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the same throughout a survey leg, if we go to another leg, using a different tool or to another well then we would expect that error source to be independent over that range.
So our the error sources are independent from each other, but a given error source might be independent at all times, or correlated from station to station within a survey leg or from survey leg to survey leg with a well or from well to well within a field.
Therefore the model defines four propagation modes for the errors:
Random
R S
always independent
Systematic Well by Well
correlated from survey station to survey station
W
correlated from leg to leg correlated over all wells
Global
G
The propagation mode is a property of the error source and is defined in the tool model. In practise, most error sources are systematic or random and only a limited few well by well or global sources have been identified.
To combine all the error sources, we need to create a sum over all survey legs, survey stations and error sources which apply to a particular well. When doing this summation the propagation mode is used to define at what step in the summation arithmetic addition is used and at what stage RSS addition is required.
𝑺𝒖𝒓𝒗𝒆𝒚 𝑳𝒆𝒈𝒔
𝑺𝒖𝒓𝒗𝒆𝒚 𝑺𝒕𝒂𝒕𝒊𝒐𝒏
𝑬𝒓𝒓𝒐𝒓 𝑺𝒐𝒖𝒓𝒄𝒆
𝒕𝒐𝒕𝒂𝒍 𝒆𝒓𝒓𝒐𝒓 = ∑
∑
∑ 𝒆 𝒍,𝒊.𝒌
𝒍=
𝒊=
𝒌=
The mathematical details of this process can be found in Appendix A. The final output of the summation is a 3x3 covariance matrix, which describes the error ellipse at a particular station. In the nev -axes, the covariance matrix is:
𝝈 𝑵 𝟐
𝝈
𝑵 𝝈 𝑬 𝝈 𝑵 𝝈 𝑽
𝝈 𝑬 𝟐
[𝑪] 𝒏𝒆𝒗 =[
]
𝝈 𝑵 𝝈 𝑬
𝝈
𝑬 𝝈 𝑽
𝝈 𝑽 𝟐
𝝈 𝑵 𝝈 𝑽 𝝈 𝑬 𝝈 𝑽
Here 𝜎 𝑁 2 is the variance in the north-axis and the uncertainty in north axis (at 1-standard deviation) is ±√𝝈 𝑵 𝟐 . In the same way, the other terms on the lead diagonal are uncertainties along the other principle axes. The 𝜎 𝑁 𝜎 𝐸 , 𝜎 𝑁 𝜎 𝑉 and 𝜎 𝐸 𝜎 𝑉 terms are the covariances and give the skew or rotation of the ellipse with respect to the principle axes. Ap1.2.7 Transformation to Borehole Axes The covariance matrix above is expressed in the earth-centred nev -axes, this can be transformed to the borehole reference frame, hla by pre- and post-multiplying the covariance matrix with the nev -to- hla direction cosine matrix, [𝑇] ℎ𝑙𝑎 𝑛𝑒𝑣 .
𝒏𝒆𝒗 𝑻 [𝑪]
𝒏𝒆𝒗 [𝑻] 𝒉𝒍𝒂 𝒏𝒆𝒗
[𝑪] 𝒉𝒍𝒂 = [𝑻] 𝒉𝒍𝒂
The direction cosine matrix can be obtained a rotation in the horizontal place to the borehole azimuth, followed by a rotation in the vertical to the borehole inclination and is given by:
𝒄𝒐𝒔𝑰𝒄𝒐𝒔𝑨 −𝒔𝒊𝒏𝑨 𝒔𝒊𝒏𝑰𝒄𝒐𝒔𝑨 𝒄𝒐𝒔𝑰𝒔𝒊𝒏𝑨 𝒄𝒐𝒔𝑨 𝒔𝒊𝒏𝑰𝒔𝒊𝒏𝑨 −𝒔𝒊𝒏𝑰 𝟎 𝒄𝒐𝒔𝑰 ]
𝒏𝒆𝒗 =[
[𝑻] 𝒉𝒍𝒂
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