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Open AccessArticle
Peer-Review Record

Comparison of FOSS4G Supported Equal-Area Projections Using Discrete Distortion Indicatrices

ISPRS Int. J. Geo-Inf. 2019, 8(8), 351; https://doi.org/10.3390/ijgi8080351
Reviewer 1: Miljenko Lapaine
Reviewer 2: Christian Willmes
ISPRS Int. J. Geo-Inf. 2019, 8(8), 351; https://doi.org/10.3390/ijgi8080351
Received: 30 April 2019 / Revised: 30 July 2019 / Accepted: 7 August 2019 / Published: 9 August 2019
(This article belongs to the Special Issue Open Science in the Geospatial Domain)

Round 1

Reviewer 1 Report

Our world is like a sphere. Continuity in a map is extremely important, and it is far more useful to decrease the number of interruptions by arranging the map within a single flat shape, such as an oval, rectangle, or circle. There is then only one interruption, the outside edge.

In 1923, after working since 1916 with the interruption of the two projections along various meridians in his effort to reduce the distortion at their outer edges, Goode developed the homolosine projection. In addition to being interrupted the projection is a combination of two equal-area projections, the Sinusoidal (better shapes in the equatorial areas) and the Mollweide (better shapes in the high latitudes), in order to minimize the total amount of inevitable distortion. So, it is rather strange to prove something known about 100 years ago.

The additional interruptions violate even further the inherent continuity of the surface of the globe. Distortions may be considerably reduced by interrupting the surface of the globe map in many places.The question of whether the disadvantage of lack of continuity is worth the advantage of less distortions is one that must be decided in terms of the use to which the map will be put. (See: Special Publication No. 2 of the American Cartographic Association has been published in 1988 by the American Congress on Surveying and Mapping).


Title:

Comparison of FOSS4G supported equal-area projections using discrete Tissot indicatrices

The paper do not use Tissot indicatrices at all.

 

Line 3

Tissot indicatrices were positioned on the ellipsoid surface

Tissot indicatrix is an ellipse, and as a plane curve can not be positioned on the curved ellipsoid surface. Moreover, Tissot indicatrix or ellipse of distortion lies in the plane of projection and is created by mapping a differential small circle.

 

Line 5 and further

The term distortion has not the usual meaning as it has in the theory of map projections. It should be emphasized that it is a distortion as defined by the authors of the article.

 

Line 22 and further

Instead of the term cartographic projection the term map projection is recommended because of its common use.

 

Line 24

Mercator projection expand the surface area of the Earth by 200%.

This is wrong, because Mercator projection expand the surface of the Earth into infinity.

 

Line 29

Modern equal-area projections … often do not find support among the core FOSS4G libraries …

Not correct, see the newest Equal Earth map projection.

 

Line 26

Why using an equal-area projection will minimize the size of datasets?

 

Line 89

Other equal-area projections are supported by FOSS4G but do not present enough unique characteristics to set them markedly apart from the quintet analyzed.

This statement should be explained.

 

Line 94

The indicatrix is itself a simple object: a circle defined on the surface of the sphere or the ellipdoid.

This is not true, Tissot indicatrix is na ellipse in the plane of projection. And, there is no circle defined on the surface of the ellipsoid, except parallels of latitude if the ellipsoid is rotational one.

 

Line 97

In pseudo-cylindrical equal-area projections Tissot's indicatrices tend to acquire an elliptic shape.

This is rather strange statement, because Tissot's indicatrices are always ellipses. It seems that the authors did not read or understand the Tissot approach, although they cite his book.

 

Line 99

Whereas Tissot originally defined his indicatrices as infinitesimal circles…

This is not true. Tissot's indicatrices are not infinitesimal circles. Tissot's indicatrices are ellipses which can be explained as images in the plane of projection of infinitesimal circles.

 

Line 100

Each discrete Tissot indicatrix …

These are not Tissot's indicatrices but something invented by the authors of the paper.

 

Line 120

The indicatrices used in this study were positioned on the globe …

Again, these are not indicatrices.

 

Line 130

Angular and distance distortions were computed as follows.

It should be emphasized that distortion in the paper is not defined in usual way. It is defined by the authors of the article.

 

Line 236

ISRIC has recently adopted the Homolisine projection..

This statement requires references.


Author Response

The authors would like to thank the reviewer for the time spent on this article. The questions raised are pertinent and helped improved the article. Reactions to the various comments are found in-line bellow.


> Our world is like a sphere. Continuity in a map is extremely important, and it is far more useful to decrease the number of interruptions by arranging the map within a single flat shape, such as an oval, rectangle, or circle. There is then only one interruption, the outside edge.


Interruptions may or may not be an inconvenience, depending on the kind of data and analysis. For instance, a computation relying on cell neighbourhood that encompasses the oceans should be carried out in the counter-domain of a projection with only one interruption. If instead only the land-masses are object of study then interruptions over oceans are no hindrance. Also, it is important to consider that the projection used at computation time may not necessarily match the projection(s) used to present or serve the end result. For instance, a global raster computed on the Homolosine projection may be served through a WCS supporting various projections, even non-equal area ones. The discussion on this issue in section 4.1 was expanded.


