Summary

This study evaluates the impact of implementing different vertical temperature sampling strategies within ice boreholes on accurately representing the temperature profile, which subsequently affects the reliability and representativeness of borehole climate reconstructions. The widely used linear and exponential sampling strategies are compared with a greedy optimal sampling approach introduced by the authors. Their results show a remarkable reduction in sampling error with the optimal sampling technique compared to the linear and exponential strategies. This is particularly noteworthy when taking into account the contribution of the sensor device error. In this scenario, a smaller number of sensors placed using the optimal greedy approach outperforms a larger number of sensors positioned according to linear or exponential sampling in terms of sampling error. The authors demonstrate that the results are not sensitive to surface temperature conditions but are instead determined by the nature of heat diffusion and advection.

General overview

I believe this work introduces a novel perspective by highlighting that sensor placement is a source of uncertainty in retrieving accurate single-time and continuous borehole temperature measurements. This adds to other well-known sources of uncertainty, such as device error or thermal perturbations during the drilling process. In my opinion, the paper is well-written and structured, the results are presented clearly, and the discussion and conclusions are concise. However, I have a series of comments that I think the authors should address before the manuscript is accepted for publication in GI.

(1)    In the introduction, I missed a mention of other sources of uncertainty that affect the subsequent ground surface temperature reconstructions from borehole inversions, such as the impact of borehole depths (Beltrami et al., 2015), or the uncertainty in soil/ice thermal properties (e.g., Shen et al., 1995; Cuesta-Valero et al., 2022), which are known to be extremely heterogeneous in space and depth, at least in boreholes over land (e.g., Smerdon et al., 2004; García-Pereira et al., 2024). 

Beltrami, H., Matharoo, G. S., & Smerdon, J. E. (2015). Impact of borehole depths on reconstructed estimates of ground surface temperature histories and energy storage. Journal of Geophysical Research: Earth Surface, 120(4), 763–778. https://doi.org/10.1002/2014JF003382

Cuesta-Valero, F. J., Beltrami, H., Gruber, S., García-García, A., & González-Rouco, J. F. (2022). A new bootstrap technique to quantify uncertainty in estimates of ground surface temperature and ground heat flux histories from geothermal data. Geoscientific Model Development, 15, 7913–7932. https://doi.org/10.5194/gmd-15-7913-2022

García-Pereira, F., González-Rouco, J. F., Schmid, T., Melo-Aguilar, C., Vegas-Cañas, C., Steinert, N. J., Roldán-Gómez, P. J., Cuesta-Valero, F. J., García-García, A., Beltrami, H., & de Vrese, P. (2024). Thermodynamic and hydrological drivers of the soil and bedrock thermal regimes in central Spain. SOIL, 10, 1–21. https://doi.org/10.5194/soil-10-1-2024

Shen, P. Y., Pollack, H. N., Huang, S., & Wang, K. (1995). Effects of subsurface heterogeneity on the inference of climate change from borehole temperature data: Model studies and field examples from Canada. Journal of Geophysical Research, 100(B4), 6383–6396. https://doi.org/10.1029/94JB03136

Smerdon, J. E., Pollack, H. N., Cermak, V., Enz, J. W., Kresl, M., Safanda, J., & Wehmiller, J. F. (2004). Air-ground temperature coupling and subsurface propagation of annual temperature signals. Journal of Geophysical Research, 109, D21107. https://doi.org/10.1029/2004JD005056

(2)    Are these uncertainties greater than the differences associated with different sampling techniques? Is that the reason why “the topic of sensor placement in boreholes has not received much attention in the field of borehole thermometry” (line 37, page 2)? Even though the authors did not perform borehole inversions, I think discussing this in Section 4 would be valuable to readers.

(3)    Why do the authors impose a Dirichlet instead of a Neumann boundary condition at the ice bottom of the form T’(t,H) equals to the geothermal heat flux? The Neumann condition implies a non-linear increse in temperature from the ice bottom to the top (Robin, 1955; Moreno-Parada et al., 2024), and is the usual approach in ice sheet modeling (e.g., Larour et al., 2012; Lipscomb et al., 2019, Robinson et al., 2020). While the synthetic boreholes in this study are much shallower (200 m) than the ice sheet thicknesses at EDML (2782 m) and GRIP (3029 m), using a Dirichlet condition could slightly alter the borehole temperature profile. Given that the sampling error is on the order of mK, this effect might be of similar magnitude to the device error.

