Acoustic velocity measurements for detecting the crystal orientation fabrics of a temperate ice core

The crystal orientation fabric (COF) in ice cores provides detailed information, such as grain size and distribution and the orientation of the crystals in relation to the largescale glacier flow. These data are relevant for a profound understanding of the dynamics and deformation history of glaciers and ice sheets. The intrinsic, mechanical anisotropy of the ice crystals causes an anisotropy of the polycrystalline ice of glaciers and affects the velocity of acoustic waves propagating through the ice. Here, we employ such acoustic waves to obtain the seismic anisotropy of ice core samples and compare the results with calculated acoustic velocities derived from COF analyses. These samples originate from an ice core from Rhonegletscher (Rhone Glacier), a temperate glacier in the Swiss Alps. Point-contact transducers transmit ultrasonic P waves with a dominant frequency of 1 MHz into the ice core samples and measure variations in the travel times of these waves for a set of azimuthal angles. In addition, the elasticity tensor is obtained from laboratorymeasured COF, and we calculate the associated seismic velocities. We compare these COF-derived velocity profiles with the measured ultrasonic profiles. Especially in the presence of large ice grains, these two methods show significantly different velocities since the ultrasonic measurements examine a limited volume of the ice core, whereas the COFderived velocities are integrated over larger parts of the core. This discrepancy between the ultrasonic and COF-derived profiles decreases with an increasing number of grains that are available within the sampling volume, and both methods provide consistent results in the presence of a similar amount of grains. We also explore the limitations of ultrasonic measurements and provide suggestions for improving their results. These ultrasonic measurements could be employed continuously along the ice cores. They are suitable to support the COF analyses by bridging the gaps between discrete measurements since these ultrasonic measurements can be acquired within minutes and do not require an extensive preparation of ice samples when using point-contact transducers.


Seismic velocities from COF
The hexagonal crystal structure of an ice monocrystal causes an anisotropy in its elastic parameters and therefore affects the propagation velocity of seismic waves. As a result of the crystallographic symmetry, the acoustic velocity parallel to the c-axis, which corresponds to the optical axis perpendicular to the basal planes of the ice crystal lattice (see e.g. Cuffey and Paterson, 2010), differs significantly from the velocity in direction of the basal plane. This seismic anisotropy of an ice crystal is fully 95 described by the fourth order elasticity tensor C ijkl , i,j,k,l = 1, 2, 3 (e.g. Aki and Richards, 2002). The velocity of an acoustic wave with any inclination and azimuthal direction can be calculated analytically (Tsvankin, 2001), provided the mass density of ice is known.
Due to the symmetry relations (Voigt, 1910) the 81 unknown elements of the tensor can be reduced to 21 elements. The hexagonal symmetry of ice further reduces the number of independent constants to five for a monocrystal. For the determination 100 of a representative elasticity tensor for a polycrystalline medium, we follow the approach of Kerch et al. (2018).A detailed description on calculating the polycrystalline tensor can be found there.
The theoretical framework calculates the effective elasticity tensor and derives the seismic velocities from this tensor. Then, the velocities are derived by solving the Christoffel Equation (e.g. Tsvankin, 2001, Ch. 1.1.2). According to Maurel et al. (2016), this approach for an effective elasticity tensor provides more accurate results (at least for some specific textures) than the 105 complementary velocity averaging method (i.e. calculating the velocities for the individual crystals and computing the average velocity for the polycrystalline medium afterwards). Here, we only summarise the key points for calculating the effective elasticity tenor.
-This approach is based on an earlier study of Diez and Eisen (2015). However, the framework of Diez and Eisen (2015) relies on particular COF patterns, such as a thick and partial girdle or a single maximum structure and their representation 110 through the eigenvalues of the orientation tensor. Kerch et al. (2018) do not presume specific COF patterns, which makes it most suitable for our dataset.
-It then considers the elements of a monocrystal tensor C m precisely determined in laboratory experiments. In our study, we used the elasticity tensor of Bennett,  Bennett (1968) for T = -10 • C. This provides the best agreement between our calculated and measured data in both our study and in earlier experiments (Diez et al., 2015, Tab. 1).
-For each ice grain i the monocrystal tensor C m is transformed into where R ϑ is the rotational matrix around the vertical axis, R ϕ is the rotational matrix around geographic north and ϑ 120 and ϕ are azimuth and colatitude angle of the grain i, respectively. This aligns the elasticity tensor with the coordinate system of the ice core (transformation of the coordinate system).
-The rotated monocrystal tensors are summed up elementwise thereby assuming a superposition of all n G grains and their respective properties. The relative grain sizes are used as 125 weighting factors w G (i) for each grain. The resulting polycrystalline tensor does not have a hexagonal structure anymore, but a triclinic structure with 21 independent elements.
An ultrasonic point-contact transducer transmitted an acoustic signal into the ice. This signal was recorded by a second transducer on the opposite side of the core. In the current experimental setup only measurements parallel and perpendicular to the 150 vertical axis of the ice core (colatitude ϕ = 0/90 • ) were considered. The azimuthal coverage for ϕ = 90 • was ∆ϑ = 15 • between 0-345 • . Figure 1a shows the experimental setup that consists of a pulse generator, an oscilloscope, and a set of point contact transducers. The pulse generator (LeCroy wave station) was employed to generate a pulse with a dominant frequency of 1 MHz 155 and a repetition rate of 10 ms. This electric signal was amplified (amplifiers not shown in Fig. 1a) and a point contact (PC) transducer converted it into an acoustic signal and transmitted it into the ice. This transducer was manufactured in-house at ETH Zurich and provides a stable and highly repeatable sources over a wide range of radiation angles due to its broadband instrument response. This instrument response was calculated in advance using the capillary fracture methods described in Selvadurai (2019). A second transducer (type KRNBB-PC) received and converted the acoustic signal into an electronic pulse, 160 which was transferred to a digital oscilloscope (LeCroy Wavesurfer 3024). For each measurement, we stacked at least 20 individual waveforms to enhance the signal-to-noise ratio. Since the amplifiers caused delays, we determined the actual zero-time of the entire system by a regression through repeated measurements on steel cylinders with precisely determined lengths. These calibration measurements were performed at least twice a day under identical temperature conditions.

