WKT construction for local site scaling in x and y

Question

I need to build a simple Esri compatible proj string to simply represent scaling northing and easting from a point of origin central to a survey?

I've been able to work this out fine with a derivedprojcrs and the affine parametric transformation, however that seems to have some compatibility problems with some software.

Effectively the .prj should scale as follows:

X’new = (SCALE_FACTOR * (X’old – ORIGIN_X)) + ORIGIN_X
Y’new = (SCALE_FACTOR * (Y’old – ORIGIN_Y)) + ORIGIN_Y

My current Esri prj that isn't scaling correctly is:

COMPD_CS["Custom_Transverse_Mercator_with_NAVD88_and_GEOID18",
    PROJCS["Custom_Transverse_Mercator",
        GEOGCS["NAD83",
            DATUM["North_American_Datum_1983",
                SPHEROID["GRS 1980",6378137,298.257223563]],
            PRIMEM["Greenwich",0.0],
            UNIT["Degree",0.0174532925199433]],
        PROJECTION["Transverse_Mercator"],
        PARAMETER["False_Easting",2814946.878],
        PARAMETER["False_Northing",1359674.745],
        PARAMETER["Central_Meridian",-106.14951023],
        PARAMETER["Scale_Factor",1.0004354904],
        PARAMETER["Latitude_Of_Origin",38.81910429],
        UNIT["US survey foot",0.304800609601219]],
    VERT_CS["NAVD88 (GEOID18)",
        VERT_DATUM["North_American_Vertical_Datum_1988 with GEOID18",2005],
        UNIT["US survey foot",0.304800609601219],
        AXIS["Gravity-related height",UP]]]

My test points are as follows:

Name    Latitude    Longitude
1   38.81867707 -106.15170219
2   38.82023696 -106.15255798
3   38.82234018 -106.14698871
4   38.81947480 -106.14686198
0   38.81910429 -106.14951023 (Point of origin)

Input 'Grid' Points (these are the State Plane ESPG 6428 N/E for the lat/lon's above)
Name    Northing    Easting
1   1,359,523.631   2,814,321.258
2   1,360,093.468   2,814,081.516
3   1,360,848.077   2,815,673.670
4   1,359,804.295   2,815,702.344
0   1,359,674.745   2,814,946.878 (point of origin)

Ground scale factor: 1.0004355056

Grid Distance to PoO 
Name    Northing    Easting
1   -151.1140   -625.6200
2   418.7230    -865.3620
3   1173.3320   726.7920
4   129.5500    755.4660
0   0.0000  0.0000


Ground Distance to Pt of Origin
Name    Northing    Easting
1   -151.1798   -625.8925
2   418.9054    -865.7389
3   1173.8430   727.1085
4   129.6064    755.7950
0   0.0000  0.0000

Grid to Ground - Scaled from Point 0
Name    Northing    Easting
1   1,359,523.565   2,814,320.986
2   1,360,093.650   2,814,081.139
3   1,360,848.588   2,815,673.987
4   1,359,804.351   2,815,702.673
0   1,359,674.745   2,814,946.878

However the poor prj above is providing the following scaled points.

Name    Northing    Easting
1   1,359,519.086   2,814,322.049
2   1,360,087.471   2,814,078.121
3   1,360,853.818   2,815,665.613
4   1,359,809.759   2,815,701.766
0   1,359,674.745   2,814,946.878

Answer 1

The original projection and parameters for NAD83 (2011) State Plane Colorado Central that matter for a grid-to-ground adjustment are:

projection: Lambert conformal conic (2 standard parallels)
false easting: 3000000 US survey feet
false northing: 1000000 US survey feet

The Esri projection engine used by ArcGIS has an odd Lambert conformal conic definition. If you define it from scratch as a new custom projected coordinate system, you'll see 2 standard parallels, latitude of origin, central meridian, false easting and northing, AND a scale factor.

Create a new projected coordinate system using:

(same GCS and linear units)
projection: Lambert conformal conic
false easting: 2814946.878 US survey feet
false northing: 1359674.745 US survey feet
scale factor: 1.0004355056
central meridian: -105.5
standard parallel 1: 38.45
standard parallel 2: 39.75
latitude of origin: 37.83333333333333

The conversion method appears to take off the original false easting and false northing, multiplies those xy values by the scale factor, and then applies different false easting and false northing values. That's what I'm trying to replicate with the new PCS definition.

Transverse Mercator <> Lambert conformal conic so I'm hoping that where the not-quite-matching results are coming from.

Note: usually I see a grid-to-ground / combined scale factor applied to the final PCS values. Mathematically, that's equivalent at the parameter level by multiplying any existing scale factor (like in transverse Mercator) AND the existing false easting and false northing values by the grid-to-ground value. The results are used as new parameter values in a custom PCS.

Disclosure: I work for Esri.