Cross-borehole seismic tomography is a powerful tool to investigate the subsurface with a very high spatial resolution. In a set of boreholes, comprehensive three-dimensional investigations at different depths can be conducted to analyse velocity anisotropy effects due to local changes within the medium. Especially in glaciological applications, the drilling of boreholes with hot water is cost-efficient and provides rapid access to the internal structure of the ice. In turn, movements of the subsurface such as the continuous flow of ice masses cause deformations of the boreholes and complicate a precise determination of the source and receiver positions along the borehole trajectories. Here, we present a three-dimensional inversion scheme that considers the deviations of the boreholes as additional model parameters next to the common velocity inversion parameters. Instead of introducing individual parameters for each source and receiver position, we describe the borehole trajectory with two orthogonal polynomials and only invert for the polynomial coefficients. This significantly reduces the number of additional model parameters and leads to much more stable inversion results. In addition, we also discuss whether the inversion of the borehole parameters can be separated from the velocity inversion, which would enhance the flexibility of our inversion scheme. In that case, updates of the borehole trajectories are only performed if this further reduces the overall error in the data sets. We apply this sequential inversion scheme to a synthetic data set and a field data set from a temperate Alpine glacier. With the sequential inversion, the number of artefacts in the velocity model decreases compared to a velocity inversion without borehole adjustments. In combination with a rough approximation of the borehole trajectories, for example, from additional a priori information, heterogeneities in the velocity model can be imaged similarly to an inversion with fully correct borehole coordinates. Furthermore, we discuss the advantages and limitations of our approach in the context of an inherent seismic anisotropy of the medium and extend our algorithm to consider an elliptic velocity anisotropy. With this extended version of the algorithm, we analyse the interference between a seismic anisotropy in the medium and the borehole coordinate adjustment. Our analysis indicates that the borehole inversion interferes with seismic velocity anisotropy. The inversion can compensate for such a velocity anisotropy. Based on the modelling results, we propose considering polynomials up to degree 3. For such a borehole trajectory inversion, third-order polynomials are a good compromise between a good representation of the true borehole trajectories and minimising compensation for velocity anisotropy.

Cross-borehole travel-time tomography, based on seismic or ground-penetrating radar (GPR) waves, is widely used to investigate small-scale variations
of the subsurface, especially when surface-based experiments suffer from poor resolution. The main advantage is the much higher ray coverage within the
target area, allowing a more detailed analysis of small-scale variations of geological heterogeneities. The first experiments with seismic sources
were described by

In contrast to surface-based experiments, the sources and receivers are usually not directly accessible. In general, the borehole geometry is assumed
to be well-known, which allows a correct calculation of distances between sources and receivers. For this purpose, inclinometer and caliper
measurements are required to describe the actual borehole geometry, and centralisers have been used to precisely position the tools in the centre of
the boreholes. Nevertheless, there are some applications for which such estimates are not feasible. For example, for glaciological applications as
described in

In general, the majority of tomography measurements only consider 2D applications. This requires the calculation of a 2D tomographic plane that
contains the boreholes.

However, for 3D inversions, borehole trace corrections are more complex due to the additional degrees of freedom.

This issue becomes even more complicated when considering a subsurface with a certain velocity anisotropy. In glaciers, the ice crystals interact
with the flow of the ice mass. The strain conditions in the glacier force the ice crystals (that is, the

In this study, we propose an approach for travel-time tomography that is explicitly designed for glaciological experiments and apply this to a
multi-cross-hole seismic data set acquired on Rhonegletscher as a case study. The continuous but not necessarily linear movement and ice melt of the
glacier during data acquisition caused a deviation of the holes, leading to uncertainties about the particular source and receiver coordinates.
Inclinometer measurements at the beginning and end of the campaign provide some constraints considered in the inversion. At the same time, we found
that the precision of different instruments was not sufficient under the given measurement conditions on glaciers. In some cases, significant
deviations occurred between the different instruments, although measurements were carried out one after the other. These observations
initially triggered our investigations for a joint velocity and coordinate inversion. However, instead of inverting for the individual source and
receiver positions, we assume that the borehole trajectories can be described with two perpendicular 2D polynomials

The inversion scheme that we apply in this study consists of two parts. The first step is a three-dimensional velocity inversion based on common
ray tracing performed on a finite difference grid. Each cell

With the measured travel times from cross-borehole experiments

The second part of our inversion scheme contains a coordinate inversion that accounts for uncertainties about the borehole trajectories. Although
inclinometer data are usually available, specific conditions such as a variable borehole diameter or a moving subsurface incorporate uncertainties
that significantly affect the velocity inversion and may lead to artefacts complicating an interpretation. An inversion scheme for the individual
coordinates

There are two options for implementing the two parts of the inversion. The first one is a sequential inversion. During each iteration of the inversion, the updates of velocity and borehole trajectories are computed independently. The second option is an extended system of equations that considers both parts in one large Jacobian matrix. We have tested both options with identical settings and updated coordinates in each step of iteration and could not find significant differences between the two methods for synthetic data sets. However, the sequential inversion seems to be numerically more stable. Furthermore, the most recent update of the velocities is already considered in the current step of iteration, and additional constraints determining whether a borehole trajectory inversion should be performed are easier to evaluate. It also provides more flexibility to decide if an update of the coordinates in the current step of iteration is beneficial and thus applied or skipped. Therefore, we calculated the results in the next sections with the sequential inversion.

