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Research Object and Relevance. Inclinometer measurements occupy an important place in geotechnical monitoring of hydraulic structures. Borehole inclinometers are used on tailings dams - they allow you to determine horizontal displacements, shifts, curvature of the casing pipe throughout its entire depth. An important part of the processing of initial data is the identification of various systematic errors. The authors of the article faced the problem of interpreting the displacements at the tailings dam of The Erdenet Mining Corporation - after the standard linear correction, some boreholes had arcuate displacements inexplicable from the point of view of geomechanics throughout their depth. Our first assumption was that this was caused by unaccounted for measurement errors. In March 2022, four series of inclinometer measurements were carried out within one day at the deepest borehole of the observation station in the main direction. Based on the results of the obtained data, non-linear systematic errors were identified. Research Methods. The article shows the functional dependencies of the main sources of systematic errors in inclinometric measurements: sensitivity drift, bias shift, rotation error, depth positioning error. We used a parametric adjustment with the inclusion of a systematic error model. This approach allows not only to determine the quantitative values of systematic errors, but also to evaluate their quality and check for statistical significance using a t-test. Results. As a result of the study, statistically significant systematic non-linear errors of inclinometer measurements were revealed. Statistically, this error is well described by dependence on the square of the interval number. They can also be partly attributed to the rotation error and the depth error (it may be better to define these errors as rotation sensitivity and sensitivity to the installation of the pulley on the casing head). Also in the article, we show graphs of residual errors and Q-Q plots for borehole position error without correction and with the corrections discussed in the article. The linear component of the systematic error was up to 0.065 mm per measurement interval. The non-linear systematic error was up to 0.018 sq. mm. Conclusion. As shown in this article, finding and correcting only a linear error does not allow one to be satisfied with the result (clearly seen on the Q-Q plots). It should be noted that if linear systematic errors affect the determination of displacements and shifts, then non-linear errors also distort information about changes in the curvature of the borehole. At the same time, in order to apply the method proposed by the authors for identifying nonlinear systematic errors between series of observations, strict consistency with the geomechanical model of potential deformation of the monitored object is required.
ABSTRACT
Research Object and Relevance. Inclinometer measurements occupy an important place in geotechnical monitoring of hydraulic structures. Borehole inclinometers are used on tailings dams - they allow you to determine horizontal displacements, shifts, curvature of the casing pipe throughout its entire depth. An important part of the processing of initial data is the identification of various systematic errors. The authors of the article faced the problem of interpreting the displacements at the tailings dam of The Erdenet Mining Corporation - after the standard linear correction, some boreholes had arcuate displacements inexplicable from the point of view of geomechanics throughout their depth. Our first assumption was that this was caused by unaccounted for measurement errors. In March 2022, four series of inclinometer measurements were carried out within one day at the deepest borehole of the observation station in the main direction. Based on the results of the obtained data, non-linear systematic errors were identified.
Research Methods. The article shows the functional dependencies of the main sources of systematic errors in inclinometric measurements: sensitivity drift, bias shift, rotation error, depth positioning error. We used a parametric adjustment with the inclusion of a systematic error model. This approach allows not only to determine the quantitative values of systematic errors, but also to evaluate their quality and check for statistical significance using a t-test.
Results. As a result of the study, statistically significant systematic non-linear errors of inclinometer measurements were revealed. Statistically, this error is well described by dependence on the square of the interval number. They can also be partly attributed to the rotation error and the depth error (it may be better to define these errors as rotation sensitivity and sensitivity to the installation of the pulley on the casing head). Also in the article, we show graphs of residual errors and Q-Q plots for borehole position error without correction and with the corrections discussed in the article. The linear component of the systematic error was up to 0.065 mm per measurement interval. The non-linear systematic error was up to 0.018 sq. mm.
