Introduction

Whenever a deep hole is drilled into the surface of the Earth, there is uncertainty about its trajectory. Asymmetries in the drill, layering in the material being drilled and shortcomings in drilling technique may cause the borehole to stray from its intended path (e.g. Reference WalkerWalker, 1986). Inclinometry tools are designed to remove spatial uncertainty by mapping the deviation of the borehole. Typically, two pieces of information are collected at a series of stations, or locations, along the length of the hole: the tilt of the inclinometer from vertical and the azimuth of that tilt in a geographical coordinate system. Of the many techniques available for making these measurements, we present a brief overview of those that have been used for glaciological work. Sources for some of this inclinometry equipment are given.

In this paper, we attempt to formalize some aspects of inclinometry-data analysis and, at the same time, introduce a new inclinometry tool that was recently assembled at the University of British Columbia (UBC). We describe the calibration procedure for the UBC inclinometer, formulate a general theory for interpretation of inclinometer data and examine the sensitivity of this theory to error in the data.

Historical Overview

Inclinometry of glacier boreholes has long been associated with field investigations of the flow law for ice (Perutz, Reference Perutz1947, Reference Perutz1949, Reference Perutz1950; Reference Gerrard, Perutz and RochGerrard and others, 1952; Reference SharpSharp, 1953; Nye, Reference Nye1952, Reference Nye1953, Reference Nye1957; Reference MillerMiller, 1958; Reference Paterson and SavagePaterson and Savage, 1963; Reference Savage and PatersonSavage and Paterson, 1963; Reference ShreveShreve and Sharp, 1970; Raymond, Reference Raymond1971b, Reference Raymondc; Reference HookeHooke, 1973; Reference Garfield and UedaGarfield and Ueda, 1976; Reference PatersonPaterson, 1983; Reference Hooke and HansonHooke and Hanson, 1986; Reference Hooke, Holmlund and IversonHooke and others, 1987). Many of these researchers used inclinometry data to infer the internal velocity field of the glacier and thereby deduce parameters of the flow law. Increasing interest in basal processes has created a new application for inclinometry, that of accurately determining the position of englacial and subglacial sensors. As an example, the measurement of subglacial resistivity (Reference Brand, Pohjola and HookeBrand and others, 1987) requires knowledge of the spatial position of current and voltage electrodes within adjacent boreholes.

Electronic Inclinometry

In recent years, advances in electronic technology have enabled the development of inclinometry systems that transmit position information directly to the surface. Compared with earlier techniques, the time and labour savings of electronic inclinometry are appreciable; with appropriate equipment in place at the borehole site, the trajectory of the borehole can even be computed and displayed as the data are collected.

The UBC Inclinometer

In the spring of 1989, the glaciology group at the University of British Columbia, in collaboration with Slope Indicator Canada Limited (Richmond, British Columbia) and Narod Geophysics (Vancouver, British Columbia), developed a prototype inclinometer. This instrument is similar to the Hydro-Tech instrument in that it uses the Earth’s magnetic field as an azimuth reference, but the field-sensing device is a fluxgate magnetometer rather than a mechanical compass. Two force-balance tilt transducers and the magnetometer are enclosed within a non-magnetic stainless-steel pressure tube (see Fig. 1). Brass centralizing springs help keep the inclinometer centred in the borehole and prevent the instrument from spinning when on station. The tube is 1.5 m long and has an outside diameter of 2.54 cm (1.00in). The tilt transducers are Slope Indicator devices, identical to those used in their tracked inclinometers. The magnetometer is a Narod Geophysics miniature ring-core fluxgate magnetometer. (Reference WyckoffWyckoff (1948) gave an overview of the principals of operation for a fluxgate magnetometer; more detailed treatments related to this magnetometer can be found in: United States Patent 3 800 213 (26 March 1974), UK Patent 2 044 460 (16 October 1979), Reference Russell, Narod and KollarRussell and others (1983), Reference Narod and RussellNarod and Russell (1984) and Reference Narod, Bennest, Strom-Olsen, Nezil and DunlapNarod and others (1985)). The sensing element of this three-component magnetometer is a 12.7 mm (0.50 in) cube. As the tilt sensors contain permanent magnets, the magnetometer is mounted as far from the tilt sensors as possible. Ideally, the magnetometer would be placed at the bottom of the instrument, far from any secondary magnetic fields, but the tilt sensors completely fill the interior of the pressure tube so that no wires can pass by them; thus the tilt sensors must be positioned at the bottom and the magnetometer at the top. The space between these sensors is occupied by the electronics package. The instrument cable attaches to the top of the instrument. This cable carries power to the instrument and analogue data to the surface, and has a steel strength member. The magnetic bias created by the steel cable and the electric currents flowing in the cable is compensated by the magnetometer calibration.