X-ray measurements for air content estimation
In addition to the ultrasonic measurements, the porosity (i.e. the volume of air within the ice) was analysed by X-ray microcomputer tomography (CT) scans. For the scanning and analysis, we followed the same procedures previously adopted for bubbly ice from Dome C (  Therefore, we classify the images as two-phase systems with air bubbles in ice.

Acoustic velocities inferred from COF
Seven ice core samples, obtained from 2, 22, 33, 45, 52, 65 and 79 m depth, were analysed. The corresponding COF patterns (presented in Figs. 2a -2g) are obtained from a set of ice core thin sections from two adjacent ice core segments. They exhibit 195 clear multi-maxima patterns in all samples, consisting of four (five for 65 m) significant clusters of c-axes. These clusters always form a "diamond shape" pattern and have been found to be typical for temperate ice with branched, large ice grains.
We employed a spherical k-means clustering algorithm (Nguyen, 2020) (Table 1) and reaches a maximum value of at 79 m between the global maximum (around vertical direction) and minimum velocity values (Fig. 8n).
The p-wave velocity for vertically incident waves (parallel to z-axis of the core) increases with depth, especially for the deepest parts, where the cluster is centered around the vertical axis ( Fig. 3a blue line). The p-wave velocities for a colatitude of ϕ = 90 • 215 (horizontal direction) are shown in Fig. 3b (mean value per sample) and Fig. 4. The largest azimuthal variations appear at 2 m since the c-axes of the grains cluster around a horizontally oriented centroid (ϕ c = 88.6 • ). The maximum horizontal anisotropy is 1.4 %. Table 1. Mean, minimum, maximum :::::::: calculated p-wave velocity :::::: (without :: air ::::::::: correction) : and grade :::: degree : of anisotropy for each