Conclusion. As shown in this article, finding and correcting only a linear error does not allow one to be satisfied with the result (clearly seen on the Q-Q plots). It should be noted that if linear systematic errors affect the determination of displacements and shifts, then non-linear errors also distort information about changes in the curvature of the borehole. At the same time, in order to apply the method proposed by the authors for identifying nonlinear systematic errors between series of observations, strict consistency with the geomechanical model of potential deformation of the monitored object is required.
Keywords: Geotechnical monitoring; Systematic errors; Borehole inclinometer; Hydraulic structure; Structural health monitoring
(ProQuest: ... denotes formulae omitted.)
INTRODUCTION
The tailings dam of The Erdenet Mining Corporation is a technogenic earthwork. On the one hand, this makes it possible to simplify the construction of such a hydraulic structure, on the other hand, a simple design should not be misleading - the operation of that a large structure is a very responsible task. An integral part of safe operation is geotechnical monitoring, including observations of displacements inside the dam body and base rocks.
With the advent of borehole inclinometers in the 50s of the last century, geotechnicians have the opportunity to determine horizontal displacements: having information about the inclinometer probe inclination and its length, it is easy to calculate horizontal increments (Incremental Deviations); the sum of such increments makes it possible to calculate the borehole position in the measurements direction (Cumulative Deviations); comparison of the borehole position in different series of observations makes it possible to determine horizontal displacements (Incremental and Cumulative Displacements).
Despite the fairly good accuracy of the tilt sensor (even at the time of their first appearance), practical researchers immediately drew attention to the exactingness of this technology for good planning projects, proper installation, diligent reading, correct data processing, and intelligent scrutiny and interpretation [1,2]. There is an explanation for this: the sources of errors are the accelerometer, probe body, cable, readout, casing. In addition to this, each source of errors has several causes for their occurrence. For example, an accelerometer sensor has a sensitivity error, an error caused by linear displacement and rotation inside the probe.
P. Eric Mikkelsen did a great deal of work in advanced inclinometer data correction in his synthesis work Advances in inclinometer data analysis in 2003 [3]. In it, P. E. Mikkelsen notes the significant influence of systematic errors in the total error: on a thirty-meter interval, a random error is 1.24 mm, a systematic error is 6.60 mm. He also highlights the main systematic errors that can be significantly reduced as a result of data processing: bias-shift (differences between opposite readings traverses in a data set), sensitivity drift (calibration of the instrument constant), rotation error (rotation MEMS sensor inside probe), depth-positioning error (primary interval position error).
It is also possible to improve the accuracy of inclinometer measurements by statistical processing of a large array of initial data. In this way, the random measurement error can be reduced. This possibility was demonstrated by M. Lovisolo and A. Della Giusta in their work devoted to the assessment of the accuracy of Differential Monitoring of Stability system [4]. True, it is worth noting that the system under consideration implies a stationary installation of a chain of inclinometers in a borehole. This method of monitoring allows you to control the position of the borehole in real time, but it is difficult to implement with large monitoring objects and, accordingly, a large number of inclinometric boreholes.
Some features of the proper installation and the influence of these factors on the analysis of data during observations of landslides are described in the work of Timothy D. Stark and Hangseok Choi [5]. For example, the effect of backfilling a borehole, a competent choice of the depth of a borehole bottom, etc. Also, an interesting work on the influence of the casing material and its installation was carried out by Ralf J. Plinninger, Michael Alber and Jan Düllmann [6]. Do not bypass the developments in the field of tilt sensors themselves. The paper by P.O.P. Septimiu, Dan Pitica and Ioan Ciascai shows the possibility of reducing the systematic error caused by the error of the resistor [7]. Separately, it is worth highlighting a difficult to detect, but very insidious error caused by the displacement of the first interval [3]. It has already been marked above as a depth positioning error. This is noted in many works, including fairly recent articles [8].
