Analytical Solution for the Influence of Irregular Shape Loads Near the Borehole Strain Observation
 ^{1}Institute of Geology, China Earthquake Administration, Beijing 100026, China
 ^{2}China Earthquake Networks Center, Beijing 100045, China
 ^{3}Ningxia Seismological Brueau, Yinchuan 750001, China
 ^{1}Institute of Geology, China Earthquake Administration, Beijing 100026, China
 ^{2}China Earthquake Networks Center, Beijing 100045, China
 ^{3}Ningxia Seismological Brueau, Yinchuan 750001, China
Abstract. Based on the analytic displacement solution caused by the punctate load model, we derived the calculation formulas of peripheral strain Field. This method can provide a theoretical basis for the quantitative calculation of the load influence of borehole strain observation. On this basis, using the superposition principle, we present a method for calculating the strain effect of twodimensional and threedimensional irregular shape loads. The results show that: (1) To solve the problem of twodimensional irregular shape load model, we can calculate it by vector superposition after load scattering. (2) To solve the problem of threedimensional irregular shape load model, we can use the twodimensional irregular shape load method to calculate with assigning different weights to the scattered points after the load scattering. (3) There are obvious convergence processes in the vector superposition process after scattering of twodimensional and threedimensional irregular loads, which shows the correctness and feasibility of the calculation method. (4) The calculation method introduced in this paper can provide a research basis for quantitative analysis of influence of disturbance of peripheral load in borehole strain observation.

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Wei Yan and Zirui Li
Interactive discussion
Status: closed

RC1: 'Comment on npg202045', Anonymous Referee #1, 14 Jun 2021
In this manuscript the authors attempted to present an analytical solution for elastic response to surface irregular shape loading in a uniform semihalf space. Analytical solutions have been previously derived successfully for surface loading with loading area being a point, line, rectangle, circle and elliptic, respectively. The derivation of an exact analytical solution for an irregular shape surface loading could be a big, significant contribution. However, seems in this work the authors focused on discussion of how to simplify the model of irregular shape surface loading to the one of surface punctate loading, and didn’t directly obtain the exact analytical solution. The title of this manuscript and the wording “exact analytical solution” in context seems very ambiguous and misleading. The hypothesis made for this simplification probably need to be explicitly emphasized and highlighted in the abstract.
I would like to see the strong argument for the hypothesis made by the authors, probably using the comparison to other semianalytical and numerical solutions. The discussions of convergence in Section 3.1 and 3.2 are very interesting, unfortunately the authors didn’t give any details on choosing the grid points ( Fig. 3 (b)). From my perspective, the selection of grid points seems to be important for converging analysis. Since at some cases, for example where grid points are chosen elaborately as shown in Fig. 3(b), the converging results indicated in Figs. 4 and 8 may likely only tell us the selfconsistent convergence of the simplified punctate loading solution but nothing about the convergence to the real solution. The authors probably need to demonstrate why and how good the simplified model can be used to approximate the original model.
Some other minor comments:
 I am kind of confused by the description of two and threedimensional surface loading in Section 3.1 and 3.2. Seems it is better to categorize the models to be uniform and nonuniform surface loading, where nonuniform surface loading could be a function of density, height and coordinates.
 Page 3, line 71, “the coordination” to be changed to “the coordinates”?
 Page 4, line 94, “Fig. 2ab” to be changed to “Fig. 2a, 2b”?
 Page 5, line 106, and page 7, line 157, what is “the total load”? particularly what is meant by “total load” in nonuniform loading (three dimensional loading).
 Page 8, line 160, equation (12), can the authors explain why the sum of P_i is not equal to P?
 Page 5, line 121, the point M (z=0.2m) is above surface according to the diagram in Fig.1. Should be z = 0.2m.
 Page 7, Fig 5, for the annotations, change x/m and y/m to x(m) and y(m)?
 Page 12, lines 268 and 271, “19.19” and “20.20” should be “19.” and “20.”, respectively.

RC2: 'Comment on npg202045', Anonymous Referee #2, 28 Jul 2021
The authors are addressing a long standing problem that relies on inclusion of Bousinesq's 1885 solution for a vertical point load on a halfspace. If I understood the mathematics correctly, they hoped to simply superpose adjacent and scaled point forces over irregular areas in order to determine the strains and stresses induced in the halfspace below by simply adding the contribution of each point force (to attempt to simulate a load on the surface). Regrettfully, I do not think that this manuscript warrants publication at the current time for a number of reaons. First, the authors seem to not know about the very extensive literature on this problem that has existed for decades. While I had dabbled with use of this equation for purposes of material property determination I was not up to date, but even a cursory Google search brought up two recent papers (BOUSSINESQ DISPLACEMENT POTENTIAL FUNCTIONS METHOD FOR FINDING VERTICAL STRESSES AND DISPLACEMENT FIELDS DUE TO DISTRIBUTED LOAD ON ELASTIC HALF SPACE, January 2017, Electronic Journal of Geotechnical Engineering 22(15):56875709; and DOI:10.1061/(ASCE)SC.19435576.0000567) that addressed these issues. These refer back to texts such as (Kachanov M.L, Shafiro B. and Tsukrov I. Handbook of Elasticity Solutions. Springer Science and Business Media Kluwer Academic Publishers Dordrecht The Netherlands, 2003.) that go into detail in some of the complexities required to solve such problems. As such, I am concerned that what appears to be a simplifed superposition may be insuffucient to appropriately solve this problem. I am further concerned that simply applying a force proportaional at a given point that is proportional to the topography (or equivalently the amount of mass) over the point of the loading mass may not appropriately account for the distruution of the load. I do not have the time to check on this, but I would urge the authors to at least check their solution against these earlier results.
Finally, it seems to me that often strains are measured in deeper boreholes, but the effects shown are at very shallow depths. I expect one needs to consider a St. Venant approach in which there would be a near and far field approach to analysis of the problem. This would need to be described. It would be good to perhaps show how you would intend to actually apply these results to show how they might influence a deeper strain observation so that the reader would understand better the utility. It seems to me that it needs to be emphasised that this only really would be important when the load on the surface is changed.
One other area the authors should invesitage is whether there are already more relaistic poroelastic solutions that might be relevant to your problem. Certainly this has been solved for tidal corrections and even atmosphieric loading, so I would ruge that more work is required to better understand this problem
Interactive discussion
Status: closed

