[PROJ] Is 16-bit quantization of values / (sub-)millimetric error in grids (sometimes) acceptable ?

Sun Dec 8 08:20:09 PST 2019

In my understanding (derived from metrology), "accuracy" means "how close
to an actual measured or measurable quantity is the number our system
gives?"
and "precision" means "how many decimal places does the system give?". If
the claimed precision is much more than the actual accuracy, what you get
is a
lot of meaningless numbers (insignificant digits, if you will). 1 mm
accuracy is hard, 0.1 mm attainable only under very special circumstances
and over short
distances.

However, for the purpose of checking algorithms against each other, the
"accuracy" becomes "does my algorithm give the same number as the one I'm
comparing
against?" (which might be some "official" one). Since it is all arithmetic,
the accuracy can in principle be zero, and so (in principle) you want very
high precision.

So, when talking about the level of precision that is appropriate, I think
it is important to separate mathematical and physical accuracy.

Duncan Agnew

On Sun, Dec 8, 2019 at 7:10 AM Even Rouault <even.rouault at spatialys.com>
wrote:

> Hi,
>
> I'm experimenting with encoding grids with 16-bit integer values, with an
> offset and scale, instead of using IEEE 32-bit floats. There's some
> connection
> with my previous thread about accuracies of NTv2...
>
> Let's for example take the example of egm08_25.gtx. It is 149.3 MB large
> If converting it into a IEEE 32-bit float deflate-compressed (with
> floating-
> point predictor) tiled geotiff, it becomes 80.5 MB large (the compression
> method and floatin-point predictor are fully lossless. There's bit-to-bit
> equivalence for the elevation values regarding the original gtx file)
> Now observing that the range of values is in [-107,86], I can remap that
> to
> [0,65535] with an offset of -107 and scale of (86 - -107) / 65535 ~= 0.0029
> The resulting deflate-compressed (with integer predictor) tiled GeoTIFF is
> now
> 23.1 MB large !
>
> Looking at the difference between the unscaled quantized values and the
> original ones, the error is in the [-1.5 mm, 1.5 mm] range (which is
> expected,
> being the half of the scale value), with a mean value of 4.5e-6 metre (so
> centered), and a standard deviation of 0.85 mm
>
> After that experimentation, I found this interesting page of the
> GeographicLib
> documentation
> https://geographiclib.sourceforge.io/html/geoid.html
> which compares the errors introduced by the gridding itself and
> interpolation
> methods (the EGM model is originally a continuous model), with/without
> quantization. And one conclusion is "If, instead, Geoid were to use data
> files
> without such quantization artifacts, the overall error would be reduced
> but
> only modestly". Actually with the bilinear interpolation we use, the max
> and
> RMS errors with and without quantization are the same... So it seems
> perfectly
> valid to use such quantized products, at least for EGM2008, right ?
>
> Now looking at horizontal grids, let's consider Australia's
> GDA94_GDA2020_conformal.gsb. It is 83 MB large (half of this size due to
> the
> error channels, which are set to a dummy value of -1...)
> Converting it to a compressed tiled Float32 tif (without those useless
> error
> channels), make it down to 4.5 MB.
> And as a quantitized uint16 compressed tif, down to 1.4 MB (yes, almost 60
> times smaller than original .gsb file). The maximum scale factor is 1.5e-7
> arcsecond, hence a maximum error of 2.3 micrometre... I'm pretty sure
> we're
> several order of magnitudes beyond the accuracy of the original model,
> right ?
> In EPSG this transformation is reported to have an accuracy of 5cm.
> The fact that we get such a small scale factor is due to GDA94 -> GDA2020
> conformal being mostly a uniform shift of ~1.8 m and that the grids is
> mentioned to "Gives identical results to Helmert transformation GDA94 to
> GDA2020 (1)"
>
> If we look at the France' ntf_r93.gsb, which has shifts of an amplitude up
> to
> 130m, the maximum error introduced by the quantization is 0.6 mm. I would
> tend
> to think this is also acceptable (given the size of that particular file
> is
> small, compression gains are quite neglectable, but this is mostly to look
> if
> we can generalize such mechanism). What puzzles me is that in
> https://geodesie.ign.fr/contenu/fichiers/documentation/algorithmes/notice/
> NT111_V1_HARMEL_TransfoNTF-RGF93_FormatGrilleNTV2.pdf