It is also worth noting interesting developments in the field of robotic inclinometric measurements. Unlike work [4], the article by Paolo Allasia, Danilo Godone, Daniele Giordan, Diego Guenzi and Giorgio Lollinodeals deals with multi-repeated automated inclinometric measurements with a movable probe [9]. This system allows the use of one probe per borehole, and at the same time perform measurements at a high frequency compared to manual techniques. The authors note the high measurement accuracy: 0.4 mm in the main direction and 1.2 mm in the perpendicular direction. We dare to note that it is not entirely correct to compare these values with the total error of 7.8 mm [3], since 0.4 mm [9], judging by the description of the measurement technique and processing, refers only to a random measurement error (no information on correcting for systematic errors).
The works noted above show the need for further research on inclinometric measurements, methods for their processing, and, in particular, the identification of systematic errors. In our study, we tried to find out if there are systematic errors in the actual data obtained in March 2022 at the tailings dam of The Erdenet Mining Corporation. Inclinometric equipment in geotechnical monitoring at this object is used by DGSI Slope Indicator. Since the measurements were taken on the same day, no geomechanical displacements are expected. At its core, we are trying to isolate systematic errors by a mathematical method in the cumulative deviations of the borehole.
RESEARCH METHODS
First, let us describe the effect of some systematic errors on inclinometer measurements. MEMS-sensor puts out a signal proportional to the sine function (as a function of the vertical slope angle a). For simplicity, the output value is multiplied by an instrument constant к, for example, for DGSI Slope Indicator inclinometers in metric measurement к = 25000. Sensitivity drift of the sensor can be compared with the presence of a systematic error 0fc. in this constant к = к + ek. Let's write down the value of the inclinometer at the z-th interval in the main direction (A0¿), taking into account this error:
...
Consider the following possible systematic error - bias shift (0b = eb). This error can be characterized as a tilt of the sensor inside the probe along the main reading direction. This error is easily eliminated by double reading (A0-A180), since it has a different sign when measured in the direction of 0 and 180. It is for this reason that we can talk about inclinometer measurements in the directions 0 and 180 as an analogue of measuring the angles by «Face Left» and «Face Right» with Total Station. The bad news here is that there may be an additional slight offset when measuring «Face Right». Then the residual part of this systematic error (0bs¿ = ebs) will lead to a single measurement bias, as well as an additional linear cumulative bias. Note also that this additional error occurs not only between A0 and A180, but also between series of measurements. Let's write it down a bit in a simplified form (similar to [3]):
...
...
...
A small rotation of the sensor inside the probe introduces another systematic error known as rotation error (0rot). Usually, this error is mentioned when changing the probe in different monitoring series - this error will manifest itself precisely in the comparison of measurements between different probes [3]. At small angles of rotation (y), the error in values A and B can be written as follows:
...
where A¿, B¿ - are the interval readings in the rotated positions; aA, aB - slightly different small rotation angles of accelerometer axes corresponding directions A and B.
One more insidious error is the depth positioning error (0^ре). It occurs when the reading is taken at a different depth than intended [8]. For example, the length of our cable has changed between series, the bottomhole or holehead elevation has changed between series of measurements. This error can be characterized as an error that is directly proportional to the curvature (c) of the casing and the systematic depth position shift (edps):
...
In view of the foregoing, we will show the formula for the averaged z-th reading:
...
The formula for the cumulative deviation will take the following form:
...
where An, Bn - are the initial cumulative readings of the probe in the main and perpendicular directions; n - number of the interval relative to which the cumulative displacements are calculated; ... accumulative casing curvature.
...
...
...
...
Since ... guided by the principle of negligible influence, size of the Jacobian matrix size will be jn by n+4j. If we make reasonable assumptions that the systematic errors of sensitivity drift, rotation inside the probe, and depth position could not occur on the same day of measurements and using one probe, then the size of the Jacobian matrix will be jn by n+j. Then the Jacobi matrix will take the form:
...
where lnn - is identity matrix; n - is interval number vector; A - is the matrix of coefficients of the equations of amendments (represents the random component of the error); G - is the matrix defining the model of the systematic errors action.
