the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 17 Oct 2022
A borehole trajectory inversion scheme to adjust the measurement geometry for 3D travel time tomography on glaciers
Sebastian Hellmann
Melchior Grab
Cedric Patzer
Andreas Bauder
Hansruedi Maurer
Abstract. 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 obtained 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 exacerbate 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 on 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 and heterogeneities in the velocity model can be imaged similar to an inversion with 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 such a velocity anisotropy. Therefore, for such a borehole trajectory inversion, polynomials of degree three are a good compromise between a good representation of the true borehole trajectories and avoiding compensation for velocity anisotropy.
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Preprint
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Journal article(s) based on this preprint
Sebastian Hellmann et al.
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2022-1069', Anonymous Referee #1, 02 Dec 2022
The author presents the cross-borehole travel time tomography results for glaciological application. They propose sequential inversion which inverts both velocity structure and deviations of borehole trajectory. Especially, they propose to use two orthogonal polynomials to stabilize inversion instead of using individual source and receiver coordinates. They evaluate the method using synthetic and field data and conclude that their method could reduce artefacts due to the uncertainty of source and receiver positions. They also discuss the effect of anisotropy on their algorithm.
The ideas, methods, and data are unique and will contribute to scientific progress within the scope of Solid Earth. Their approach and applied methods are valid. The manuscript is well organized, clear, and interesting and I think it is good to be published with minor/moderate revision.
My comments/corrections are listed below:
Â
- Section 3
- It is better to show the day of data acquisition because its timing is discussed in Section 5. It would be easier for the reader to understand and evaluate the results in this manuscript.
- Section 3, line 169
- It is not clear what the author trying to describe by “Geophones at the surface of the glacier complemented the measurement setup.”. It would be better to mention the purpose of geophone and their usage in this study (included in the inversion?).
- Section 4, lines 252 - 254
- It is difficult to understand which part of Fig. 7b is mentioned in this sentence. It would be better if the author describes the more detail of the figure and explain which part is polynomial coefficients one through 4th degree. Also, it is difficult to see which parts are a value > 0.99 since the color bar is monotone above the value > 0.2.
- Sections 4 and 5
- Although the number of polynomials in the inversion is discussed in 6.2, the author does not show the value in their application to synthetic and field data. Its value is important for anisotropic effect according to 6.2. Please include the number of polynomials that are used for inversions in the manuscript.
- Section 5, line 304
- This sentence explains that the author used two starting models for trajectory inversion. However, the result of interpolated trajectory case (second starting model) is already described in Fig.8b in line 289. It is easier for a reader to understand if the prerequisite is described before or just after its results are shown.
- Section 5, line 305
- According to the caption of Figure 8b, it is the result of using interpolated trajectories as an initial model. However, this sentence says that Figure 8b is the result using vertical boreholes as a starting model. It is inconsistent. The results of using vertical wells as an initial model are not shown in this manuscript. Please add the results.
- Section 6.1 line338
- “lower that” should be “lower than”.
- Figures 3, 4, 8 (a)
- Legends in this figure are confusing since these are velocity inversion results without coordinates updated. The expression “updated coordinates” seems not right.
- Figures 3, 4, 5
- It seems black asterisks and triangles in Figures 3 to 5 are coordinates for geophones. Please describe what “start coordinates” in the legends mean. The term “start coordinates” is also used in Figure 6, but it seems different from figures 3 to 5.
Citation: https://doi.org/10.5194/egusphere-2022-1069-RC1 - AC1: 'Reply on RC1', Sebastian Hellmann, 26 Apr 2023
- Section 3
-
RC2: 'Comment on egusphere-2022-1069', Anonymous Referee #2, 17 Feb 2023
In the manuscript a 3D-crosshole inversion method is presented that incorporates borehole position as a variable to account for inaccurate positioning, which is especially relevant for surveys in a glacial environment. To reduce the number of inverted parameters the borehole trajectory is approximated by two perpendicular polynomials. The inversion is sequenced in velocity and trajectory inversion, which makes it more robust and computationally efficient.
