Mathias,

Thanks for pointing out the reverse conversion is the one in question.  The results are:

 lat lon alt x y z karney lat err karney lon err karney alt err sky lat err sky lon err sky alt err k lat - s lat k lon - s lon k alt - s alt

 min -2.1e-14 -3.67254e-09 -9e-06 -0.99846 -8.9e-06 -2.8e-14 3.51e-06 max 2.1e-14 3.68982e-09 8.94e-06 -3.5e-06 9.05e-06 2.8e-14 0.998459 mean 2.3e-18 -3.85265e-12 -2.1e-08 -0.3197 2.11e-08 6.46e-17 0.3197 std. dev. 3.85e-15 9.30808e-10 3.67e-06 0.256233 3.67e-06 8.09e-15 0.256233

The method was to pick psuedo random (lat,lon,alt) points as before, convert to ecef (x,y,z) via GeographicLib (which was previously shown to be well matched with lla2ecef), and then do the reverse transform either via GeographicLib (karney lat err, ...) or ECEF_to_LLA (sky lat err, ...).  The columns karney lat err, ... sky lat err, ... are the difference between the reverse transform output and the original (lat, long, alt) data.   The final 3 columns are the difference between the karney and skytraq (ECEF_to LLA) reverse transform output.  I didn't summarize the longitude differences because the longitude is easy to compute in the reverse transform, the errors are small, and I would have had to clean up 360 degree reporting differences.  The units I used are degrees and meters.

As I would expect the reverse transform in GeographicLib gives us the back the original point with a very high degree of accuracy, all differences were less than 4nm.  ECEF_to_LLA showed errors up to 1m.   I point out that there are errors of similar magnitude in latitude with ECEF_to_LLA.

From my reading of the original works ECEF_to_LLA appears to be based on
Bowring BR (1976) Transformation from spatial to geodetic coordinates.
Surv Rev 23: 323–327
which I have not been able to obtain.

The excellent documentaion of GeographicLib states

The conversion from geographic to geocentric coordinates is straightforward. For the reverse transformation we use

Several changes have been made to ensure that the method returns accurate results for all finite inputs (even if h is infinite). The changes are described in Appendix B of

The Vermeille reference gives the Bowring forumlas and compares the results of the Vermeille method with those of Bowring for a limited number of points and shows errors in height with the Bowring approximations of up to 0.25m.  These seem consistent with the magnitude of the differences I found between GeographicLib and ECEF_to_LLA.

Vermeille also cites
Heikkinen M (1982) Geschlossene Formeln zur Berechnung
ra¨ umlicher geoda¨ tischer Koordinaten aus rechtwinkligen Koordinaten.
Z Vermess 5: 207–211
and says
Borkowski (1987), (1989), Heikkinen (1982) and
Lapaine (1991) have proposed their own solutions
based also on an equation ofdegree 4. The resulting
formulae are more complicated than the method
described here.
Joesef cites two presentations
(1) http://www.joachim-bolz.de/projekte/gps/gps.html,
(2) http://www.fsd.mw.tum.de/Homepage-Daten/Lehre%20WS11-12/FSD1/FSD_I_Trafo_WGS.pdf
(1) appears to follow Bowring.  (2) appears to follow Heikkinen, but I am not able to read German.

Josef also says he believes that he thinks the methods (1) and (2) differed in height by approximately 68m.  This result is unexpectedly large.  It is hard for me to believe that the various solutions Vermeille cites have differences this large, but I haven't compared them or found a reference that does so.  I think Vermeille would have pointed out these errors instead of mentioning complexity.   I suspect one of the follow may explain this large difference:
2) There is an error in (2), perhaps typographic
3) Different assumptions are used in (2) that we haven't appreciated
4) ???

However Josef also states he believes the results from (2) more closely match the results from ntrip and the gpsvisulaizer elevation data base.  While I wouldn't consider either of these a primary reference I can't explain the discrepancy.  It also unclear to me what points Josef is using for his comparison.   Are his points near lat=0.177 (units?), lon=0.162(units ?), alt 395 (presumably m)?
in an area of a variation in
lat/lng/alt  of  0.177277/0.162193/394.858827

Is it possible that the gps data Josef used just had large elevation errors?  I know my old Garmin has a hard time estimating elevation.

At this time I cannot support Josef's desire to change to the method of (2).  I don't think the original issue is understood.

