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In Southern California, where I live, multiple sources of information seem to say that the conversion between NAD27 and NAD83 is almost purely in the east-west direction (about 80 meters), with almost no correction (only a few meters) along north-south. For example, the gdal utility cs2cs seems to verify this, by showing almost no change in the latitude and a much larger change in the longitude:

$ echo "33.760268  -116.682251" | cs2cs -rsf "%.6f" +proj=latlong +datum=NAD27 +to +proj=latlong +zone=11 +datum=NAD83 -
33.760288   -116.683102 0.000032

But I'm running into weird discrepancies that I don't understand. Here are the bottom left corners of two USGS topos, one from 1988 (left) and one from 2018 (right):

comparison of USGS topos

The text in the 1988 topo says:

1927 North American Datum. To place on the predicted North American Datum 1983 move the projection lines 80 meters east as shown by dashed corner ticks.

However, there is a clearly visible additional shift of about 200 meters in the y direction. For example, the y=35 km line coincides with a road in the old topo, but is far south of the same road in the new one.

I'm also seeing this with a handheld GPS unit. If I set it to read out in NAD27 and take the UTM coordinates on my front porch, I get this (x y):

0410910 3750091

If I then set it to NAD83 and repeat, I get this:

0410830 3750288

The sign and magnitude of the x offset are exactly as calculated by cs2cs and as claimed on the topo maps. However, there is an additional y offset of almost 200 meters.

What am I not understanding here?

If anyone can help me with this, feel free to come by my UTM coordinates for a beer. Just make sure to resolve the ambiguity first, or you'll end up at my neighbor's house up the street to the north.

[EDIT] It seems that each NAD datum has its own conversion between lat-lon and UTM:

$ echo "33.760268  -116.682251" | cs2cs -rf "%.6f" +proj=latlong +datum=NAD83 +to +proj=utm +zone=11 +datum=NAD83 -
529425.663755   3735620.830534 0.000000
$ echo "33.760268  -116.682251" | cs2cs -rf "%.6f" +proj=latlong +datum=NAD27 +to +proj=utm +zone=11 +datum=NAD27 -
529426.322142   3735427.157052 0.000000

This can also be seen on the USGS maps, where the corners of the map are defined by lat-lon, but the UTM grid is shifted north-south relative to the corners.

So I guess this sort of clears it up for me, in the sense that it shows there are really four different coordinate systems: (NAD27,lat-lon), (NAD27,utm), (NAD83,lat-lon), and (NAD83,utm). I would still be happy to get an answer explaining this further. Are there two things that have to be specified, an ellipsoid and a projection?

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You'll see different offsets depending on the coordinate reference system. The north-south 200 m offset is classic when working in UTM and comparing comparing coordinates based on NAD27 versus NAD83. Part of the issue is the different ellipsoid sizes and shapes between Clarke 1866 (used for NAD27) and GRS80 (used for NAD83). The mathematics of the transverse Mercator projection magnifies the differences between the two ellipsoids.

If you used the older NADCON application, it would report different offset values because it worked strictly in latitude-longitude values.

I remember the dashed tick line statement on the topos as referring to the graticule--move the latitude-longitude lines not the "projection" lines. Considering that the topo has both State Plane and UTM lines and ticks, I wonder which one it could refer to?

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