> In 1923, after working since 1916 with the interruption of the two projections along various meridians in his effort to reduce the distortion at their outer edges, Goode developed the homolosine projection. In addition to being interrupted the projection is a combination of two equal-area projections, the Sinusoidal (better shapes in the equatorial areas) and the Mollweide (better shapes in the high latitudes), in order to minimize the total amount of inevitable distortion. So, it is rather strange to prove something known about 100 years ago.


As exposed in the Introduction, the primary goal of this study is to quantify and locate distortion using FOSS4G. This is naturally limited to the projections supported by this software. As also stated in the Introduction, equal-area projections proposed in the late XX century are known to yield the lowest distortions when measured with cartometric methods. However, none is supported by FOSS4G.


> The additional interruptions violate even further the inherent continuity of the surface of the globe. Distortions may be considerably reduced by interrupting the surface of the globe map in many places.The question of whether the disadvantage of lack of continuity is worth the advantage of less distortions is one that must be decided in terms of the use to which the map will be put. (See: Special Publication No. 2 of the American Cartographic Association has been published in 1988 by the American Congress on Surveying and Mapping).


The authors fully agree with these observations. While results point towards the Homolosine, section 4.1 states its shortcomings and points towards the Eckert IV and the Mollweide as alternatives.


> Title:

> Comparison of FOSS4G supported equal-area projections using discrete Tissot indicatrices

> The paper do not use Tissot indicatrices at all.


Indeed it does not. As the reviewer well notes, the indicatrices employed in this study are discrete constructions that allow the computation of distortions using FOSS4G. These indicatrices were named "discrete Tissot indicatrices" which can cause misinterpretations. The title of the article was thus modified and the remainder of the text corrected to properly distance the method used from that proposed by Tissot.

 


> Line 3

> Tissot indicatrices were positioned on the ellipsoid surface

> Tissot indicatrix is an ellipse, and as a plane curve can not be positioned on the curved ellipsoid surface. Moreover, Tissot indicatrix or ellipse of distortion lies in the plane of projection and is created by mapping a differential small circle.


See comment above. Tissot's indicatrix is referenced as a circle by various authors, including Mulcahy, since that is the shape it acquires in the counter-domain of a conformant projection.  


> Line 5 and further

> The term distortion has not the usual meaning as it has in the theory of map projections. It should be emphasized that it is a distortion as defined by the authors of the article.> 


The discrete nature of this study was emphasized further in section 2.2. 


> Line 22 and further

> Instead of the term cartographic projection the term map projection is recommended because of its common use.


Modified as requested. 


> Line 24

> Mercator projection expand the surface area of the Earth by 200%.

> This is wrong, because Mercator projection expand the surface of the Earth into infinity.


As stated in section 4.2, in a FOSS4G (and most other GIS programmes) the counter-domain of any projection is always infinite. However, the raster data format is by definition finite in area. Therefore a raster defined on the Mercator projection is forcefully limited. In fact, this limit is arbitrary, and the value initially reported was tied to a particular programme and not meaningful. This segment of the Introduction was modified to properly convey these aspects. 


> Line 29

> Modern equal-area projections … often do not find support among the core FOSS4G libraries …

> Not correct, see the newest Equal Earth map projection.


The Patterson projection is a recent projection, but not exactly modern. It presents the same kind of compromises found in the projections of the late XIX and early XX centuries. Snyder's projections of the late XX century, for instance, are considerably more modern. The text was modified to clarify this concept.

 

> Line 26

> Why using an equal-area projection will minimize the size of datasets?


It reduces the number of raster cells. The modifications to the Introduction should make this clear.


> Line 89


> Other equal-area projections are supported by FOSS4G but do not present enough unique characteristics to set them markedly apart from the quintet analyzed.

> This statement should be explained.


This segment of the text was expanded. 


> Line 94

> The indicatrix is itself a simple object: a circle defined on the surface of the sphere or the ellipdoid.

> This is not true, Tissot indicatrix is na ellipse in the plane of projection. And, there is no circle defined on the surface of the ellipsoid, except parallels of latitude if the ellipsoid is rotational one.


Please see comments above. 


> Line 97

> In pseudo-cylindrical equal-area projections Tissot's indicatrices tend to acquire an elliptic shape.

> This is rather strange statement, because Tissot's indicatrices are always ellipses. It seems that the authors did not read or understand the Tissot approach, although they cite his book.


Please see comments above.  


> Line 99

> Whereas Tissot originally defined his indicatrices as infinitesimal circles…

> This is not true. Tissot's indicatrices are not infinitesimal circles. Tissot's indicatrices are ellipses which can be explained as images in the plane of projection of infinitesimal circles.


Please see comments above. 


> Line 100

> Each discrete Tissot indicatrix …

> These are not Tissot's indicatrices but something invented by the authors of the paper.


