誤差、位移、速度：由全球至測站區域卡式坐標系之轉換

Error, Displacement, Velocity: Transforming from Global to Local Cartesian Coordinates on Station

「全球導航衛星系統」(Global Navigation Satellite System, GNSS)，在單點定位時，其解算直接所得坐標為在「地心地固」(Earth Centered Earth Fixed, ECEF)卡式坐標系中。當投影坐標為最後之應用需求時，ECEF 坐標須先行轉換至大地坐標，繼而經由地圖投影程序，轉換為指定之投影坐標系統中。由ECEF 坐標轉換至大地坐標，需經過迭代，地圖投影如橫麥卡托，計算中須使用級數，均屬於複雜的計算。經由協變方計算的誤差傳播，雖然並不需要考量數值的技巧，但仍是複雜而不易的。此一問題不限於誤差指標，對於以ECEF 計算之速度與位移，亦有著同樣的需求。為能提供一個有效的解決方案，由ECEF 的全球卡氏坐標系轉換至當地之測站區域卡氏坐標系，是目前通用的方式。本文目的在解說此一過程，並以實例說明其應用。

Global Navigation Satellite System (GNSS) are mostly elaborated in the Earth Centered Earth Fixed (ECEF) Cartesian coordinate system while the observations of a single station are utilized. These coordinates have to be transformed into geodetic coordinates, if they are demanded to be converted into plane coordinates with a specific map projection. The transform from ECEF coordinates to geodetic coordinates requires iterative procedures, and in the map projection, Taylor series expansion is frequently applied. These processes are quite technically demanding. Although the error propagation of covariance involves only straight forward numerical techniques, which is still complicated and difficult. This transform concerns not only the error propagation, but also the displacement, velocity, and other quantities derived in ECEF, where rotations converting the global Cartesian coordinate system into a station local Cartesian coordinate system are the primary mechanism. This note introduces this procedure which is illustrated with numerical examples.