# Cluster Averages

#### John Thompson

##### Active Member
I'm sure I've read this somewhere, but I can't find it now. How are points weighted when clusters are averaged?

I have two shots, taken one right after the other. Both were 120 seconds. One had 592 epochs and the other had 594. I rotated the rover pole 180° between them. One has a horizontal residual of 0.034' and the other is 0.018' from the averaged coordinates. I would expect more balanced residuals for identical shots.

#### Matt Johnson

##### Well-Known Member
5PLS
Here is Vladimir Prasolov's explanation of it.

Currently the averaging algorithm is doing the following things:

1. For grid/local systems we are averaging in that space. For other systems we use temporary Oblique Stereographic projection as usual.
2. The RTK ECEF covariance matrix is transformed to topocentric Cartesian system.
3. We scale that topocentric covariance matrix to have a-posteriori topocentric covariance matrix, that corresponds HRMS and VRMS.
4. For averaging we use the inverse of that matrix as weights. The topocentric averaged covariance matrix is estimated as inversed sum of weights (individual inversed covariance matrix).
5. We calculate averaged HRMS and VRMS as square root of sum of corresponding diagonal elements of averaged topocentric covariance matrix.
6. We transform averaged topocentric covariance matrix to geocentric ECEF for storing it in averaged position.
7. The epoch counters are accumulated.
8. The GPS/GLONASS satellites' numbers are averaged with weights of epoch counters.
9. The DOPs are averaged as inversed squared values with weights of epoch counters. It is due to fact DOPs are square roots of corresponding elements of Cofactor matrix.

#### Jim Frame

##### Active Member
Is there a similar weighted process for averaging the epoch-by-epoch positions of an RTK point, or is that a simple mean?

• Nate The Surveyor

#### John Thompson

##### Active Member
Wow, that's a lot more math than I was doing. Help me translate that to English for a guy who 20 years ago failed linear algebra on the first attempt, and barely scraped by on the second...

Roughly speaking, it sounds like coordinates are weighted by HRMS and VRMS with a couple coordinate transformations/re-projections in the process.

Here are some more particulars on the shots.

Name 8104 8104A
Horiz. Residual 0.034ft 0.018ft
Time 10:39:40 10:43:17
Epochs 592 594
GPS 6 6
GLONASS 7 7
HRMS 0.008ft 0.007ft
VRMS 0.014ft 0.015ft
HDOP 0.760 0.758
VDOP 1.404 1.351
TDOP 0.943 0.907
Confidence 14.00 13.00
Consistency 874.50 880.00
Duration 121.8s 121.8s

It looks to me like the cluster average has weighted one shot approximately twice as much as the other, but I don't see much difference between shots. The spread between shots is small, so maybe it's just from accumulated rounding errors?

• Nate The Surveyor

#### Shawn Billings

##### Shawn Billings
5PLS
Is there a similar weighted process for averaging the epoch-by-epoch positions of an RTK point, or is that a simple mean?
Simple average

#### Shawn Billings

##### Shawn Billings
5PLS
Wow, that's a lot more math than I was doing. Help me translate that to English for a guy who 20 years ago failed linear algebra on the first attempt, and barely scraped by on the second...

Roughly speaking, it sounds like coordinates are weighted by HRMS and VRMS with a couple coordinate transformations/re-projections in the process.

Here are some more particulars on the shots.

Name 8104 8104A
Horiz. Residual 0.034ft 0.018ft
Time 10:39:40 10:43:17
Epochs 592 594
GPS 6 6
GLONASS 7 7
HRMS 0.008ft 0.007ft
VRMS 0.014ft 0.015ft
HDOP 0.760 0.758
VDOP 1.404 1.351
TDOP 0.943 0.907
Confidence 14.00 13.00
Consistency 874.50 880.00
Duration 121.8s 121.8s

It looks to me like the cluster average has weighted one shot approximately twice as much as the other, but I don't see much difference between shots. The spread between shots is small, so maybe it's just from accumulated rounding errors?
Good summation

• Nate The Surveyor

#### David M. Simolo

##### Active Member
Here is Vladimir Prasolov's explanation of it.

Currently the averaging algorithm is doing the following things:

1. For grid/local systems we are averaging in that space. For other systems we use temporary Oblique Stereographic projection as usual.
2. The RTK ECEF covariance matrix is transformed to topocentric Cartesian system.
3. We scale that topocentric covariance matrix to have a-posteriori topocentric covariance matrix, that corresponds HRMS and VRMS.
4. For averaging we use the inverse of that matrix as weights. The topocentric averaged covariance matrix is estimated as inversed sum of weights (individual inversed covariance matrix).
5. We calculate averaged HRMS and VRMS as square root of sum of corresponding diagonal elements of averaged topocentric covariance matrix.
6. We transform averaged topocentric covariance matrix to geocentric ECEF for storing it in averaged position.
7. The epoch counters are accumulated.
8. The GPS/GLONASS satellites' numbers are averaged with weights of epoch counters.
9. The DOPs are averaged as inversed squared values with weights of epoch counters. It is due to fact DOPs are square roots of corresponding elements of Cofactor matrix.
That's exactly the way I would have done it.... (insert sarcasm font). It would take me a long time to truly understand all of that. Pretty impressive.

• Nate The Surveyor

#### Nate The Surveyor

##### Well-Known Member
He he he David, I'd not have done it the way the Javad bunch did either!
But, when it works so well....

I had a very productive day yesterday.
Someday, I'd like to shake the hand of some of these programmers.

N