We will form the weight matrix (P) according to the following formulas (about the type of mean square error (m) and approximate values can be found in the work - Bannikov A. and Gordeev V. Accuracy With Borehole Inclinometers At The Tailings Dam Of The Erdenet Mining Corporation. 21st International Multidisciplinary Scientific Ge°Conference SGEM 2021. P.305-312. DOI: 10.5593/sgem2021/2.1/s09.40):
...
...
Taking into account only the systematic error of the bias shift, the weight matrix will have a diagonal form:
...
The corrections to the approximate values of the parameters (t) are found by the formula (only bias shift errors), where I - is the initial amendment vector (average values):
...
Then the vector of cumulative bias shift systematic errors (0fts) can be found by the formula:
...
The vector of corrections (v) compensating for the influence of random errors is calculated by the formula:
...
...
The mean square error of a unit of weight is found by the formula: ...
...
More details on the adjustment technique taking into account systematic errors are described in our work [10].
RESULTS
We have analyzed four complete series of inclinometric measurements on borehole VI-10i with a depth of 95 meters (188 intervals). Each series was performed twice with the probe rotated 180 degrees in the borehole (FL&FR) carried out on the same day.
In the graph below, we will show the residual errors without any adjustments in the main direction A (Figure 1-a). Also, let's graphically show the discrepancy with the normal distribution of error residuals by means of a quantile-quantile plot (Figure 1-b). Once again, we draw your attention to the fact that the presented errors are measurement errors and do not apply to geomechanical displacements. The Q-Q plot (Figure 1-b) clearly shows the presence of a systematic error in each series - not a single median coincides with the zero value. This is a good start to try to identify various systematic errors.
Since all measurements were taken by the same probe on the same day, there is no reason to assume sensitivity drift errors, rotation error, depth positioning error. Let us find and exclude the bias-shift error by the method described above. This systematic error is proportional to the interval number for which the cumulative deviation of borehole is calculated. In other words, it has a linear nature of influence. Bias-shift errors are unambiguously identified - the cumulative Student's probability distribution for these errors in all series of measurements is close to one. The standard error of the weight unit equal to 2.04 mm.
Let's build similar graphs of residual errors, taking into account linear adjustment (Figure 2). As can be seen from the graph of residual errors, the absolute values became noticeably smaller, but the Q-Q plot shows a significant deviation from the theoretical distribution for the first and fourth series of measurements.
We assumed that in different series of measurements there may be additional errors in the sensitivity drift, the sensitivity of the MEMS sensor to rotation inside the probe, the effect of the pulley installation on the borehole head (which will lead to a depth positioning error). After the calculations, the following probabilities of detecting systematic errors were obtained (Table 1).
As you can see, the sensitivity drift is not detected, but there are a rotation errors in three series, and a pulley installation errors in two series. After excluding insignificant models of systematic errors from the adjustment calculations, the standard error of the weight unit was 1.11 mm. The graphs below show the residual errors and the fit of the residuals to a normal distribution (Figure 3). On the presented plot of residuals (Figure 3-a, the scale of errors is kept for visual comparison with the graphs in Figures 1 and 2), we see a significant improvement in the results of inclinometric measurements - the maximum errors reach only one mm. The graph shows a fairly good approximation of the actual distribution of residuals to the theoretical one.
If we do not assume the presence of errors in the sensitivity to rotation and position in depth (all series were carried out on the same day), then it is necessary to assume the presence of some other systematic error. One option may be a quadratic dependence on the measurement interval. Such a model of systematic error (together with the linear component) is detected in all series with a confidence probability greater than 0.97, the standard error of the weight unit was 0.64 mm. Figure 4-a shows the residual errors, the correspondence to the theoretical distribution is shown in Figure 4-b.
CONCLUSION
Without confidence in the measurements, there can be no confidence in the calculated displacements caused by geotechnical factors. The inclinometric data needs to be corrected, and our study clearly demonstrates this. In this article, we have shown the capabilities of the mathematical apparatus for detecting systematic errors, and also identified them using a specific example.