Fig. 1.

A block diagram of the peripheral devices attached to the UBC inclinometer (inclinometer not drawn to scale). The power module provides power to the instrument from a small 6 Ah sealed lead-acid battery. Analogue signals are fed to the data logger which, on command from the handheld control box, records data from the tilt sensors and magnetometer into the storage module. The operator keys in depth information on the control box.

Data are collected by a data logger mounted inside a weather-tight plastic case. For each hole, the inclinometer is stepped down and up at 1 m intervals, resulting in a two-fold redundancy in the measurements. The depth of the inclinometer at each station, as marked on the cable, is coded manually on a digital potentiometer that is mounted in a hand-held control box. After the operator closes a momentary switch, indicating that the inclinometer is on station, the data logger records the time, the digital values of the tilt and magnetometer signals, and the voltage on the potentiometer. The data logger copies the data into the storage module. At the end of the day, the storage module is carried to the field camp where the data are transferred to a computer for analysis.

This inclinometer system is very efficient. A single operator can make two inclinometry passes through a 70 m hole at 1 m depth intervals in 25 min.

Coordinate Systems

Three right-handed Cartesian coordinate systems are required to process data from a tracked or gyroscopically oriented inclinometer. Two additional coordinate systems associated with the Earth’s magnetic field and the magnetometer are required to process data from a magnetically oriented inclinometer (Fig. 2). The coordinate system introduced by Reference NyeNye (1957), and used in several subsequent papers by Nye and others, is aligned with the flow direction of the glacier. This is natural for studies of internal deformation but, because our primary concern is locating sensors placed within boreholes, we choose a system related to map coordinates.

Fig. 2.

The five righthanded inclinometer coordinate systems displayed as a stereo pair. Top: the geographical coordinate system U and the geomagnetic coordinate system H. The angle “d” between the grid north axis y_U and the magnetic north axis x_H is the magnetic declination. Bottom: the tilt sensor (T), case (C), and magnetometer (B) coordinate systems share a common z axis pointing downward along the body of the inclinometer.

The geographical coordinate system U is tied to the Universal Transverse Mercator (UTM) coordinate system used on Canadian topographic maps. The x_U y_U and z_U axes point east, north and upward, respectively. The geomagnetic coordinate system (labelled Η after the standard designation for magnetic field) is tied to the local declination of the Eart’s magnetic field. The Z_H axis is positive downward and the x_H axis points along the magnetic declination. This definition follows the international convention for geomagnetic coordinate systems and results in positive Z_H readings in the Northern Magnetic Hemisphere. The remaining three coordinate systems have z-axes that point downward along the axis of the inclinometer. The tilt-sensor coordinate system Τ has its x_T and y_T axes aligned with the sensitive axes of the tilt sensors. The case coordinate system C has the x_C axis aligned with some feature on the inclinometer. In the case of a tracked inclinometer, this would be the alignment mechanism. The UBC instrument has its x_C axis aligned with a machined facet on the inclinometer case (the use of this facet is discussed below). The magnetometer coordinate system (labelled Β after the standard designation for magnetic flux density) represents the three axes of sensitivity of the fluxgate magnetometer. For the UBC inclinometer, the three instrument-based coordinate systems are distinct, and the rotational angles between the x_T , x_C and x_B axes must be known in order to process data. For other instruments, these three systems may or may not coincide.

Data Analysis

The analysis of inclinometer data proceeds in three distinct steps: (1) The instrument calibrations are used to compute the two tilt angles and the azimuth at each station. (2) The tilt and azimuth values are transformed into vectors representing the orientation of the inclinometer at each station. (3) An inversion scheme computes the continuous trajectory of the borehole based on the discrete set of orientations.