Acoustic velocities from ultrasonic experiments
We measured the acoustic velocities on five of the above-mentioned ice core samples. The ice core samples were taken from 220 2, 22, 33, 45 and 65 m depth and usually from the upper of the two ice core segments that have been used for the COF analysis (cf. Fig. 1c). The distance between the uppermost COF thin section and the ultrasonic sample is between 5 and 15 cm (with an exception for 65 m with an offset of 60 cm). For each sample, we carried out three individual horizontal measurements on three different levels (indicated as z 1 , z 2 , z 3 in Fig. 1a). For the vertical measurements, we obtained one measurement per sample. As there was a half cylinder of ice from the lowermost depth (79 m) available, we also measured the vertical velocity 225 for this sample. The ultrasonic measurements were conducted using different pieces of ice than used for the COF analysis and therefore, the actual grain size and distribution remain unknown. The positions of the ultrasonic measurements are marked in In a first step, the recorded traces were shifted to correct for zero-time t 0 and the p-wave arrivals (example shown in Fig. 1b) were picked. Additionally, the ice core diameter for each azimuth was measured and since the core was not perfectly round, 230 the diameter varied by a few millimeters. The velocities for each azimuth were calculated using the ice core diameter and the p-wave travel time. ::: We :::::::: measured ::: the ::: ice :::: core :::::::: diameter ::: for :::: each :::::::::::: measurement :::::::::: individually ::: and ::: we :::::: found :: no :::::::::: dependence ::: of :::::::: calculated ::::::: seismic :::::::: velocities :: on ::: the :::::::: measured ::: ice :::: core ::::::::: diameters.
To ensure data consistency, the reciprocal travel times were compared for quality checks. Rays with opposing azimuths (ϑ and ϑ+180 • ) are reciprocal and the velocity should be identical. Larger deviations (> 30 m s −1 ) for individual measurements 235 were considered incorrect and these measurements were removed from the final dataset (in total, 7 out of 315 traces). Finally, the reciprocal traces for the individual horizontal and vertical measurements were combined and an average velocity for each azimuth was calculated. That is, we only consider an azimuthal range of 0-180 • and therefore, the horizontal results show a periodicity of 180 • . This processing scheme was applied to all 5 samples and the results are summarised in Fig. 4. Minimum and maximum velocity within the stack of repeated measurements for each azimuth are shown as reddish coloured areas. in ::: 45 m :::::: maxima ::: of ::: one :::::: profile ::::::: coincide ::::: with : a :::::::: minimum ::: of ::: the ::::: other ::::::: (Fig. 4d). : The measured velocity profiles show higher amplitudes between maximum and minimum compared to the calculated COF-derived profiles, but the latter are rather smooth.

Porosity from X-ray tomography
The X-ray CT images provide porosity information in the vicinity of the horizontal ultrasonic measurements (summarised in Table 2). The porosity is governed by air bubble layers in the ice. These air bubble layers show a preferentially horizontal dis- spherical bubbles as second phase (cf. Fig. 3a (dashed magenta line) as an example for uncorrected values). Since we have a relatively low porosity (<1 %), but do not know the exact size and position of the individual air bubbles, we used a correction for spherical inclusions at very low volume fraction, where the effective elastic moduli can be calculated exactly Torquato (2002, p. 499). The CT and LASM images indicate that the majority of air bubbles not associated with grain boundaries ::: (and :::::::: therefore ::: not :::::: pinned :: to ::: and ::::::: affected ::: by ::: the :::::::: boundary ::::::::: pathways) are spherical and do not show any elongation in certain directions and 265 therefore confirm our assumption. We retrieved the required bulk and shear moduli of the ice matrix from the corresponding elements of the computed polycrystal elasticity tensor. With these bulk and shear moduli, the CT-derived porosity values and the mass densities of air (ρ = 1.3163 kg m −3 at T = −5 • ) and ice (ρ = 918 kg m −3 ), we obtained the mean velocities of such a two-phase material. Finally, the difference between this mean velocity and the calculated mean velocities of pure ice was subtracted from the individual velocity values. This correction was applied to the COF-derived profiles (blue curves) in Figs. 4 270 and 3. The porosity correction causes a shift of the average velocities (see Fig. 3 blue dashed vs. solid lines), but does not affect the shape (i.e. maxima and minima) of the horizontal profiles at the individual depths (Fig. 4).