Larour, E., Seroussi, H., Morlighem, M., & Rignot, E. (2012). Continental scale, high order, high spatial resolution, ice sheet modeling using the Ice Sheet System Model (ISSM). Journal of Geophysical Research, 117, F01022. https://doi.org/10.1029/2011JF002140

Lipscomb, W. H., et al. (2019). Description and evaluation of the Community Ice Sheet Model (CISM) v2.1. Geoscientific Model Development, 12, 387–424. https://doi.org/10.5194/gmd-12-387-2019

Moreno-Parada, D., Robinson, A., Montoya, M., & Alvarez-Solas, J. (2024). Analytical solutions for the advective–diffusive ice column in the presence of strain heating. The Cryosphere, 18, 4215–4232. https://doi.org/10.5194/tc-18-4215-2024

Robin, G. de Q. (1955). Ice movement and temperature distribution in glaciers and ice sheets. Journal of Glaciology, 2(18), 523–532. https://doi.org/10.3189/002214355793702028

Robinson, A., Alvarez-Solas, J., Montoya, M., Goelzer, H., Greve, R., & Ritz, C. (2020). Description and validation of the ice-sheet model Yelmo (version 1.0). Geoscientific Model Development, 13, 2805–2823. https://doi.org/10.5194/gmd-13-2805-2020

(4)    I found reading and understanding the methodology quite challenging, especially Section 2.2.1. The manuscript would benefit from a clearer presentation of the sampling error calculation, avoiding notation that is not referenced in the results. Perhaps moving the formal algorithm to an appendix (also for Section 2.2.2) and providing a step-by-step explanation in the main text, connecting the different subsets of sensors mentioned to what is shown in Fig. 1, would enhance clarity. 

(5)    Fig. 4 compares the error of the three sampling strategies with and without device error, but the manuscript does not explicitly state whether the assumed device error values are typical for borehole thermistors. Are device error values generally larger than the reduction in sampling error achieved through greedy optimal sampling? This would be worth mentioning in the discussion.

Other suggestions that the author may want to consider are:

(6)    Page 2, lines 51-55: I think this paragraph would better fit the rationale of the introduction if placed before the paragraph starting in line 37: “Despite its significance …”.

(7)    Page 3, lines 81-82, lines 84-86: the naming of the parameters of the heat diffusion equation should appear when first shown the equation, in line 76 after “as a function of time t and depth z (positive downwards)”.

(8)    Table 1, foot note: watch out the parenthesis convention here, e.g., “Hammer and Dahl-Jansen (1999)(GRIP)” would better be “GRIP (Hammer and Dahl-Jensen, 1999)”. 

(9)    Page 8, line 2: “and 757 for GRIP …”.

(10)    Page 8, line 169: “with respect to the mean value in…”

(11)    Page 9: Why is the number of sensors limited to 20? Is this limit based on economic reasons?

(12)    Page 10, Fig. 2 caption: “200 m borehole”

(13)    Page 11, line 200: how did the authors calculate the significance in the differences here? What p-value do the authors consider as a threshold for significance?

(14)    Page 11, lines 208 and 209: the device error is simultaneously referred to as epsilon d and sigma d. Please, be consistent with the notation.

(15)    Page 14, Fig. 6 caption: “Greedy optimal sensor placements sensitivity to the surface temperature time series…”

(16)    Page 14, Fig. 6 caption: w.r.t should be “with respect to”.

(17)    Page 14, line 285: by how much do the optimal sensor locations shift?

(18)    General comment: I would humbly suggest the authors to talk about “ice boreholes” instead of simply “boreholes” to distinguish them from terrestrial borehole used in subsurface borehole climatology. Perhaps I am biased here, so it is just a suggestion.

(19)    Fig. 5: I think thicker lines and bigger symbols would improve visibility.

(20)    Fig. 6: The figure could overall be bigger by occupying the full text width. Panel (a) would also benefit from more intense colors for the boxplots for enhanced visibility.

This manuscript by Shaju et al. examines the optimal positioning of temperature sensors in ice boreholes. This field of research is quite far from my own field, but it was a pleasure to read a very well written paper that I could easily follow despite my lack of prior knowledge. The introduction is well crafted, and the results and discussions are easy to follow.

Reviewer 1 has already presented a very detailed, in-depth expert review and I agree in the comments presented there. In particular, sections 2.2.1 and 2.2.2 are difficult material to comprehend without background knowledge.

One little thing that the authors probably already have considered and might want to add to the discussion section is the question of sensor failure. With the greedy algorithm, the sensor positions are optimized, and fewer sensors are needed. However, this must come at the cost of a large penalty if one sensor fails? Or phrased differently, could/should some sort of redundancy be built in?