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A2 - Details of the Error Summation Method Following on from the discussion in section Ap1.2.6, we have the example equations for adding error arithmetically or by root-sum-squaring: 𝒆 𝒕𝒐𝒕𝒂𝒍 = 𝒆 𝟏 +𝒆 𝟐
𝟐 +𝒆
𝟐 𝟐
𝒆 𝒕𝒐𝒕𝒂𝒍 =√𝒆 𝟏
If we square both sides then we get:
𝟐 =(𝒆
𝟐
𝟏 +𝒆 𝟐 )
𝒆 𝒕𝒐𝒕𝒂𝒍
𝟐 = 𝒆
𝟐 +𝒆
𝟐
𝒆 𝒕𝒐𝒕𝒂𝒍
𝟏
𝟐
o f w ’ only cons d d summ ng wo o sou c s, and remembering that for the error model at a particular point, the e variables will actually be 3x1 vectors and the squaring these would actually be e i . e i T then these equations become: o lu s.G n lsng h sums o nclud mo h n wo
𝑬 𝒕𝒐𝒕𝒂𝒍 =(∑𝒆 𝒊 )(∑𝒆 𝒊 ) 𝑻 𝑬 𝒕𝒐𝒕𝒂𝒍 =∑𝒆 𝒊 𝒆 𝒊 𝑻
Where now E total is a 3x3 covariance matrix, whose lead diagonal terms are the 𝑒 𝑡𝑜𝑡𝑎𝑙 2 values along the principle axes. Returning to the error model, when we come to create the final error summations, we need consider three summations – over the error sources, the survey legs and the survey stations and use the propagation modes to determine at which steps arithmetic summation is appropriate and at which RSS summation is required. The overall summation of random, systematic and global/well by well error sources is
𝒔𝒗𝒚
𝒔𝒚𝒔𝒕
𝒓𝒂𝒏𝒅
+ ∑ [𝑪] 𝒊,𝑲 𝒘𝒆𝒍𝒍 𝒊∈{𝑾,𝑮}
[𝑪] 𝑲
=∑[𝑪] 𝒊,𝑲
+∑[𝑪] 𝒊,𝑲
𝒊∈𝑹
𝒊∈𝑺
The individual terms for the various groups of error sources are given below. In these equations: e i,l,k
is the vector contribution of i th error source, in the i th survey leg at the k th survey station (3x1 vector) e * i,l,K is the vector contribution of i th error source, in the i th survey leg at the last survey point of interest i.e. the K th survey station (3x1 vector) i s h summ ono o souc sfom1… I k s h summ on of su y s ons f om 1… K : the current survey station l is the summation over survey legs from 1.. L : the current survey leg
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