Comparing COF-derived velocity and ultrasonic measurements
The results for COF-derived velocities and the ultrasonic velocity profiles are compared in Fig. 3 and Fig. 4. As presented in  (Table 2), but the air bubble content cannot fully explain the observed discrepancy between the two velocity profiles shown in Fig. 4. These discrepancies could be caused by the differences in the grain size distribution within the individual samples, since we did not conduct both 285 measurements on exactly the same pieces of ice.
Seismic waves have a band-limited frequency content resulting in a finite range of wavelengths. As indicated in Sect. 2.3, the dominant wavelength for the ultrasonic measurements was approximately 3.8 mm. As a consequence, the seismic waves are not just affected by the medium along an infinitely thin ray path connecting the source and receiver, but by a finite volume surrounding the ray path. This volume can be estimated with the first Fresnel volume path (e.g. Williamson and Worthington,290 1993). Assuming a homogeneous medium including source position S and receiver position R, a point D is considered to be within the first Fresnel volume, when where l is the direct ray path between source and receiver, n is the order of the Fresnel zone, and λ is the dominant wavelength. The ice grains within this Fresnel volume influence the velocity that is derived from the corresponding ultrasonic 295 measurement. To illustrate the situation, we superimposed in Fig. 5 a Fresnel zone computed from Eq.

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Therefore, the velocities of COF analysis and the ultrasonic measurements are expected to be different in the presence of large grains. Conversely, a good match can be expected, when a large number of small grains is involved. To investigate this further, we computed grain size distributions (Fig. 6) using all thin sections prepared for the COF analysis (Hellmann et al., 2020b) ::::::::::::::::::: (Hellmann et al., 2021). Clearly, the sample from : at : 22 m depth shows the largest number of grains and thus the smallest mean grain size. Considering the previous discussion, it is therefore not surprising that we observe a relatively good match in Fig. 4 335 for this sample and larger discrepancies for the remaining samples.
As a result of the previous discussion, we also assume that a larger amount of ultrasonic measurement levels should lead to a better match with the statistically averaged profile from the COF analysis. Additional ultrasonic measurements are available for the sample of 33 m. These measurements were obtained on the neighbouring core segment just below the analysed COF samples (the original ultrasonic measurements, shown again in Fig. 7a, were acquired above the COF samples). When 340 considering the additional measurements the differences between the mean velocity profiles derived from COF analyses and ultrasonic measurements further decrease (Fig. 7b). In turn, when considering only a subset of thin sections (here only the four horizontal thin sections) to derive the velocity profile, it also converges to the ultrasonic profile (Fig. 7c).
To conclude, ultrasonic and COF analyses complement each other. The first is a deterministic approach allowing a detailed analysis of a particular ice core volume :: of :: a ::: few : cm 3 . The latter is a statistical approach that provides an integrated COF 345 pattern derived from several centimetres (up to 50 cm) long ice core samples and thus an averaged velocity. However, both methods are most likely comparable, when the numbers of grains are similar in both samples. Hence, both methods should be combined and ultrasonic measurements may become a valuable technique to support the existing method.

Ambiguities with other COF patterns
In this study, the COF patterns are assumed to be known a priori :::::: a-priori : and the ultrasonic results could be correlated with 350 this known COF. The question arises, if ultrasonic measurements are a suitable method to determine unambiguously unknown COF patterns?
To address this question, we consider the sample at 22 m depth. Its resulting COF and the associated velocity distribution, already shown in Figs. 2b and 2i, are shown again in Figs. 8a and 8b. For this sample, the small grain size prerequisite is met, leading to a good match between COF-derived and ultrasonic velocity profiles (Fig. 4b, shown again in Fig. 8e without uncer-355 tainty ranges). We now compare the results with a small circle girdle structure (Fig. 8c), which is a common COF pattern for compressional deformation in combination with recrystallisation (Wallbrecher, 1986). Its corresponding velocity distribution is shown in Fig. 8d. When only considering the horizontal orientations, as measured with our ultrasonic experimental setup (marked with black dots in Figs. 8b and 8d), the azimuthal velocity variations of both, the actual "diamond shape" pattern (blue curve in Fig. 8e) and the small circle girdle (dashed green curve in Fig. 8e), are compatible with the measured ultrasonic data 360 (orange curve in Fig. 8e). Obviously, there exist several COF structures that explain the ultrasonic data equally well, thereby leading to ambiguities and uncertainties in interpreting the ultrasonic data. Similar ambiguities can be observed for COF patterns typical for polar ice such as single maximum vs. girdle fabrics. Adding the additional vertical measurement parallel to the core axis (black dot in the centre of Fig.8b and 8d) does not remove this ambiguity for the small-girdle example.
To further reduce this ambiguity, it would be required to add additional ultrasonic measurements, spanning a range of az-365 imuths and inclinations, such that the area of the stereoplots would be sampled more regularly. With modern point-contact transducers, it seems to be feasible to implement such an experimental layout with reasonable expenditure of time when using a multi-channels recording system.
These ambiguities show, that COF analyses will also be required in the future, but ultrasonic measurements can support this analysis and bridge the gaps between the discrete COF samples. Finally, ultrasonic measurements on ice cores and in borehole 370 provide the link between COF and surface geophysical velocities (Bentley, 1972;Gusmeroli et al., 2012;Diez et al., 2014).