The most common systematic error and at the same time quite easily detected is bias-shift error. Figure 2-b clearly shows that such an adjustment may not be final - in our case, the residuals clearly do not obey the normal or close to it distribution law. We have shown that a simple quadratic dependence on the number of the measurement interval gives a somewhat better result (the estimates of the standard of the weight unit were: for a linear correction - 2.04 mm, for a nonlinear correction caused by the sensitivity of the sensor to rotation and a pulley installation error - 1.11 mm, for a nonlinear correction with a quadratic interval number - 0.64 mm). Some ambiguity of the results is due to the fact that there are a lot of sources of errors in inclinometric measurements, and their cumulative or individual influence is not always clear. This is a good reason to discuss this problem at the SGEM thematic conference.
It is worth mentioning separately that if linear systematic errors affect the definition of displacements and shifts, then non-linear errors also distort information about changes in the curvature of the borehole. It is also worth noting that before identifying systematic errors in measurements, one should have a clear understanding of the geological, geomechanical, geotechnical factors that affect the displacement of the object of observation. Otherwise, you can get carried away with the processing of measurements and miss critical deviations.
REFERENCES
[1] Mikkelsen P. E. and Wilson S. D., Field Instrumentation: Accuracy, Performance, Automation, and Procurement In Proceedings, International Symposium on Field Measurements in Geomechanics, Vol. 1 (K. Kovari, ed.), A. A. Balkema, Rotterdam, The Netherlands, 1983, pp. 251 272.
[2] Green, G.E.; Mikkelsen, P.E. Deformation measurements with inclinometers. In Transportation Research Record 1169. Transportation Research Board: Washington, DC, USA, 1998; pp. 1-15.
[3] Mikkelsen P. E., Advances in inclinometer data analysis', in F Myrvoll (ed.), Proceedings of the 6th International Symposium on Field Measurements in Geomechanics, A. A. Balkema, Rotterdam, 2003, pp. 555 567. 10.1201/9781439833483.ch73.
[4] Lovisolo, M. & Della Giusta, A. Precision Of D.M.S. Columns From Real Time In place Measurements And Improvement In Micro-movements Analysis With Early Warning Function, WIT Transactions on Modelling and Simulation, Vol 41, pp.177-186. 10.2495/CMEM050171
[5] Stark, T.D., Choi, H. Slope inclinometers for landslides. Landslides 5, 339 (2008). https://doi.org/10.1007/s10346-008-0126-3
[6] Düllmann, J. & Plinninger, Ralf & Alber, Michael. (2010). On the influence of casing and backfilling on inclinometer measurements along shear planes in hardrock. 33. pp. 152-156.
[7] Septimiu, P.O.P. & Pitica, Dan & Ioan, Ciascai. (2012). Identifying and Removing Systematic Error due to Resistance Tolerance from Measurement System of Inclinometer. Journal of Electrical and Electronics Engineering. 5. pp. 199-202.
[8] Mikkelsen P. E. and DiBiagio E., Depth position errors in inclinometer surveys and false displacement results, in PM Dight (ed.), Proceedings of the Ninth Symposium on Field Measurements in Geomechanics, Australian Centre for Geomechanics, Perth, 2015, pp. 117 123. 10.36487/ACG_rep/1508_03_Mikkelsen
[9] Allasia, Paolo & Godone, Danilo & Giordan, Daniele & Guenzi, Diego & Lollino, Giorgio. (2020). Advances on Measuring Deep-Seated Ground Deformations Using Robotized Inclinometer System. Sensors. 20. 3769. 10.3390/s20133769.
[10] Bannikov A., Gordeev V., The accuracy improvement of angular measurements by the means of systematic errors models incorporating, 20th International Multidisciplinary Scientific Ge°Conference SGEM 2020. Sofia, 2020. P. 53-62. 10.5593/sgem2020/2.2/s09.007
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