> <https://geodesie.ign.fr/contenu/fichiers/documentation/algorithmes/notice/NT111_V1_HARMEL_TransfoNTF-RGF93_FormatGrilleNTV2.pdf>
> where they compare the
> NTv2 approach regarding their native 3D geocentric correction approach,
> they
> underline in red a sample point where the difference between the 2 models
> is
> 1.2 mm, as if it had really some importance. For that test point, using
> the
> quantized approach would increase this difference to 1.3 mm. But when
> looking
> at the accuracy reported in the grid at that point it is 1.6e-3 arc-second
> (which is the minimum value for the latitude error of the product, by the
> way), ie 5cm, so it seems to me that discussing about millimetric error
> doesn't make sense.
> In EPSG this transformation is reported to have an accuracy of 1 metre
> (which
> is consistent with the mean value of the latitude shift error)
>
> Now, let's look at the freshly introduced BWTA2017.gsb file. 392 MB large
> As a Float32 compressed geotiff: 73 MB (5.4x compression rate)
> As a Int16 compressed geotiff: 26 MB (15x compression rate)
> Maximum error added by quantization for latitude shift: 0.25 mm
> Minimum error value advertized for latitude shift: 1.61e-5 arc-second (not
> completely sure about the units...), ie 0.5mm
> Mean error value advertized for latitude shift: 6.33e-5 arc-second, ie
> 1.9mm
> Interestingly when looking at the ASCII version of the grid, the values of
> the
> shifts are given with a precision of 1e-6 arcseconds, that is 0.03 mm !
>
> For Canadian NTv2_0.gsb, on the first subgrid, the quantization error is
> 0.9 mm for the latitude shift. The advertized error for latitude is in [0,
> 13.35 m] (the 0.000 value is really surprising. reached on a couple points
> only), with a mean at 0.27 m and stddev at 0.48 m. In EPSG this
> transformation
> is reported to have an accuracy of 1.5 metre
>
> ~~~~~~~~~~~~~
>
> So, TLDR, is it safe (and worth) to generate quantized products to reducte
> by
> about a factor of 2 to 3 the size of our grids compared to unquantized
> products, when the maximum error added by the quantization is ~ 1mm or
> less ?
> or will data producers consider we damage the quality of products they
> carefully crafted ? do some users need millimetric / sub-millimetric
> accuracy
> ?
>
> Or do we need to condition quantization to a criterion or a combination of
> criteria like:
> - a maximum absolute error that quantization introduces (1 mm ? 0.1 mm ?)
> - a maximum value for the ratio between the maximum absolute error that
> quantization introduces over the minimum error value advertized (when
> known)
> below some value ? For the BWTA2017 product, this ratio is 0.5. For
> ntf_r93.gsb, 0.012. For NTv2_0.gsb, cannot be computed given sothe min
> error
> value advertized is 0...
> - or, variant of the above, a maximum value for the ratio between the
> maximum
> absolute that quantization introduces  over the mean of the error value
> advertized (when known). For the BWTA2017 product, this ratio is 0.13. For
> ntf_r93.gsb, 5.5e-4. For NTv2_0.gsb, 3.3e-3
> - and perhaps consider that only for products above a given size (still
> larger
> than 10 MB after lossless compression ?)
>
> ~~~~~~~~~~~~~~~~
>
> Jochem mentionned in the previous thread that the Netherlands grids have
> an
> accuracy of 1mm. I'm really intrigued by what that means. Does that mean
> that
> the position of control points used to build the grid is known at that
> accuracy, both in the source and target systems, and that when using
> bilinear
> interpolation with the grid, one remains within that accuracy ? Actually
> both
> the rdtrans2008.gsb and rdtrans2018.gsb grids report an accuracy of 1mm,
> but
> when comparing the positions corrected by those 2 grids, I get differences
> above 1mm.
>
> echo "6 53 0" | cct -d 9 +proj=hgridshift +grids=./rdtrans2008.gsb
>    5.999476046    52.998912757
>
> echo "6 53 0" | cct -d 9 +proj=hgridshift +grids=./rdtrans2018.gsb
>    5.999476020    52.998912753
>
> echo "52.998912757 5.999476046 52.998912753 5.999476020" | geod -I
> +ellps=GRS80
> -104d18'21.49"  75d41'38.51"    0.002
>
> That's a difference of 2 mm. I get that difference on a few other "random"
> points.
>
> If applying the quantization on rdtrans2018.gsb, we'd add an additional
> maximal error of 0.6 mm. The grid being 1.5 MB uncompressed, and 284 KB as
> a
> losslessly compressed TIFF, quantization isn't really worth considering
> (reduces the file size to 78 KB)
>
> Even
>
> --
> Spatialys - Geospatial professional services
> http://www.spatialys.com
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