In the following two sections, we discuss in detail the calibration and transformation procedures for the UBC inclinometer. Although these procedures vary from instrument to instrument, all share the common goal of determining the vertical unit vector and the orientation unit vector in an inclinometer coordinate system (Fig. 3). For a magnetically oriented inclinometer, m represents the Earth’s magnetic field vector and, for a tracked inclinometer, represents the orientation of the track grooves. The UBC inclinometer yields the true magnetic vector, whereas an instrument that uses a gimbal-mounted compass yields the magnetic vector projected on to the gravitationally horizontal plane.

Fig. 3.

A stereogram showing the two unit vectors measured by an inclinometer and the tilt coordinate system Τ in which they are resolved. The till sensors yield and the orientation apparatus yields in .

Calibration

The tilt sensors are calibrated using a calibration frame that allows positioning of the inclinometer at precise tilt angles (to within 1 min of arc) throughout its operating range of ±30° of tilt from vertical. Repeated calibration of the instrument between field seasons indicates negligible drift in the calibration. The output voltage from the tilt sensors is proportional to the sine of the tilt angle from vertical. In the Τ system, the components of are given by

(1)

where I_x and I_y are the two tilt angles.

The calculation of the orientation vector would be trivial if the three axes of the magnetometer were to have zero offset and equal sensitivity but, in practice, the design of the magnetometer sensor makes it very difficult to realize these characteristics, particularly with regard to voltage offsets. A calibration is required. If a series of measurements of the Earth’s magnetic field is taken with the magnetometer positioned at many orientations within the field, then the measured points will map out a surface in the Β system. If we assume that the axes of sensitivity of the magnetometer (with their non-zero offsets and unequal sensitivities) are mutually orthogonal, then the surface is a translated ellipsoid. The calibration function we seek transforms this ellipsoid into a unit sphere centred on the origin; applying the calibration to any given point will produce the corresponding orientation vector.

The calibration data are best collected on the glacier surface where the magnetic environment can be controlled. The operator, stripped of magnetic clothing, holds the inclinometer at a succession of orientations. At each position, the inclinometer is held steady while the three magnetometer outputs are recorded. The orientations are chosen such that the Earth’s magnetic field intersects the magnetometer sensor from as many directions as the patience of the operator allows. The specific calibration orientations are not important, so a hand-held calibration can be used. In addition to instrument error, reading errors result from fluctuations in the Earth’s magnetic field during the calibration and movement of the inclinometer as data are recorded (the band width of the magnetometer is 5 Hz and the three components are measured sequentially within a span of 114 ms).

Typically, about 200–500 calibration triplets are collected. A calibration transformation that maps a data point M(_B) on this ellipsoid on to a unit sphere centred on the origin is given by

(2)

where Τ is an offset vector, S is a scaling matrix, and m_(B) is the transformed point. The six free parameters (S_xx)S_yy,S_zz,T_x,T_y,T_z) are determined by minimizing the objective function

(3)

for the n transformed calibration triplets (S_xx, S_yy and S_zz are non-negative). An algebraic approach to solving the minimization is algebraically horrific, if not intractable, so we use an iterative six-dimensional simplex algorithm (Reference Press, Flannery, Teukolsky and VetterlingPress and others, 1986, p. 289). For the UBC inclinometer, the transformed calibration data fit a unit sphere to within 2%. Removing the constraint on magnetometer-axis orthogonality by allowing non-zero off-diagonal terms in calibration matrix S results in a negligible improvement in fit, at the expense of greatly increasing the geometrical complexity of data analysis.

Transformations

In this section, we discuss the transformations that are applied to data from the UBC inclinometer. Some of these transformations are generic in nature and can be used with all inclinometers.