Application on synthetic and field data show good fits and a reduction of velocity artefacts in well determined areas. A priori information are needed to estimate optimal inversion parameters. An interdependence between anisotropy and borehole trajectory adjustments is discussed. To avoid overfitting of borehole trajectories, that may compensate for anisotropy effects, a polynomial degree of x^3 is proposed for a compromise.Â
The study is well done and concept and the used method are of interest for tomographic applications. With modifications to improve the comprehensibility, the manuscript can be published.
Abstract:
In the abstract it sounds like no a priori information are necessary anymore, but as stated in the conclusion, selection of damping parameter and polynomial order is still based on deviation measurements.
Line 20: I suggest to replace “Therefore” with “Based on the modelling results we propose/determined”, because this is not the conclusion of the previous explanation, but of the investigation you described in the manuscript.
Line 22: Compensation for velocity anisotropy is minimized not avoided.
Â
Section 3:
It is not clear when the seismic acquisition happened and if the deviation measurements were done on the same days. An overview in form of e.g. a small table would help. As I understood there was ~ 10 days between the acquisition days. It does not get clear how the inversion incorporates borehole deviation for the same borehole on different days. As I understand the polynomial coefficients are adjusted with the whole dataset, but the deviation is different for the subsets collected on different acquisition days. Please give insight on how this is done.
The model includes anomalies that are not resolved by the tomography, because there is no ray coverage in this area. When an anomaly like this is incorporated in the model this should be mentioned in the discussion of the results to avoid confusion.
Line 169: How many geophones are installed at the surface? In which setup? It is also not clear whether the data from the surface geophones is used in the inversion. This would also make the model setup in line 187 more understandable (“…and added a total of 828 receivers…”)
Line 172: “…explained in more detail below.” State where it is explained “…explained in more detail in section 4.”
Line 201: The damping factor of 0.1 is selected. In Line 300 a damping factor between 1000 and 10000 is recommended. It is not clear if these are different factors and why the magnitude changes.
Table 1: For evaluation of the combined inversion adding the RMSE of velocity inversion with true trajectory should be a benefit.
Line 220: Here add a reference to the supplementary video material [Hellmann,2022a]
Line 215 and 224: I do not agree with the statements, that in Figure 4 the lower channel “could not be recovered”, while in the combined inversion in Fig. 5 “the upper and especially the lower channel are correctly resolved”. The anomaly associated with the channel is well visible in both results. I agree, that there is less artefacts and the channel is better resolved in the combined inversion results which supports the papers objective. Please clarify this.
Line 247 / Fig. 7b: It is not clear which values are in the range of >0.99 since the colorbar ends at 0.2. The allocation of the polynomial coefficients on x- and y-axis is not intuitive. Label the polynomial coefficients on the axes. For better visibility only a fragment of the matrix could be shown in order to gain the space for labels.
Section 5:
279: Is 200ms correct? A 200ms window around the estimated arrival time seems very long since the highest expected traveltimes at v=3800m/s and max. distance of 100m (90m depth, 40m borehole spacing) are ~25ms.
Line 312: Deviations of 0.6m and 1m are seen as not realistic, while having a displacement speed of 0.06m/d and 11-14 days. This results in 0.06m/d*14d = 0.84m which seems realistic in this context.
Section 3 and 5:
What is the estimation of the borehole deviation measurement error? How does it compare to the difference between measured borehole trajectory and fitted trajectory after inversion? The comparison is important to assess the plausibility of the trajectory fitting.
Section 6:
Line 328: “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.” Is it avoided? At the end of Section 2 Line 140 you explain that sequenced inversion and the inversion with extended set of equations do not show significant differences is the results. Could you please explain this.
Section 6.2: This section gives important insight on trade off between trajectory optimization and anisotropy. A lot of inversion results are presented without visualizing any of them. An additional Figure is recommended. Â Â Â If to many Figures are already in the manuscript than add significant Figures to the supplement.
Section 7:
Line 439: Video Supplement is in [Hellmann, 2022a], not [Hellmann,2022b]
If possible give an outlook on how to determine inversion parameters (damping, …) without a priori information from inclinometer measurements. If these are always required the advantages of the combined inversion are mitigated.