Initially we encountered the issue of deviating boreholes when acquiring cross-borehole seismic data for anisotropy-related investigations on
Rhonegletscher (Rhone glacier), a temperate glacier (ice temperature

Field data measurement geometry.

The continuous displacements and deformations of the glacier ice body are still imposing problems in our attempt to invert simultaneously for subsurface velocities and the borehole trajectories. In principle, the borehole trajectories would have to be estimated for every single source–receiver borehole pair, but this would lead to a poorly constrained inversion problem. To avoid this problem, we have set up our experimental schedule such that every borehole is only occupied for a relatively short time span (up to 4 d). We then made the assumption that the changes in the trajectories are acceptably small within this short time span (i.e. we have inverted for a single trajectory for each borehole). This is certainly a limitation of our methodology, but it is, in our view, an unavoidable compromise that needs to be made.

The drilling of the boreholes was stopped about 15–20

A 5

For sufficiently dense data coverage, we selected a shot interval of 1

To demonstrate the effect of our borehole correction, we first applied the approach to a synthetic cross-hole seismic data set. For this purpose, we
defined a set of nine boreholes with a length of 80

True model of the synthetic example data set with a homogeneous background velocity of 3800

For the investigations, we defined a heterogeneous velocity model (shown in Fig.

The results of the velocity inversion with correct source and receiver positions are shown in Fig.

Results of the velocity inversion with a synthetic example data set and correct borehole trajectories:

RMSEs for the velocity, the combined inversion, and a reference calculation for a velocity inversion with correct trajectories.

In a next step, the information about the true borehole inclination was ignored and we started with straight vertical boreholes. We ran the inversion without and with borehole trajectory inversion and stopped the inversions after seven iteration steps. Table

Results of the velocity inversion with a synthetic example data set and initially straight borehole trajectories.

The calculations for the inversion were repeated with an additional borehole trajectory inversion. Each borehole is approximated by a set of two
mutually perpendicular polynomials of degree 4. Figure

Results of the sequential (velocity and borehole trajectory) inversion with a synthetic example data set and initially straight borehole trajectories.

Deviations of the individual boreholes (locations shown in Fig.

The borehole trajectories, obtained from the trajectory inversion, are shown in Fig.

Resolution matrix for the coupled velocity and borehole trajectory inversion.

The issue of resolving the individual model parameters can be further investigated with the resolution matrix

For the borehole inversion parameters, the relationship between estimated and true parameters is much stronger, especially for the polynomial
coefficients of degree 1 and 2. As shown in Fig.

The resolution matrix is also a useful tool to investigate whether the model parameters of the velocity inversion also affect the borehole trajectory
inversion and vice versa. If there is a dependency between these parameters we expect to see off-diagonal elements in the resolution matrix. As
shown in Fig.

For the field data set, we determined the travel times of the recorded P waves with a cross-correlation algorithm between repetitive measurements (in
general three repetitions) within a window with a length of up to 4

Inversion results for field data from Rhonegletscher.

RMSEs for the velocity and the combined inversion applied to the field data.

Maximum degree of the two polynomials for each borehole.

Results of borehole trajectory adjustment for field data. Green profiles show initial (dashed line) and final (solid line) borehole trajectories for a setup considering a priori inclinometer data; blue profiles show the borehole trajectories estimated from initially vertical (straight) boreholes.

The rms values of both inversion schemes are similar, as shown in Table

Nevertheless, boreholes BH07 and BH11 show large deviations. Their maximum deviation at the bottom is 0.6 and 1

In summary, our inversion algorithm provides the option to adjust borehole trajectories such that artefacts in the velocity inversion can be removed. For a good three-dimensional adjustment, a reasonably high number of data points along these trajectories for different directions is required. For our specific experimental setup on a continuously moving glacier, this inversion scheme provides improved results.