Please see comments above. 


> Line 120

> The indicatrices used in this study were positioned on the globe …

> Again, these are not indicatrices.


Please see comments above. 


> Line 130

> Angular and distance distortions were computed as follows.

> It should be emphasized that distortion in the paper is not defined in usual > way. It is defined by the authors of the article.> 


This aspect was clarified in section 4.1. 


> Line 236

> ISRIC has recently adopted the Homolisine projection..> 

> This statement requires references.


This is the first article reporting this work.


Reviewer 2 Report

Thank you for this very interesting study.

In the following my review comments:

- This is a very interesting assertion, that "big spatial data researchers must undertake as first task a re-projection with an equal-area projection." Please provide some more metrics and arguments, comparing equal area projections to commonly used ones, such as Platte-Carre, for WGS84 Geographic Coordinate Grids (EPSG:4326), or Web-Mercartor (EPSG:3857) for WebMaps.

- Would it be more suitable to store for example gridded Climate Data, for example simulation-outputs, in equal area projections? If yes, please provide some arguments/metrics. As far as I see, this aspect is not really adressed by climate modellers, may it because of the model design, with a discrete finite number of grid cells, so that projection would have no effect on file size?  

- Do you know about this kind of map projection considerations in the context of existing big spatial data infrastrucutres, such as for example Copernicus Access Hub (https://scihub.copernicus.eu/) ? If not, can you probably come up with an explanation to this "oddity" in your conclusion or discussion section?

- And please adress the issue of possible information loss, by reprojecting a dataset to reduce the number of raster cells, as asserted in l.239 "was reduced in more than 1 billion cells".

- The linked repository (https://gitlab.com/ldesousa/projections-compare) is/was access restricted, when I tryed to access the repo.


Minor typo:
- l. 150-151: "Figure 4 and Figure 4 convey..."

Author Response

> Thank you for this very interesting study. 


The authors would like to thank the reviewer for the time spent on this article. The questions raised are pertinent and helped improved the article. 


> In the following my review comments:


> - This is a very interesting assertion, that "big spatial data researchers must undertake as first task a re-projection with an equal-area projection." Please provide some more metrics and arguments, comparing equal area projections to commonly used ones, such as Platte-Carre, for WGS84 Geographic Coordinate Grids (EPSG:4326), or Web-Mercartor (EPSG:3857) for WebMaps. 


More detailed metrics on raster area and number of cells were added tothe Introduction.


> - Would it be more suitable to store for example gridded Climate Data, for example simulation-outputs, in equal area projections? If yes, please provide some arguments/metrics. As far as I see, this aspect is not really adressed by climate modellers, may it because of the model design, with a discrete finite number of grid cells, so that projection would have no effect on file size?  


While none of the authors is a climate scientist, the principles that lead to this study are also valid in that field. Equal-area projections not only reduce the storage space and computation time required in processing, they also portrait more acurately spatial patterns and processes. E.g. when mapping changes in temperature, a non equal-area projection lends an artificial importance to more distorted areas that is undesirable. Reference institutions such as Climate.gov [1] or the University of Alabama [2] publish data in equal-area projections (primarilly Mollweide). Another example are the data obtained with the MODIS sensor, which are published by NASA also in the Sinusoidal projection [3]. The Climate science community seems awere of these issues. 


> - Do you know about this kind of map projection considerations in the context of existing big spatial data infrastrucutres, such as for example Copernicus Access Hub (https://scihub.copernicus.eu/) ? If not, can you probably come up with an explanation to this "oddity" in your conclusion or discussion section?


The Copernicus infrastructure uses the Universal Transverse Mercator system, composed by 60 zones defined with the Gauss-Krüger projection. This system can be regarded as a single projection with 61 interruptions. While not equal-area, the UTM system greatly reduces distortion and the number of cells in the rasters it supports.


Another popular cloud computing infrastructure is the Google Earth Engine (GEE). It relies on GeoTools and therefore can use internally any projection supported by that library. However, GEE uses the Marynus of Tyre projection by default, and any different projection requires an re-projection computation. The reason for this can only be object of speculation. A cloud-computing service that charges its costumers by the number of raster cells, or computation cycles, will naturally be more profitable using a non equal-area projection that artificially expands the area of computation. The authors would rather not dive into such speculation in the article. 


> - And please adress the issue of possible information loss, by reprojecting a dataset to reduce the number of raster cells, as asserted in l.239 "was reduced in more than 1 billion cells".


This point in now addressed in the Introduction.


> - The linked repository (https://gitlab.com/ldesousa/projections-compare) is/was access restricted, when I tryed to access the repo.


Access rights to the repository were reviewed. It is now accessible from anywhere and by everyone.


[1] https://www.climate.gov/news-features/understanding-climate/climate-change-global-temperature-projections


[2] https://www.nsstc.uah.edu/climate/


[3] https://modis-land.gsfc.nasa.gov/MODLAND_grid.html


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