In order to simplify the analysis, the scaled magnetic vector m_(B) must be rotated about the inclinometer axis from the Β system into the Τ system (Fig. 2). The angle Ω, measured from the x_B axis to the x_T axis, is calculated as the sum of two intermediate angles. The angle between the x_T and the x_C axes can be measured accurately on the tilt calibration frame. The angle between the x_C and x_B axes is not as well constrained. Repeated sightings on known geographical reference points are made through a telescope clamped to the machined facet on the body of the inclinometer. By reconciling the magnetic declination published on the 1:50 000 topographic map (Energy, Mines and Resources, Canada map sheet 115 F/1, Edition 1, 1987; covers Trapridge Glacier, Yukon) with the coordinates of the reference points and the coordinates of the observation site, the value of this angle can be computed to within ±0.5°. Weather permitting, these reference sightings are made before and after each hole is logged. This allows correction for temporal variations in magnetic declin-ation caused by geomagnetic disturbances.

The calibrated magnetic vector m_(B), from Equation (2), is nominally a unit vector, but errors in the data will result in a slightly erroneous vector magnitude. If the magnitude is not corrected, then the components of m_(B), and vectors derived from it, cannot be used as direction cosines. Therefore, the unit magnetic vector is defined by rotating and normalizing m_(B) using the equation

(4)

where m_(B) denotes the scalar length of vector m_(B).

Examination of Figure 3 reveals that, in the case where is the full magnetic field vector, there is redundancy in the data. The three degrees of freedom representing the orientation of the inclinometer body are constrained by the four degrees of freedom embodied in the components of and

, which in turn are derived from the five instrument readings (I_x, I_y, m_x(B), m_y(B) and M_z(B)). The two additional degrees of freedom governed by the instrument readings are the magnitude of the magnetic field vector (lost in normalizing

) and the spin orientation of the inclinometer on its axis (in our analysis, this information is discarded). For the UBC inclinometer, the value of is far better determined than

, so we use to define the inclination angle of the z_T axis. Vector is used to constrain the azimuth of the z_T axis about vector . The functions of and can be exchanged but, were used as the inclination reference, accuracy would be lost. Note that data from magnetically oriented inclinometers become indeterminate when and are parallel or anti-parallel.

Our objective now is to find a set of Eulerian transformation angles ϕ, θ and ψ (see Appendix A) which will map

, as expressed in the Τ system, on to the U system such that the horizontal projection of rii has the proper magnetic declination d. By casting the transformation in this way, we avoid directly implicating the Η system. The transformation must also map

, as expressed in the Τ system, on to the vertical in the U system. We will treat the U system as unprimed and the Τ system as primed (Appendix A, Equation (A2)).

The angle θ defines the net tilt of the z_T axis with respect to the vertical z_U axis. This net tilt is simply

(5)

The angle ψ controls the relative contribution of the two tilt angles to the net tilt, defined as

(6)

The angle ϕ rotates the Τ system about the z_U axis; in other words, ϕ is the Eulerian angle that controls the declination of the transformed magnetic vector. Given θ and ψ, the value of θ must satisfy

(7)

where 0 < F ≤1 is the magnitude of when projected on to the horizontal plane and d is the magnetic declination east of UTM north. This system of equations has an explicit solution for ϕ and F, but the algebra can be simplified considerably by first solving for the case where d =0 and subsequently adjusting the solution. This trick results in an indirect association with the Η system. The equation to solve becomes

(8)

and the solution is

(9)

(10)

The non-zero declination d is reinstated by modifying Equation (9) to give , m_x cos φ) — my sin φ

(11)

Equations (9) and (11) are curiously insensitive to the value of m_z in the sense that a value of m derived from a gimbal-mounted compass (i.e. is horizontal) will give the same value for ϕ as a value of representing the full Earth field, but this insensitivity is expected. Both gimbal-mounted and full-field magnetometers are equally adept at determining magnetic bearing.

As the final step in the data-transformation procedure, the orientation vector of the inclinometer (which corresponds to the z_T axis), as expressed in the U system, is computed by the transformation

(12)

The Eulerian transformation matrix A is defined in Table 1 and by Equations (5),(6) and (11). For this analysis the value of F is irrelevant.

Table 1.

The Eulerian transformatio matrix

Universal Application of Transformations

Note that Equations (5), (6), (9) and (12) apply to any inclinometry data that can be expressed in terms of a tilt vector ft and an orientation vector . This includes data from tracked and gyroscopically oriented inclinometers.