The methods of GeographicLib could reduce significant but not overwhelming errors, i.e. ~1m, and these differences are explainable as I discussed above, but these are small relative to the original reported issue.  It would seem to me that the gpsbabel project could add more value by concentrating on things central to the project like format translations instead of pretending we are geodesy experts, i.e. I think we should build on the work of acknowledged experts in geodesy instead of trying to do it ourselves.  However, I still don't have an overwhelming example to support GeographicLib usage although errors approaching 1m seem like something we might want to correct.

Steve

On 1/4/2013 4:59 AM, Mathias Adam wrote:
```Hi Steve,
hi Robert,

Am Fr, 4.01.2013, 03:32 schrieb tsteven4:
```
```I wrote a test driver to compare lla2ecef from skytraq.c with the
Forward method of GeographicLib::Geocentric.  I used approximately
uniform distributions of:
i) latitude from -90 to 90 deg
ii) longitude from -360 to 360 deg
iii) elevation from -9000 to 9000m
I also changed lla2ecef to use doubles instead of long doubles.  I also
changed a few integer constants to double constants, e.g. 1 -> 1.0, but
that's just me.  The results, for a 10,000 point simulation, are
x y                               z
minimum difference (m)              -3.72529E-09 -7.68341E-09
-6.51926E-09
maximum difference (m)               5.82077E-09 7.33417E-09
6.51926E-09
mean difference (m)                  2.21595E-10 -9.21225E-12
3.29291E-11
standard deviation of difference (m) 1.22922E-09 1.25931E-09
3.84619E-09

Thus, for psuedo random data:
1) The differences between GeographicLib and lla2ecef are irrelevant for
us.
2) there is no need to use long doubles in lla2ecef.
Perhaps there is some pathological input that would show a greater
difference, but I don't think lla2ecef explains the bug report Mathias
cited.

Steve
```
```
thanks for trying that out!  I agree there's no need to replace this code
with GeographicLib (at least not unless the lib might be incorporated
anyway someday).

Did you also try the other direction ECEF_to_LLA()? In fact, that's the
conversion Josef put on discussion afair.

```
```On 1/3/2013 7:18 PM, Robert Lipe wrote:
```
```It might make an interesting comparison point, but I'd still rather
not pull it in if it's "just" a bug fix in a formula for one format.
```
```the reason for my previous mail was to a) just mention that skytraq might
be a candidate to profit from GeographicLib, so if it would turn out
someday that there are several modules which will use it one can decide
whether it's worth incorporating it then.   And b) as a reminder to myself
to try it once I find the time to investigate the conversion  ;-)

```
```Didn't we go through several different formulas for ECEF to WGS84 a
year or so ago?
```
```Hmm, don't remember that, at least not that we found a conclusion. I do
have that conversion on my list of things to check since the first release
though...

```
```Oh, and we do use WGS-84 as that's by far the most common.
```
```
Regards,
Mathias

```
```
On Mon, Dec 31, 2012 at 1:11 AM, Mathias Adam

Hi,

Am So, 30.12.2012, 20:12 schrieb Robert Lipe:
>> I remain incredibly impressed with GeographicLib.  The
documentation,
references, and methodology are some of the finest I have ever seen
in
a
>> software package.  I thought it was
> It's well regarded.  I'm just not sure it solves interesting
problems
for
> us.   The huge percentage of the data we target is WGS84.  If
you look
at
> the files doing projection conversions, you'll find the formats
doing
projection conversion are either centered around paper maps,
location
specific, or just into masochism and allowing internal
representations
in
> arbitrary projections.
> I would not be unhappy forgetting about most of those projection
conversions and leaving it to tools specializing in such things,
like
GeographicLib and Proj.4.  I think I'd rather do less projection
than
> more.

documentation: I think it might be helpful for the skytraq module
too.
This format does a conversion from ECEF coordinates to gpsbabel's
internal
representation (WGS84, if I remember correctly).

The current implementation of this conversion seems to need some
investigation:
<http://old.nabble.com/Help-in-converting-ECEF-%3C-%3E-Lat-Lng-td34134547.html>

Now, GeographicLib claims to provide an ECEF conversion:
<http://geographiclib.sourceforge.net/html/classGeographicLib_1_1Geocentric.html>

I think it's at least worth a try -- and if it turns out to give
better
results than my quick "well, seems to work" solution it would be an
additional argument for including this library (as long as the
licensing
issue can be sorted out, of course).

Unfortunately I haven't found the time to look into it now (and
probably
won't in the next couple of weeks).

Best Regards,
Mathias

```
```
```
```
```