As is well known and from theoretical proofs, proper weighting of the least squares solution of a leveling network is inversely proportion to the distances between the leveling points or the height difference between pairs of points. In other words, the uncertainties or the relative contribution of different observations in the least squares solution of the leveling networks are expressed in terms of distances. Sometimes the uncertainties of the existing control information are expressed in terms of their standard deviations, which were estimated from their previous dispersion or variance-covariance matrices. Although this type of expression or representation of uncertainties is very common, it makes the integration of the uncertainties of the existing control information to a new leveling network a non-trivial task. Now the question is:
What is the best weighting strategy for the observations and the control points? In some cases the control points will be introduced as fixed constraints, which can be viewed as some sort of a non optimal pseudo inverse solution.
I would use the inverse of the squared distances as weighting. As the reading error is usually constant for leveling equipment (unless you are using trigonometric leveling) and the height difference is derived from your measurements so mixing it in the weighting process would be unwise, in my opinion as it will result in dependencies that I am not sure about their affect on the least squares process
For setting up a stochastic model, the resolution for the height measurement of your used instrument is of importance. for example a Leica DNA03 has a resolution of 0.01 mm, and a standard deviation of 0.3 mm for a 1 km double nivellement. In Geodesy we often use this stochstic model:
sigma_height [mm] = 0.002 * d[m], d ist the distance from your instrument to the levelling rod and 0.002 a internal accuracy term
Sometimes this constant value is a bit too optimistic, because you could have some not taken into account influences of refraction and etc. But after your free height network adjustment and outlier removal you can see on the so called global test, if you were too optimistic or too pessimistic with your choice.
PS: Also important is to take into account a possible inclination of line of sight or too choose equal rod distances for the levelling setup.
Regards
Mathias
Thank you very much for the comment. This problem came to my attention very recently and it seams to me it requires some understanding.
Dear Mathias,
.Thank you very much for your comments. In fact, we are dealing with a very classical problem but it requires some sort of revisit.
In L.S. adjustment it's mainly talking about the random errors that happen during our observations, and the weights that we assuming for the observations must be in some way expressed the quality (degree of confidence) of those observations. Thus you can calculate the weights of observations using two approaches:
1- Using the inverse length of the survey line (because the length of survey line is an expression of the number of observations (levelling readings)).
2- Using the inverse of variance (S.D. square), since the variance is an expression of precision of the measured values.
I think both of the above approaches can precisely evaluate the observed values and consequently give the best weighting.
Regards
Dr Mohammed
We recommend to use a combined stochastic model, i.e. an individual uncertainty (for each observation/waypoint) and a length-depending uncertainty, see Eq (24). The length of the survey line ignoresthe terrain profile and a levelling loop of e.g. 100 m in a horizontal terrain profile has the same weight as a levelling loop of 100 m in the mountains but different numbers of waypoints. If the number of waypoints differs from each other, the uncertainties cannot be the same.
http://geodaesie.info/sites/default/files/privat/zfv_2017_1_Loesler_Eschelbach_Haas.pdf
Dear Michael
First of all, thank you very much for the feedback. Your suggestion is very interesting and we will test it.
Regards,
Gamal
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