Inverse Problem

In this section, we develop a general theory, applicable to all inclinometers, for computing a continuous borehole trajectory based on discrete measurements of borehole depth and inclinometer orientation. Before considering the details of this step, it is appropriate to consider the relationship between the inclinometer orientation and the borehole axis.

The simplest assumed relationship is that the inclinometer is always parallel to the borehole axis. Although this presumption is erroneous, without additional data describing the borehole geometry and centring device geometry, it is difficult to improve upon. At any given station along the borehole, the inclinometer axis lies along a straight line drawn between the centring devices. If the two centring devices operate identically, the borehole has smooth walls and constant diameter, and the curvature of the borehole is a circular arc, then the inclinometer axis will lie parallel to the borehole axis at a point midway between the centring devices. In practice, these three conditions are rarely satisfied. It is here that tracked inclinometers exhibit improved performance over instruments such as the UBC tool; the groove-tracking wheels can accurately position the inclinometer in the centre of the casing and the casing provides both smooth walls and regular borehole geometry. The trajectory computation method described below will assume that the borehole and inclinometer axes coincide; the sensitivity of the method to positioning error will be discussed at the end of the paper.

The task of computing a continuous borehole trajectory from a set of discrete tilt and azimuth measurements constitutes an inverse problem. An infinite number of possible trajectory solutions exist that will satisfy a given data set exactly. An infinite number of additional solutions exists that do not satisfy the data exactly, but which do fall within the error bounds of the data. A single solution of the first type can be isolated by placing appropriate geometrical constraints on the trajectory; the solution to the inverse problem becomes an interpolation scheme. All treatises of which we are aware, this paper included, use geometrical constraints to choose an exact solution that will hopefully reflect the true trajectory. This method for solving an inverse problem differs from traditional methods in that an a priori structural model is solely responsible for isolating one of the possible solutions. Traditional methods can find solutions of the second type by setting an objective function, such as minimizing the rate of change of borehole tilt, and by allowing for assessment of error in the data. Data smoothing is another possible approach to finding inexact trajectory solutions.

Many methods for interpolating slope angles have been discussed in the petroleum industry literature. The “terminal-angle tangential method” is often mentioned as the best-known interpolation method: over any station interval, the orientation of the inclinometer axis is assumed to be equal to that of the lower station. In effect, no interpolation is performed and the slope of the computed trajectory is discontinuous at every station. Because of its faulty representation of inclinometer motion, this method results in appreciable error. Reference Walstrom, Harvey and EddyWalstrom and others (1972) discussed five different methods for interpolating the tilt angle and azimuth between stations. The underlying premise of many of their mathematical models is that tilt angle and azimuth can be treated as independent quantities. This premise does not always hold true, as is revealed by considering the radius-of-curvature method developed by Reference WilsonWilson (1968) and later expanded by Reference RiveroRivero (1971). Their method maps the borehole trajectory on to the surface of a vertical cylinder such that the desired azimuth and tilt are preserved at the end points of the spiral segment. The resulting trajectory is not independent of the coordinate system in which the projection is made, indicating that in this case the tilt angle and azimuth cannot be treated separately. Angle-interpolation methods are also cluttered with special treatment for the multivalued nature of the inverse trigonometric functions.

We assume that the path of the inclinometer between stations can be described by a series of circular arc sections (Fig. 4). Figure 5 shows one of these arcs in detail. The end points of the arc (P1, P2) are tangential to the normalized orientation vectors (

), and the length of the arc L equals the measured depth increment on the cable. Successive arcs join in a smooth manner with no discontinuities in slope, although discontinuities in the direction of slope change are present. If the position of P₁ is known, then the position of P₂ or any intermediate point P₃ along the arc can be computed. This method has been discussed by Reference ZarembaZaremba (1973), but Zaremba derived an unnecessarily complicated solution.

Fig. 4.

A perspective view of the circular-arc interpolation model. Between pairs of measurement stations, the borehole is assumed to follow a circular trajectory. The arc length is constrained by the measured distance between stations, and the plane of the arc is defined by the instrument-orientation vectors at the end points of the arc. These orientation vectors are tangential to the arc.

Fig. 5.