Citation: https://doi.org/10.5194/egusphere-2022-1069-RC2 - AC2: 'Reply on RC2', Sebastian Hellmann, 26 Apr 2023
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2022-1069', Anonymous Referee #1, 02 Dec 2022
The author presents the cross-borehole travel time tomography results for glaciological application. They propose sequential inversion which inverts both velocity structure and deviations of borehole trajectory. Especially, they propose to use two orthogonal polynomials to stabilize inversion instead of using individual source and receiver coordinates. They evaluate the method using synthetic and field data and conclude that their method could reduce artefacts due to the uncertainty of source and receiver positions. They also discuss the effect of anisotropy on their algorithm.
The ideas, methods, and data are unique and will contribute to scientific progress within the scope of Solid Earth. Their approach and applied methods are valid. The manuscript is well organized, clear, and interesting and I think it is good to be published with minor/moderate revision.
My comments/corrections are listed below:
Â
- Section 3
- It is better to show the day of data acquisition because its timing is discussed in Section 5. It would be easier for the reader to understand and evaluate the results in this manuscript.
- Section 3, line 169
- It is not clear what the author trying to describe by “Geophones at the surface of the glacier complemented the measurement setup.”. It would be better to mention the purpose of geophone and their usage in this study (included in the inversion?).
- Section 4, lines 252 - 254
- It is difficult to understand which part of Fig. 7b is mentioned in this sentence. It would be better if the author describes the more detail of the figure and explain which part is polynomial coefficients one through 4th degree. Also, it is difficult to see which parts are a value > 0.99 since the color bar is monotone above the value > 0.2.
- Sections 4 and 5
- Although the number of polynomials in the inversion is discussed in 6.2, the author does not show the value in their application to synthetic and field data. Its value is important for anisotropic effect according to 6.2. Please include the number of polynomials that are used for inversions in the manuscript.
- Section 5, line 304
- This sentence explains that the author used two starting models for trajectory inversion. However, the result of interpolated trajectory case (second starting model) is already described in Fig.8b in line 289. It is easier for a reader to understand if the prerequisite is described before or just after its results are shown.
- Section 5, line 305
- According to the caption of Figure 8b, it is the result of using interpolated trajectories as an initial model. However, this sentence says that Figure 8b is the result using vertical boreholes as a starting model. It is inconsistent. The results of using vertical wells as an initial model are not shown in this manuscript. Please add the results.
- Section 6.1 line338
- “lower that” should be “lower than”.
- Figures 3, 4, 8 (a)
- Legends in this figure are confusing since these are velocity inversion results without coordinates updated. The expression “updated coordinates” seems not right.
- Figures 3, 4, 5
- It seems black asterisks and triangles in Figures 3 to 5 are coordinates for geophones. Please describe what “start coordinates” in the legends mean. The term “start coordinates” is also used in Figure 6, but it seems different from figures 3 to 5.
Citation: https://doi.org/10.5194/egusphere-2022-1069-RC1 - AC1: 'Reply on RC1', Sebastian Hellmann, 26 Apr 2023
- Section 3
-
RC2: 'Comment on egusphere-2022-1069', Anonymous Referee #2, 17 Feb 2023
In the manuscript a 3D-crosshole inversion method is presented that incorporates borehole position as a variable to account for inaccurate positioning, which is especially relevant for surveys in a glacial environment. To reduce the number of inverted parameters the borehole trajectory is approximated by two perpendicular polynomials. The inversion is sequenced in velocity and trajectory inversion, which makes it more robust and computationally efficient.
Application on synthetic and field data show good fits and a reduction of velocity artefacts in well determined areas. A priori information are needed to estimate optimal inversion parameters. An interdependence between anisotropy and borehole trajectory adjustments is discussed. To avoid overfitting of borehole trajectories, that may compensate for anisotropy effects, a polynomial degree of x^3 is proposed for a compromise.Â
The study is well done and concept and the used method are of interest for tomographic applications. With modifications to improve the comprehensibility, the manuscript can be published.
