I am reading 'Surveying and Geomatics Engineering' book the was just put out by the ASCE. I'm trying to wrap my head around Least Squares Adjustment with GNSS data. I can follow a least squares adjustment with a traverse, but I want to see if I am following this text correctly for GNSS.?ÿ
Chapter 5 page 141
There are two components for a least squares adjustment, functional model and stochastic model.
The functional model pretty much averages out the corrected observation generated from the RINEX file and the precise ephemeris. The software will use the data from the precise ephemeris files to improve the coordinates collected during the static observation. For this example I'll use 4 hour static observation at 15 second intervals which will give me (4 hours * 60 minutes * 4-15 sec. interval = 960 observations)
The stochastic model takes the averaged coordinates for the 960 observations from the functional model and weights the positions according to the spread of the average or the standard deviation. The marks with the higher weights will hold their position more than the marks with the lower weight. The adjustment will process the best fit until the base line distance between each mark in the network is under a specified tolerance which could be 2 cm horizontally and 5 cm vertically. (Basically move the coordinates of the lower weighted marks more than the higher weighted marks would move until the vectors or baselines make sense to one another.)
This all happens on the reference ellipsoid (NAD83 2011) then is brought up to a map projection when the vectors are adjusted.?ÿ
Let me know if I am understanding the basic concept. No need to get too detailed yet!
Thanks,
Dan
I can't tell, is that the text from the book, or your interpretation of the book? It's....sort of there.
?ÿ
The stochastic model is the weighting model, independent of the observations themselves. During adjustment, it controls how much a given observation moves.
The functional model is a set of equations enforcing a condition for the adjustment. It can be conditional or parametric; the latter is generally what is used. Conditional would be like forcing a closed loop traverse to have latitudes and departures of zero, or a triangle traverse to have 180 degrees; parametric would be fixing one or more stations (parameters) and then running the adjustment.
The functional model and the stochastic model are applied together in the mathematical model, not separately. If the stochastic model was well developed, and the functional model is a good representation of the physical site/world, AND there are no blunders or systematic errors in the data, then the residuals) obtained from the adjustment should show up as a normal distribution.
That's the real power of least squares - it lets you model the errors inherent in the instrumentation/measurements (use observations with very disparate magnitudes of errors), model the real world itself, then use as much data as you want to describe the real world as accurately as possible, plus detect blunders and outliers along the way.
?ÿ
The adjustment will process the best fit until the base line distance between each mark in the network is under a specified tolerance which could be 2 cm horizontally and 5 cm vertically. (Basically move the coordinates of the lower weighted marks more than the higher weighted marks would move until the vectors or baselines make sense to one another.)
Almost there... A least squares adjustment aims to minimize the sum of the squares of the residuals. It doesn't force every observation or baseline to be "under a specified tolerance".
The focus is on iterating through a series of adjustments until the adjusted values (typically coordinates) do not move appreciably.
The operator usually sets a threshold beyond which it's practically pointless to continue adjusting, such as 1mm or 0.01ft. There's also usually a maximum number of iterations set just in case the solution never converges to that point.
Either way, the observations themselves are not all forced to fit under a certain level - they are adjusted based upon the observations themselves (functional) and their respective weights (stochastic).
?ÿ
Hope that helps, I haven't read the book. I need to pick up a copy and read it for myself.