A detailed view of one circular arc in the borehole trajectory. Knowing the position of the starting point P₁1, the tangent vectors and

, and the length of the arc L, the position of the end point P₂ can be computed. The position of any intermediate point P₃ on the arc can also be computed. The angle γ is defined by and .

Our solution for P₂ is

(13)

where

(14)

In the limit, where (the anti-parallel case is unlikely), the circular arc will have infinite radius (γ → 0), and the solution for P₂ becomes

(15)

A full derivation of the solution is found in Appendix B. A solution for a point P₃ at an arbitrary position along the arc is also given. The trajectory computed by this circular-arc method is independent of the coordinate system orientation. The trajectory of the borehole is reconstructed by successively applying Equations (13) and (15), as appropriate, beginning at the glacier surface and working downward.

Sensitivity

The trajectory of the borehole as determined by the circular-arc method, or by any other interpolation method, is inherently incorrect — even if the borehole-orientation data are error-free. This is because the continuous borehole trajectory is sampled at a finite number of points. In the case of the UBC inclinometer, we do not believe that sampling density is a major source of error because our 1 m sampling interval is short compared to the length of the inclinometer and the drill stem used to drill the holes; we do not expect perturbations in the borehole trajectory on a scale smaller than 1 m. For this dense sampling interval, and given error-free borehole-orientation data, the divergence between the true trajectory and the computed trajectory is expected to be at most a few centimetres. Problems arise because the borehole-orientation data are not error-free.

The UBC inclinometer has logged 125 boreholes during the 1989 and 1990 field seasons at Trapridge Glacier, Yukon. Each hole was logged at least twice. By comparing the inclinometry results, we estimate that the instrument locates the bottom of a 70 m hole to within 20–30 cm. The error tends to be largest in the azimuthal sense; the radial deviation of a borehole is determined to within 15 cm. Based on this evidence, we identify two major sources of error: random-posidoning error of the inclinometer in the borehole and systematic orientation error. Superimposed on these error terms are the measurement errors of the tilt sensors and magnetometer.

Accurate positioning of a non-tracked inclinometer within a borehole is dependent on the texture of the borehole wall and the performance of the centring devices. Glacier borehole walls are not necessarily smooth and the centring devices allow the inclinometer to cant relative to the borehole axis. The discrepancy between the axis of the borehole and the axis of the inclinometer is estimated at ±0.5°. This estimate is based on the diameter of the borehole and that of the inclinometer with the centring-devices fully compressed. It is also likely that the inclinometer tends to underestimate the borehole tilt since tension in the cable and the instrument weight work to force the instrument towards a vertical orientation. As this effect depends on the lay of the cable and the local hole geometry, it is impractical to quantify; we satisfy ourselves with the ±0.5° positioning error. The tilt-sensor error (less than one part in 10000) is neglected. The magnetometer error is appreciable and amounts to ±2% on the x_B and y_B axes, and ±3% on the z_B > axis. These estimates are based on the calibration fit and the specifications for the magnetometer. Neglecting stretch in the cable, the error in positioning along the borehole is estimated at 1 cm for each 1 m interval.

The effects of the random errors can be modelled using Monte Carlo techniques. Synthetic data are generated for a sequence of stations with error terms superimposed on the “correct” data. The standard deviation of the Gaussian-distributed error for each reading is given above. Relative contributions to the net deviation can be examined by selectively removing the error terms. Figure 6 shows the results from a series of Monte Carlo tests on a 70 station hole (69 m long). The figure shows horizontal maps of the computed bottom locations for each Monte Carlo test. The underlying model is a linear hole dipping 10° to the south with the top at the origin. Each test contains 1000 runs through the hole. Figure 6a shows the effects of applying all error terms simultaneously. The cross hair indicates the location of the error-free hole bottom at 11.98 m south. Note the slight stretching of the “data cloud” in the azimuthal (east-west) sense and the offset towards the north (origin). Both of these effects are primarily due to error introduced by the magnetometer, as can be demonstrated by removing the magnetometer and depth error (Fig. 6b). This results in a more symmetric cloud, with less noticeable offset. Figure 6c shows how the magnetometer introduces these two distortions. The east-west lineation is actually a short section of a circle with its centre at the origin. This is easily seen in Figure 6d where gross errors (±50%) in the magnetometer over one station interval result in a circular scattering envelope. Again, the cross hair indicates the error-free solution. The foreshortening (drawing in towards the origin) observed in Figures 6a, c and d results from the fact that error in azimuth will always cause the radius of the interpolating circular arc to decrease; this results in a horizontal projection that is always smaller than, or equal to, the true projection. On average, error in tilt measurements also produces foreshortening (Fig. 6b). Note that, for holes of this depth, the foreshortening amounts to no more than a few centimetres. In Figure 6a, the net depth error amounts to no more than ±2 cm.