RC1: 'Comment on npg202045', Anonymous Referee #1, 14 Jun 2021
In this manuscript the authors attempted to present an analytical solution for elastic response to surface irregular shape loading in a uniform semihalf space. Analytical solutions have been previously derived successfully for surface loading with loading area being a point, line, rectangle, circle and elliptic, respectively. The derivation of an exact analytical solution for an irregular shape surface loading could be a big, significant contribution. However, seems in this work the authors focused on discussion of how to simplify the model of irregular shape surface loading to the one of surface punctate loading, and didn’t directly obtain the exact analytical solution. The title of this manuscript and the wording “exact analytical solution” in context seems very ambiguous and misleading. The hypothesis made for this simplification probably need to be explicitly emphasized and highlighted in the abstract.
I would like to see the strong argument for the hypothesis made by the authors, probably using the comparison to other semianalytical and numerical solutions. The discussions of convergence in Section 3.1 and 3.2 are very interesting, unfortunately the authors didn’t give any details on choosing the grid points ( Fig. 3 (b)). From my perspective, the selection of grid points seems to be important for converging analysis. Since at some cases, for example where grid points are chosen elaborately as shown in Fig. 3(b), the converging results indicated in Figs. 4 and 8 may likely only tell us the selfconsistent convergence of the simplified punctate loading solution but nothing about the convergence to the real solution. The authors probably need to demonstrate why and how good the simplified model can be used to approximate the original model.
Some other minor comments:
 I am kind of confused by the description of two and threedimensional surface loading in Section 3.1 and 3.2. Seems it is better to categorize the models to be uniform and nonuniform surface loading, where nonuniform surface loading could be a function of density, height and coordinates.
 Page 3, line 71, “the coordination” to be changed to “the coordinates”?
 Page 4, line 94, “Fig. 2ab” to be changed to “Fig. 2a, 2b”?
 Page 5, line 106, and page 7, line 157, what is “the total load”? particularly what is meant by “total load” in nonuniform loading (three dimensional loading).
 Page 8, line 160, equation (12), can the authors explain why the sum of P_i is not equal to P?
 Page 5, line 121, the point M (z=0.2m) is above surface according to the diagram in Fig.1. Should be z = 0.2m.
 Page 7, Fig 5, for the annotations, change x/m and y/m to x(m) and y(m)?
 Page 12, lines 268 and 271, “19.19” and “20.20” should be “19.” and “20.”, respectively.

RC2: 'Comment on npg202045', Anonymous Referee #2, 28 Jul 2021
The authors are addressing a long standing problem that relies on inclusion of Bousinesq's 1885 solution for a vertical point load on a halfspace. If I understood the mathematics correctly, they hoped to simply superpose adjacent and scaled point forces over irregular areas in order to determine the strains and stresses induced in the halfspace below by simply adding the contribution of each point force (to attempt to simulate a load on the surface). Regrettfully, I do not think that this manuscript warrants publication at the current time for a number of reaons. First, the authors seem to not know about the very extensive literature on this problem that has existed for decades. While I had dabbled with use of this equation for purposes of material property determination I was not up to date, but even a cursory Google search brought up two recent papers (BOUSSINESQ DISPLACEMENT POTENTIAL FUNCTIONS METHOD FOR FINDING VERTICAL STRESSES AND DISPLACEMENT FIELDS DUE TO DISTRIBUTED LOAD ON ELASTIC HALF SPACE, January 2017, Electronic Journal of Geotechnical Engineering 22(15):56875709; and DOI:10.1061/(ASCE)SC.19435576.0000567) that addressed these issues. These refer back to texts such as (Kachanov M.L, Shafiro B. and Tsukrov I. Handbook of Elasticity Solutions. Springer Science and Business Media Kluwer Academic Publishers Dordrecht The Netherlands, 2003.) that go into detail in some of the complexities required to solve such problems. As such, I am concerned that what appears to be a simplifed superposition may be insuffucient to appropriately solve this problem. I am further concerned that simply applying a force proportaional at a given point that is proportional to the topography (or equivalently the amount of mass) over the point of the loading mass may not appropriately account for the distruution of the load. I do not have the time to check on this, but I would urge the authors to at least check their solution against these earlier results.
Finally, it seems to me that often strains are measured in deeper boreholes, but the effects shown are at very shallow depths. I expect one needs to consider a St. Venant approach in which there would be a near and far field approach to analysis of the problem. This would need to be described. It would be good to perhaps show how you would intend to actually apply these results to show how they might influence a deeper strain observation so that the reader would understand better the utility. It seems to me that it needs to be emphasised that this only really would be important when the load on the surface is changed.
One other area the authors should invesitage is whether there are already more relaistic poroelastic solutions that might be relevant to your problem. Certainly this has been solved for tidal corrections and even atmosphieric loading, so I would ruge that more work is required to better understand this problem
Wei Yan and Zirui Li
Wei Yan and Zirui Li
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