Equal Area Cylindrical not using `k_0` scale factor in WKT form

[snowman2]

Example of problem

This is expected:

$ ./projinfo '+proj=cea +type=crs'
PROJ.4 string:
+proj=cea +lat_ts=0 +lon_0=0 +x_0=0 +y_0=0 +datum=WGS84 +units=m +no_defs +type=crs

WKT2:2019 string:
PROJCRS["unknown",
    BASEGEOGCRS["unknown",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ID["EPSG",6326]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8901]]],
    CONVERSION["unknown",
        METHOD["Lambert Cylindrical Equal Area",
            ID["EPSG",9835]],
        PARAMETER["Latitude of 1st standard parallel",0,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8823]],
        PARAMETER["Longitude of natural origin",0,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["False easting",0,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",0,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]]]

This is missing the scaling factor:

$ ./projinfo '+proj=cea +k_0=2 +type=crs'
PROJ.4 string:
+proj=cea +k_0=2 +type=crs

WKT2:2019 string:
PROJCRS["unknown",
    BASEGEOGCRS["unknown",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ID["EPSG",6326]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8901]]],
    CONVERSION["unknown",
        METHOD["Lambert Cylindrical Equal Area",
            ID["EPSG",9835]],
        PARAMETER["Latitude of 1st standard parallel",0,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8823]],
        PARAMETER["Longitude of natural origin",0,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["False easting",0,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",0,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]]]

Problem description

This defines a scaling factor. https://proj.org/operations/projections/cea.html
However, it is not being used.

Expected Output

I would expect it to have a scaling factor similar to lcc or merc:

$ ./projinfo '+proj=lcc +lat_1=2 +k_0=2 +type=crs'
PROJ.4 string:
+proj=lcc +lat_0=0 +lon_0=0 +lat_1=2 +lat_2=0 +x_0=0 +y_0=0 +k_0=2 +datum=WGS84 +units=m +no_defs +type=crs

WKT2:2019 string:
PROJCRS["unknown",
    BASEGEOGCRS["unknown",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ID["EPSG",6326]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8901]]],
    CONVERSION["unknown",
        METHOD["Lambert Conic Conformal (2SP Michigan)",
            ID["EPSG",1051]],
        PARAMETER["Latitude of false origin",0,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8821]],
        PARAMETER["Longitude of false origin",0,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8822]],
        PARAMETER["Latitude of 1st standard parallel",2,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8823]],
        PARAMETER["Latitude of 2nd standard parallel",0,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8824]],
        PARAMETER["Easting at false origin",0,
            LENGTHUNIT["metre",1],
            ID["EPSG",8826]],
        PARAMETER["Northing at false origin",0,
            LENGTHUNIT["metre",1],
            ID["EPSG",8827]],
        PARAMETER["Ellipsoid scaling factor",2,
            SCALEUNIT["unity",1],
            ID["EPSG",1038]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]]]

Environment Information

$ ./projinfo
Rel. 7.0.0, March 1st, 2020

• Ubuntu 18.04.3 LTS

Installation method

• From source master

[rouault]

Hum this is interesting... This was ported from GDAL 2.x OSRSetCEA() to PROJ.4 method, which has no scale factor. I presume GDAL implementation comes from EPSG 'Lambert Cylindrical Equal Area' method itself (https://www.epsg-registry.org/export.htm?gml=urn:ogc:def:method:EPSG::9835) that doesn't define a scale factor. Hence this is lost when mapping to WKT.
The clean way of solving this would probably when detecting +proj=cea and +k/k_0 to use a custom WKT representation based on the EPSG one but with an extra parameter...

[snowman2]

I haven't dug into the PROJ code to check, but does it use the scale factor parameter?

[rouault]

yes, the cea.cpp file uses P->k_0

[snowman2]

I added the extra parameter via WKT, but then it is ignored in the PROJ string:

./projinfo 'CONVERSION["unknown",METHOD["Lambert Cylindrical Equal Area"],PARAMETER["Latitude of 1st standard parallel",0,ANGLEUNIT["degree",0.0174532925199433],ID["EPSG",8823]],PARAMETER["Longitude of natural origin",0,ANGLEUNIT["degree",0.0174532925199433],ID["EPSG",8802]],PARAMETER["False easting",0,LENGTHUNIT["metre",1],ID["EPSG",8806]],PARAMETER["False northing",0,LENGTHUNIT["metre",1],ID["EPSG",8807]],PARAMETER["Scale factor at natural origin",1,SCALEUNIT["unity",1],ID["EPSG",8805]]]'
PROJ string:
+proj=cea +lat_ts=0 +lon_0=0 +x_0=0 +y_0=0

WKT2:2019 string:
CONVERSION["unknown",
    METHOD["Lambert Cylindrical Equal Area"],
    PARAMETER["Latitude of 1st standard parallel",0,
        ANGLEUNIT["degree",0.0174532925199433],
        ID["EPSG",8823]],
    PARAMETER["Longitude of natural origin",0,
        ANGLEUNIT["degree",0.0174532925199433],
        ID["EPSG",8802]],
    PARAMETER["False easting",0,
        LENGTHUNIT["metre",1],
        ID["EPSG",8806]],
    PARAMETER["False northing",0,
        LENGTHUNIT["metre",1],
        ID["EPSG",8807]],
    PARAMETER["Scale factor at natural origin",1,
        SCALEUNIT["unity",1],
        ID["EPSG",8805]]]

[rouault]

except that it won't receive it now, unless you use a non-CRS PROJ pipeline string

I added the extra parameter via WKT, but then it is ignored in the PROJ string

'Expected' as there's no mapping between 'Scale factor at natural origin' and k_0 for this METHOD

[snowman2]

Okay, so this won't work via WKT until the fix is applied. Makes sense.