.e., regression (23). Top right: Estimated ${\bf p}$ with the derivative along the tracks, i.e., regression (25). Bottom left: Estimated ${\bf p}$ without tracks, i.e., regression (26). Bottom right: recorder drift in model space $\bold B'\bold L\bold q$.

It is a customary matter of practice to forget the two $\epsilon$s and play with the $\lambda$.If we kept the two $\epsilon$s, the choice of $\lambda$ would be irrelevant to the final result. Since we are going to truncate the iteration, choice of $\lambda$ matters. It chooses how much data energy goes into the equipment drift function and how much into topography. Antoine ended out with with $\lambda=0.08$.

There is another parameter to adjust. The parameter $\rho$ controlling the decay of the leaky integration. Antoine found that value ${\rho=0.99}$ was a suitable compromise. Taking $\rho$ smaller allows the track drift to vary too rapidly thus falsifying data in a way that falsifies topography. Taking $\rho$ closer to unity does not allow adequately rapid variation of the data acquistion system thereby pushing acquisition tracks into the topography.

Figure (bottom-left corner) shows the estimated roughened image ${\bf p}$with $\lambda\bold L$ data-drift modeling and (top-left corner) ${\bf p}$ without it. Data-drift modeling (bottom-left) yields an image that is essentially track-free without loss of detail. Top right shows the poor result of applying the derivative $d\over ds$ along the tracks. Tracks are removed but the topography is unclear.

The bottom-right part of Figure provides important diagnostic information. The estimated instrumentation drift $\bold L\bold q$ has been transformed to model space $\bold B'\bold L\bold q$.We do not like to see hints of geology in this space but we do. Adjusting $\lambda$ or $\rho$ we can get rid of the geology here, but then survey tracks will appear in the lake image. The issue of decomposing data into signal and noise parts is dealt with further in chapter .

Figures and show selected segments of data space. Examining here the discrepancy between observed data and modeled data offers us an opportunity to get new ideas. The top plot is the input data $\bold d$.Next is the estimated noise-free data ${\bf BH^{-1}p}$.Then the estimated secular variations $\lambda\bold L \bold q$.Finally residual $\bold B \bold {H^{-1}} \bold p + \lambda \bold L \bold q - \bold d$after a suitable number of iterations. The modeled data in both Figures b and b show no remaining spikes.

 

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Figure 21 (a) Track 17 (input data) in Figure . (b) The estimated noise-free data ${\bf BH^{-1}p}$. (c) Estimated drift $\bold L\bold q$. (d) Data residual.

 

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Figure 22 (a) Track 14 (input data) in Figure . (b) Modeled data, ${\bf BH^{-1}p}$. (c) Estimated drift. (d) Data-space residual.

The estimated instrument drift is reasonable, mostly under a meter for measurments with a nominal precision of 10 cm. There are some obvious problems though. It is not a serious problem that the drift signal is always positive. Applying the track derivative means that zero frequency is in the null space. An arbitrary constant may be moved from water depth to track calibration. More seriously, the track calibration fluctuates more rapidly than we might imagine. Worse still, Figure c shows the instrument drift correlates with water depth(!). This suggests we should have a slower drift function (bigger $\rho$ or weaker $\lambda$), but Antoine assures me that this would push data acquisition tracks into the lake image. If the data set had included the date-time of each measurement we would have been better able to model drift. Instead of allowing a certain small change of drift with each measurement, we could have allowed a small change in proportion to the time since the previous measurement.

An interesting feature of the data residual in Figure d is that it has more variance in deep water than in shallow. Perhaps the depth sounder has insufficient power for deeper water or for the softer sediments found in deeper water. On the other hand, this enhanced deep water variance is not seen in Figure d which is puzzling. Perhaps the sea was rough for one day of recording but not another.