Abstract:
In the abstract it sounds like no a priori information are necessary anymore, but as stated in the conclusion, selection of damping parameter and polynomial order is still based on deviation measurements.
Line 20: I suggest to replace “Therefore” with “Based on the modelling results we propose/determined”, because this is not the conclusion of the previous explanation, but of the investigation you described in the manuscript.
Line 22: Compensation for velocity anisotropy is minimized not avoided.
Â
Section 3:
It is not clear when the seismic acquisition happened and if the deviation measurements were done on the same days. An overview in form of e.g. a small table would help. As I understood there was ~ 10 days between the acquisition days. It does not get clear how the inversion incorporates borehole deviation for the same borehole on different days. As I understand the polynomial coefficients are adjusted with the whole dataset, but the deviation is different for the subsets collected on different acquisition days. Please give insight on how this is done.
The model includes anomalies that are not resolved by the tomography, because there is no ray coverage in this area. When an anomaly like this is incorporated in the model this should be mentioned in the discussion of the results to avoid confusion.
Line 169: How many geophones are installed at the surface? In which setup? It is also not clear whether the data from the surface geophones is used in the inversion. This would also make the model setup in line 187 more understandable (“…and added a total of 828 receivers…”)
Line 172: “…explained in more detail below.” State where it is explained “…explained in more detail in section 4.”
Line 201: The damping factor of 0.1 is selected. In Line 300 a damping factor between 1000 and 10000 is recommended. It is not clear if these are different factors and why the magnitude changes.
Table 1: For evaluation of the combined inversion adding the RMSE of velocity inversion with true trajectory should be a benefit.
Line 220: Here add a reference to the supplementary video material [Hellmann,2022a]
Line 215 and 224: I do not agree with the statements, that in Figure 4 the lower channel “could not be recovered”, while in the combined inversion in Fig. 5 “the upper and especially the lower channel are correctly resolved”. The anomaly associated with the channel is well visible in both results. I agree, that there is less artefacts and the channel is better resolved in the combined inversion results which supports the papers objective. Please clarify this.
Line 247 / Fig. 7b: It is not clear which values are in the range of >0.99 since the colorbar ends at 0.2. The allocation of the polynomial coefficients on x- and y-axis is not intuitive. Label the polynomial coefficients on the axes. For better visibility only a fragment of the matrix could be shown in order to gain the space for labels.
Section 5:
279: Is 200ms correct? A 200ms window around the estimated arrival time seems very long since the highest expected traveltimes at v=3800m/s and max. distance of 100m (90m depth, 40m borehole spacing) are ~25ms.
Line 312: Deviations of 0.6m and 1m are seen as not realistic, while having a displacement speed of 0.06m/d and 11-14 days. This results in 0.06m/d*14d = 0.84m which seems realistic in this context.
Section 3 and 5:
What is the estimation of the borehole deviation measurement error? How does it compare to the difference between measured borehole trajectory and fitted trajectory after inversion? The comparison is important to assess the plausibility of the trajectory fitting.
Section 6:
Line 328: “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.” Is it avoided? At the end of Section 2 Line 140 you explain that sequenced inversion and the inversion with extended set of equations do not show significant differences is the results. Could you please explain this.
Section 6.2: This section gives important insight on trade off between trajectory optimization and anisotropy. A lot of inversion results are presented without visualizing any of them. An additional Figure is recommended. Â Â Â If to many Figures are already in the manuscript than add significant Figures to the supplement.
Section 7:
Line 439: Video Supplement is in [Hellmann, 2022a], not [Hellmann,2022b]
If possible give an outlook on how to determine inversion parameters (damping, …) without a priori information from inclinometer measurements. If these are always required the advantages of the combined inversion are mitigated.
Citation: https://doi.org/10.5194/egusphere-2022-1069-RC2 - AC2: 'Reply on RC2', Sebastian Hellmann, 26 Apr 2023
Peer review completion
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Sebastian Hellmann et al.
Sebastian Hellmann et al.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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