Hello Johann,
To answer #1 - why is vertical accuracy always less than horizontal
accuracy?
Perhaps the easiest to read explanation of GPS accuracy is at
http://www.eomonline.com/Common/Archives/May97/gilbert.htm
Scroll down near the bottom for the section titled "Why is z less
accurate than x and y?".
To summarize, the satellites that are in your line of sight will be in
different directions (often opposed). The redundant information can be
used to check / validate the measurement from other satellites. You
don't get the same level of information for vertical directions since
the ones below the earth are not in the line of sight (and you don't
get the compensating data).
There are a number of other references including:
http://users.erols.com/dlwilson/gpsvert.htm
see the notes at the end - it basically states that vertical accuracy
gets much worse when at latitudes greater than 65 degrees.
http://www.redsword.com/gps/old/sum_pos.htm
makes similar statements and the comments that vertical accuracy may
be far more important than horizontal in several applications (e.g.,
landing an airplane).
As a result of this analysis, I would expect there to be no cases
where the vertical and horizontal accuracy are equal [though I could
not find a specific reference stating this]. I also noted in one
article that some GPS devices have a bias - the vertical measurement
may average 10 m too high with a comment that the bias made good
measurements more difficult.
To answer #2 - how does a receiver use 5 (or more) satellites to do
position finding?
Going back to that first reference:
http://www.eomonline.com/Common/Archives/May97/gilbert.htm
There is a paragraph near the top that explains that you have four
equations with four unknowns (T, X, Y, Z). There are straight forward
methods to do that solution. When you have five data satellites,
methods used include:
- choose four out of five satellites (let's call them A, B, C, D, E)
to give you combinations such as
A, B, C, D A, B, C, E A, B, D, E A, C, D, E B, C, D, E
and so on. Analysis of the data would find several results that are
closely grouped, throw out the outliers, and take a geometric mean to
produce the result
- choose the "best four" out of the ones available. As mentioned in a
few of the other articles, the GPS may have filters to throw out
satellites that are too low in the sky. In this way, the GPS can get
good accuracy with less processing.
- use a least squares method. One site describes this latter method
as
p = (A^t A)^-1 A^t b - the ^ means the next letter(s) are a
superscript
This was pulled from one of several PDF reports on improving GPS
accuracy.
http://www.ra.pae.osd.mil/adodcas/slides/miller1.pdf
http://products.thalesnavigation.com/assets/techpapers/3_PositioningUS.pdf
http://einstein.stanford.edu/gps/PDF/wraim_tfw95.pdf
and so on. The vendors are not so straight forward - just stating they
use an overdetermined solution without explaining what it is.
You can find a number of other references using phrases such as:
GPS receiver estimate accuracy overdetermined
to give you a set of good references. Please let me know if you need
additional information in a clarification request.
--Maniac |
Clarification of Answer by
maniac-ga
on
08 May 2003 16:56 PDT
Hello Johann,
Hmm. A less technical answer for over-determined position finding.
Let me use a simpler example - if you have two points on a plane, you
can draw a straight line between them and get an "exact" fit. When you
have more than two points on a plane, you do not get an exact fit
unless they are all on the one line. The first figure on
http://www.krellinst.org/UCES/archive/classes/CNA/dir1.8/uces1.8.html
has an example of this. In this case, six data points are shown with
the "best fit" line that goes through them. The usual method to do
this is least squares.
Now - determining the location (X, Y, Z) and time (T) using a GPS,
using data from four satellites is the same kind of problem as the two
points on the plane. In a similar way, you can get an "exact solution"
for location and time with just four satellites. However, the data
from those four satellites have error - the papers describe a number
of sources including satellite orbit, radio wave, antenna phase, and
so on which introduces errors in the "exact solution". You use more
than four satellites to help get rid of those errors (even though you
don't have an "exact solution"). The end result is a better estimate
of location and time.
--Maniac
|
Request for Answer Clarification by
johann-ga
on
08 May 2003 18:40 PDT
Thanks for all the help Mr. maniac,
Im going to figure now how Im going to start writing my paper about
the second question, I already finish the first one, but for the
over-determined positioning finding, that one is going to be hard I
have to write between 200- 300 words.
By the way I posted another one,
Regards, johann
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