The combined inversion scheme provides the advantage of a subsequent correction for the borehole coordinates if no or limited information about the
positions of sources and receivers is available. However, there is a risk that the coordinate adjustment will suppress the appearance of real
velocity anomalies in the tomogram. We avoid this by decoupling the two parts of the inverse algorithm. Furthermore, an additional check of whether the
new coordinates reduce the RMSE of the entire data set was implemented. This led to a more cautious adjustment. For our field data set, only
two to three coordinate adjustments in the first iterations were required. Afterwards, the inversion continued with a traditional velocity inversion and
explained the rather similar rms values observed in Table

The idea for a sequential inversion scheme is mainly driven by the findings described in

Next to the issues in the velocity model, we also have to consider the ice flow and thus a changing geometry setup when obtaining the inversion.
Once again, some successful concepts were described in earlier studies. As an example,

An adjustment of the borehole coordinates provides the opportunity to account for deformations of the boreholes due to a movement of the subsurface.
This adjustment is based on the current velocity model as it considers the mean velocity between the source and receiver. However, other physical quantities
such as seismic anisotropy may introduce apparent errors in this seismic velocity values. In glaciers, macrostructural features, such as the
crevasses or englacial channels used in our synthetic data set, may cause velocity anisotropy in the seismic data. Besides this macrostructure,
there is the crystal orientation fabric that can also introduce such anisotropy. This anisotropy appears since the seismic velocity in a single
ice crystal depends on the propagation direction of the seismic wave relative to the

Borehole trajectories for different inversion schemes:

If significant velocity anisotropy is present, this could lead to flawed adjustments of the borehole trajectories, which in turn, reduce or entirely
remove the anisotropy effect. We have investigated this issue with a synthetic data set. The geometric setup consists of eight straight boreholes
(BH01, BH02, BH04, BH05, BH07, BH08, BH10, and BH11 in Fig.

Resolution matrix for a combined inversion for borehole trajectories (third-order polynomials) and four anisotropy parameters (marked with TP for Thomsen parameters). The colours show the dependence of the true model parameter on estimated values. Off-diagonal values indicate interference between different model parameters. The four sectors provide an enhanced overview to distinguish between borehole coefficients and Thomsen parameters.

An application to our field data is difficult to assess since the azimuth and inclination of the COF vary with depth as described in our previous
study:

In this paper, we presented a mathematically simple but efficient approach for a combined velocity and borehole trajectory inversion. This inversion scheme is especially useful for measurements in a deforming subsurface such as experiments in alpine glaciers or ice sheets. The inversion process consists of a typical three-dimensional velocity inversion and an additional borehole trajectory adjustment. All sources and receivers in each borehole are summarised along a trajectory that is characterised by two orthogonal polynomials. The two steps for velocity and borehole inversion are decoupled as shown by the resolution matrix, and thus we only consider further adjustments of the borehole trajectories if this reduces the entire RMSE of the system. With this sequential inversion, we could significantly reduce velocity artefacts that are a result of poorly determined source and receiver positions.

The adjustments are plausible for our field data, but we have also shown that poorly constrained boreholes, e.g. a small number of sources and receivers
along the borehole, may lead to larger deviations and potentially to an overfitting of noise and picking errors. Therefore, a data-set-dependent
damping factor, which considers a priori information, is required. A weakness of our inversion methodology includes the somewhat subjective choice of the regularisation parameters. In future investigations, this may be improved in two ways. When more advanced information on the glacier movements
is available, this could be supplied in the form of further constraints to the inversion problem. Alternatively, the various regularisation parameters could be determined in a more systematic fashion (compared with our trial-and-error approach). A possible option may include generalised
cross-validation

We have also analysed the interdependence between anisotropy and borehole trajectory adjustments. Excessive corrections of the borehole trajectories
can at least partially compensate for the inherent velocity anisotropy. A higher degree of freedom (or flexibility) in the borehole parameters (that
is, the use of higher-order polynomials) leads to a compensation for any weak ellipsoidal anisotropy. Therefore, it is highly recommended to
precisely determine the borehole trajectories during data acquisition. If there is no such a priori information available, the inversion parameters
must be selected accordingly to avoid this issue. We have shown that a polynomial degree of

The synthetic and field data are available in the open-access database ETH Research Collection (

The “Video supplement” mentioned in the text is available in the open-access database ETH Research Collection (

This study was initiated and supervised by HM and AB. SH and HM developed the framework. SH and CP wrote the code for the borehole trajectory inversion and anisotropy investigations. SH, MG, and AB planned the field campaign and acquired the borehole seismic field data. The data processing was conducted by SH and MG. The paper was written by SH, with comments and suggestions for improvements from all co-authors.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Katalin Havas, Johanna Kerch, Dominik Gräff, and Greg Church for their extensive technical and scientific support during data acquisition and processing. We acknowledge Ulrike Werban and Charlotte Krawczyk for the editorial work and the two anonymous referees for their valuable comments that helped to improve this paper.

This research has been supported by the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (grant nos. 200021_169329/1 and 200021_169329/2).

This paper was edited by Ulrike Werban and reviewed by two anonymous referees.