General comments

The authors introduced a 3D analytical solution for forwarding modeling of self-potential from sulfide deposits. Although the topic is interesting and the overall flow of the paper, from the formulation to comparison with 2D analytical solution and experimental results, is great, I can hardly say the paper is well written in general and there seems to be a lot of room for improvement of the manuscript.  

The motivation of the current model development is not compelling, especially given that (1) the system has not been proven to be applicable to the field (even though the authors attempted to validate the model with laboratory experiments, it does not necessarily mean it can be used in the field) and (2) complicated situations are likely to be more easily dealt with numerical tools/methods and there seem to be some published already.  



Formulation is poorly explained especially given that deriving analytical solution for the system should be the essential part for a model development paper.  I found it difficult to follow how the equations are derived. There can be several reasons for this, including (1) the papers cited (Li et al., 2005, He, 2012) for seemingly essential equations are not publicly available and not explained well, (2) the authors do not seem to use mathematical symbols in an organized way, including seemingly interchangeable and thus confusing use of uppercase/lowercase and/or Greek/English letters and/or vector/scalar expressions, and (3) figure to help the reader to understand formulation is lacking and/or not referenced. 

Especially, please consider making clearer explanations on (1) what is proved and how this is done in Section 2.1, (2) the mirror image method in general potentially with a specific diagram/schematic in Section 2.2 and (3) derivation of one of formulation from Table 1 in Section 2.3.

Specific comments

L9. “It is significant to fast and precise forward modeling and inversion for SMS explorations.”.  I think English grammar here is not correct.

L22. What is “AUV”?

L34. What do you mean by “affection of the field source”?

L34. “which occur difficulty in solving the Poisson equation.” I think the authors should correct English.

L58. What is the definition of H?

L79. rho_0 must be a typo of r_0?

L82. “So we get the potential distribution along the surface of a uniformly polarized sphere is equivalent to an electric dipole.” I do not understand what this means.

L92. Is the use of permittivity correct here?  In other words, do permittivity and conductivity have the same units?

L137. What do you mean by “the dipole moment is 1 D.”?

L138. 10-10 must mean 10^-10.  What are units and adjustment term exactly?

L140. Do “exceptions” mean negative/positive peaks?

L142. Why tilted dipole is related to any depth?

L147. The authors should describe exactly what redox reaction is occurring.