Inverse Absolute Height Weighting in the Highest Order Levelling Networks

Abstract

<p>The North American Vertical Datum of 1988 (NAVD88) was established by the minimum-constrain adjustment of geodetic levelling observations in Canada, USA, and Mexico. It held fixed the height of the primary tidal benchmark at Rimouski, Quebec, Canada. The NAVD88 datum was never officially adapted in Canada due its large east-west tilt of 1.5 m from the Atlantic to Pacific coast (Hayden <em>et al.</em><em>,</em> 2012). Also, a large systematic difference (ranging from -20 cm to +130 cm) was found between NAVD88 and the pure geoid gravimetric models. Using Factor Analysis it was discovered that one of the factors, which can explain the tilt of the NAVD88, is the terrain, i.e. small in the flat states but large in the mountainous areas such as in the Rockies and the Appalachians (Li, 2012). A possible reason for the tilt of the NAVD88 might be the weights used into adjustment of the network. In this study the data of two precise national levelling networks are used, e.g. the Second Levelling of Finland and the Third Levelling of Bulgaria, in order to support the above hypothesis. An iterative procedure based on the Inverse Absolute Height Weighting (IAHW) is applied. The core of this procedure is to find this value of the power parameter (p) of the weights w=H<sup>p</sup>, where H is the absolute elevation difference of the terminals in the levelling lines, that minimize the mean of the mean squared errors (MSE) of the nodal bench marks (NBM) in both networks. It has been found that p=1 and p=4.3 for the Bulgarian and the Finnish networks, respectively. Also, a similar iterative procedure based on the Inverse Distance Weighting (IDW) is performed and the best decisions for the Finnish and the Bulgarian networks are obtained. It has been found that the weights w=L<sup>-5.9</sup> and w=L<sup>-1.6</sup>, where L is the length of the levelling line, lead to the minimal MSE of the NBM for the Finnish and the Bulgarian networks, respectively. The results of both the IDW and the IAHW procedures are compared. It has been revealed that the IAHW based adjustments lead to significantly less MSE of the NBM than all variants of the IDW. It has also been shown that concerning the Bulgarian and the Finnish analyzed here data, the IAHD approach leads to physically lower adjusted heights than the IDW. In some cases these differences are more than 1.5-2 times greater than the MSE of the corresponding bench marks.</p>

Geometryczne dane niwelacyjne i niektóre błędy systematyczne w ich przetwarzaniu

“…Concerning the use of Inverse Distance Weighting in order to minimize additionally the standard errors of the nodal benchmarks in the analyzed network, there were not expected any surprises. Analogical results were obtained with the data of the Second Levelling of Finland and the Third Levelling of Bulgaria [13].…”

“…It is a common practice the weights wn to be given as a function of the length of a levelling line Ln [1, 9, 11] similar to equation ( 5) despite other weights are more statistically relevant [12,13].…”

“…Since the dawn of this method the mathematical treatment of levelling data has not been changed, supposing that the low of error and discrepancies accumulation, applied weights in the adjustment of levelling networks, etc., are absolutely clear. Some recent studies [12,13] show some cracks into the classical theory.…”

Algorytm regulacji geometrycznych sieci niwelacyjnych

“…In addition, we can minimize the impact of the shape of the network on the final standard errors of the adjusted heights or geopotential numbers of the nodal benchmarks in the network, if we apply some iterative procedures, e.g., Inverse Distance Weighting (IDW), Inverse Absolute Height Weighting (IAHW) [10], etc.…”