觀測量探討：三角高程網平差

On the Observations: A Review of Trigonometric Height Network Adjustment

三角高程（Trigonometric Height）係利用直角三角形之幾何關係，透過縱角、平距或斜距之量測，以推算兩點間高程差值之方法。當測站數量增加並形成網形（network）時，將產生多餘觀測，進而引發平差之需求。
於「傳統」處理方式中，係先依距離與縱角計算所得之高程差，類比於直接水準測量網形之平差，並應用最小自乘法原理以推估最或似值。此種作業並非直接以實際觀測量作為平差之輸入，而是以計算所得之「高程差」代之。二者主要差異，實際上即在於觀測量之界定。
傳統方法之優勢在於計算過程簡便，並可避免嚴謹平差程序中涉及之三角函數微分運算，故具普遍適用性。然而，嚴謹之平差程序除可達至最佳化結果外，對於觀測量誤差之評估亦更為直觀，具有方法論上的優勢。
本文主要依據達乾龍等（2015）之論述，以闡明「傳統」與「嚴謹」三角高程網平差方法之差異。

Trigonometric heighting is a surveying method that determines elevation differences between points through the geometric relations of right triangles, using observations of vertical angles, horizontal or slope distances. When multiple stations and observations are involved, a network may be formed, which introduces redundant observations and necessitates an adjustment procedure.
In the“traditional”approach, height differences are first computed from distances and vertical angles, and subsequently adjusted in analogy to a spirit leveling network by applying the least squares principle. In this case, the adjusted quantities are derived from calculated height differences rather than from the raw observations themselves. The fundamental distinction between the traditional and rigorous approaches thus lies in the definition of the observational quantities used in the adjustment.
The traditional approach has the advantage of computational simplicity and avoids the differential operations of trigonometric functions required in rigorous adjustment, thereby offering broader applicability. By contrast, rigorous adjustment not only achieves optimal estimation but also provides a more direct evaluation of observational errors.
This paper primarily follows the discussion of Da et al. (2015) to illustrate the methodological differences between traditional and rigorous adjustments of trigonometric heighting networks.