Review Reports

Round  1

Reviewer 1 Report

This paper addresses two issues:

1) how to locate neighbouring points to use in a semi-Lagrangian interpolation

2) how to interpolate those points

The former uses an existing scheme but I think that it is useful to see it tested in the context of various atmospheric grids on the sphere.

The latter uses a bilinear interpolation from 4 points. An alternative that is common in the literature in the context of interpolating fluxes for conservative semi-Lagrangian methods is least square fitting of polynomials. There the strategy is to choose a lower degree polynomial that can not be fit exactly (overdetermined) and minimise the misfit with the values. For example, given 4 points, one can pick a linear polynomial (3 coefficients for 4 points) and do a least squares fit. This ought to help with the problem of three points being nearly colinear discussed in the article. Since this calculation is quite simple having already located the points on the sphere, I think that it makes sense to  ask the authors to compare with this method.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

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Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

This manuscript suggested the alternative bilinear interpolation method using for spherical grid with randomly distributed points. The author shows the reliability of suggested technique in diverse gird system and however, there are several weaknesses in new method as mentioned by author. The increase of accuracy in higher order interpolation is necessary to enhance the wide application. And thus, the reviewer proposed to develop the new method for satisfying the reliability. 

Author Response

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Author Response File: Author Response.pdf

Round  2

Reviewer 1 Report

The authors comment that it seems less efficient to solve least squares problems for specific situations, but I was actually suggesting to solve the same least square problem for all point combinations whether they are colinear or not (provided that not all 4 are colinear). Thus I think a comparison is possible.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

line 117-119

I suggest to change as follows:

though the determinant is not zero. We discovered this from our experience with various interpolation experiments; in any case, we note that the bilinear polynomial can actually degenerate into a linear one in such situations, even if the interpolation data come from a quadratic (or higher degree) polynomial.

Author Response

Please see the attached  file

Author Response File: Author Response.pdf