Future technical improvements
Our measurement scheme (Fig. 1a) was built for first attempts to investigate the feasibility of ultrasonic measurements to detect the COF along an ice core and to establish a link between COF and cross-borehole or surface seismic experiments. Although we showed that there exist ambiguities, such a device provides valuable information and could directly be employed in-situ 375 on freshly drilled ice cores. As an advantage of an immediate measurement on thermally drilled ice cores, one would avoid the refreezing of meltwater and thus a much better coupling of the transducers without extra work for removing this meltwater "skin". For mechanically drilled cores with a relatively convex shape, the point contact transducers are expected to be well coupled. Furthermore, more than two transducers are recommended to obtain several inclined measurements as discussed above and the transducers should further be pressed onto the ice with a defined constant pressure. A constant pressure is 380 relevant to avoid any pressure melting effects and ensures identical coupling conditions. This enhances the comparability of the acoustic signals throughout the entire experiment.
In addition, the determination of the exact distance between source and receiver should be automated. A manual measurement of the distances, as performed in our experiments, leads to a higher uncertainty in the derived velocities. Moreover it is not feasible with several transducers. These improvements require a more comprehensive measurement device. Such a device could 385 be employed in a processing line (e.g. in polar ice core drilling projects) with existing devices such as for Dielectric Profiling (DEP) (Wilhelms et al., 1998) before cutting the ice core into sub-samples for different analyses. As it also allows for a fast data acquisition, such a device could also be employed for other purposes such as detecting the link between two neighbouring ice core segments (i.e. retrieving the actual orientation of the freshly drilled segment within the glacier).
We have performed ultrasonic experiments at ice cores from a temperate glacier, and we compared the results with those from a well-established COF analysis method. The main objectives of this study were (i) to compare the ultrasonic and COF-derived seismic velocities and (ii) to check, if ultrasonic measurements have the potential to replace or reduce the labour-intensive and destructive COF analysis. Our main findings can be summarised as follows.
-Ultrasonic and COF-derived seismic velocities are comparable, when the grain size of the ice crystals is sufficiently 395 small. However, this condition is generally not met in temperate ice. In contrast ::: As : a ::::::::::: consequence, we recommend to apply this method to cold (e.g. polar) ice cores with small grains.
-In the presence of large grains, we observe a poor correlation between the ultrasonic and COF-derived velocities. The ultrasonic measurements belong to the deterministic approaches. Each measurement samples the actual 3D volumes (Fresnel volumes) and only considers the grains therein. The COF-derived profiles provide a statistical mean value of 400 the velocities for all thin sections. Therefore, the number of measurement levels of ultrasonic measurements needs to be sufficiently large. This is especially relevant for samples from temperate ice cores.
-In the presence of a significant porosity (i.e. air bubbles), a correction needs to be applied, to make ultrasonic and COFderived velocities comparable. This requires the determination of the porosity. In this study, we have employed a CT scanner for that purpose.

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-In principle, ultrasonic measurements can be employed for determining COF patterns. However, this requires a relatively dense sampling of the ice core, including a broad range of azimuths and inclination angles. Our experimental setup, including only horizontal and vertical measurements, led to ambiguous results.
On the basis of our findings, we conclude that ultrasonic measurements are not yet an adequate replacement of COF analysis.
However, since the development of ultrasonic transducers is progressing rapidly, we judge it feasible that adequate experimental 410 layouts of ultrasonic experiments can be implemented in a foreseeable future. This would offer substantial benefits, since it would reduce the labour-intensive COF analysis. Furthermore, the ultrasonic measurements offer the significant advantage of being non-destructive, and the samples of the generally valuable ice cores would remain available for other analyses of physical properties. This also means, that the ultrasonic measurements can continuously be obtained on freshly drilled cores.
Nevertheless, a certain but reduced number of thin sections a COF analysis can still be used to calibrate the ultrasonic data and 415 to dispose of ambiguities with direct comparisons of the results of both methods on the same ice core samples.