Fig. 6.

Monte Carlo modelling of the sensitivity of the inclinometer to error in its sensors is shown as a series of scatter plots. Panels (a),(b) and (c) show the horizontal deviation of the computed hole bottom from its true position for 1000 Monte Carlo runs down a 69 m hole dipping southward at 10° (station interval of 1 m). The hole top is at the origin and the true bottom position is indicated by the cross hair, (a) The combined effect of a ±0.5° error in tilt, a 2% error in m_x and m_y, a 3% error in m_z and a 1 cm m₋₁ error in depth control, (b) The effect of tilt error alone, (c) The magnetometer error alone, (d) Scatter produced by allowing gross (50%)error in the components of over one station interval.

Systematic error is caused by the non-random orientation of the inclinometer as it moves through the borehole. The orientation of the inclinometer does not change greatly as it moves from station to station but, in separate passes through the hole, the general orientation of the inclinometer may be quite different. It follows that any offset error in determining the angle Ω between the x_T and x_B axes will accumulate differently in different inclinometer passes, and this will result in additional azimuthal error. By adjusting the value of Ω such that the azimuthal error is minimized for all holes, we can remove much of the systematic error.

Our estimated positioning error of 20–30 cm, with slightly better radial control, is consistent with the results of the Monte Carlo test in Figure 6a. This leads us to conclude that we have a good understanding of the factors influencing the performance of our inclinometer.

Discussion

Our experience with the UBC inclinometer has shown that it is an efficient and accurate instrument for glaciological work. Surveys are performed quickly and do not require any special attention to the instrument or preparation of the borehole. For these reasons, we expect that magnetically oriented inclinometers using fluxgate magnetometers will gain popularity with glaciologists. For rigorous applications, some researchers may wish to combine the better tilt accuracy of a tracked inclinometer with the additional check on orientation provided by a magnetometer.

The principal shortcoming of these instruments is that they become confused at very high magnetic latitudes. We estimate that the UBC inclinometer will not operate reliably where the dip of the magnetic field exceeds 85°. This constraint probably excludes its use throughout most of the Canadian Arctic archipelago and the Wilkes Coast and George V Land in Antarctica. We know that the accuracy of the magnetometer is being degraded by its close proximity to electrical currents inside the pressure tube. Redesigning the internal configuration of the instrument so as to avoid this magnetic noise would reduce the error observed in Figure 6c, and might allow the instrument to be used closer to the Earth’s magnetic poles.

These restrictions aside, the error in determining the position of subglacial and englacial sensors with the UBC inclinometer is small enough for all but the most exacting experiments. We are seeking to improve the performance of the instrument, primarily by redesigning the centralizing springs. If this results in reduced error in the tilt readings, then the instrument might approach the sensitivity represented by Figure 6c.

The circular-arc interpolation scheme is simple to apply and easy to visualize. On a 12 MHz IBM-PC-compatible computer equipped with a numerical coprocessor, the time required to calibrate, process and plot the data from a 70 m borehole takes a fraction of a second. The foreshortening effect of the circular-arc method is the only drawback of which we are aware, but this effect is not unique to this interpolation method.

We expect that improvements in the accuracy of inclinometry-data inversion can be obtained in at least three ways: (1) Incorporate additional borehole geometry and centring-device geometry. (2) Study the effects of centring-device spacing relative to station spacing. (3) Introduce full inversion techniques that produce trajectories having an imperfect fit, within error, to the data. Data smoothing is one strategy for finding these imperfectly fitting trajectories, but it is important that the smoothing process does not result in violation of the error bounds on the data. We invite other interested parties to consider these approaches.