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So for the interval between stations k-1 and k we can write:
𝒅∆𝒓 𝒌 𝒅𝑫 𝒌
𝒅∆𝒓 𝒌 𝒅𝑰 𝒌
𝒅∆𝒓 𝒌 𝒅𝑨 𝒌
𝒅∆𝒓 𝒌 𝒅𝒑 𝒌
=[
]
Substituting j = k and differentiating equation (A.3) we get:
[ 𝒔𝒊𝒏𝑰 𝒌−𝟏 𝒄𝒐𝒔𝑨 𝒌−𝟏 +𝒔𝒊𝒏𝑰 𝒌 𝒄𝒐𝒔𝑨 𝒌 𝒔𝒊𝒏𝑰 𝒌−𝟏 𝒔𝒊𝒏𝑨 𝒌−𝟏 +𝒔𝒊𝒏𝑰 𝒌 𝒔𝒊𝒏𝑨 𝒌 𝒄𝒐𝒔𝑰 𝒌−𝟏 +𝒄𝒐𝒔𝑰 𝒌 ]
𝒅∆𝒓 𝒌 𝒅𝑫 𝒌
𝟏 𝟐
=
[ (𝑫 𝒌 −𝑫 𝒌−𝟏 )𝒄𝒐𝒔𝑰 𝒌 𝒄𝒐𝒔𝑨 𝒌 (𝑫 𝒌 −𝑫 𝒌−𝟏 )𝒄𝒐𝒔𝑰 𝒌 𝒔𝒊𝒏𝑨 𝒌 −(𝑫 𝒌 −𝑫 𝒌−𝟏 )𝒔𝒊𝒏𝑰 𝒌 ]
𝒅∆𝒓 𝒌 𝒅𝑰 𝒌
𝟏 𝟐
=
[ −(𝑫 𝒌 −𝑫 𝒌−𝟏 )𝒔𝒊𝒏𝑰 𝒌 𝒔𝒊𝒏𝑨 𝒌 (𝑫 𝒌 −𝑫 𝒌−𝟏 )𝒔𝒊𝒏𝑰 𝒌 𝒄𝒐𝒔𝑨 𝒌 𝟎 ]
𝒅∆𝒓 𝒌 𝒅𝑨 𝒌
𝟏 𝟐
=
Putting it together:
(𝑫 𝒌 −𝑫 𝒌−𝟏 )𝒄𝒐𝒔𝑰 𝒌 𝒄𝒐𝒔𝑨 𝒌 (𝑫 𝒌 −𝑫 𝒌−𝟏 )𝒄𝒐𝒔𝑰 𝒌 𝒔𝒊𝒏𝑨 𝒌 −(𝑫 𝒌 −𝑫 𝒌−𝟏 )𝒔𝒊𝒏𝑰 𝒌
[ 𝒔𝒊𝒏𝑰 𝒌−𝟏 𝒄𝒐𝒔𝑨 𝒌−𝟏 +𝒔𝒊𝒏𝑰 𝒌 𝒄𝒐𝒔𝑨 𝒌 𝒔𝒊𝒏𝑰 𝒌−𝟏 𝒔𝒊𝒏𝑨 𝒌−𝟏 +𝒔𝒊𝒏𝑰 𝒌 𝒔𝒊𝒏𝑨 𝒌 𝒄𝒐𝒔𝑰 𝒌−𝟏 +𝒄𝒐𝒔𝑰 𝒌
−(𝑫 𝒌 −𝑫 𝒌−𝟏 )𝒔𝒊𝒏𝑰 𝒌 𝒔𝒊𝒏𝑨 𝒌 (𝑫 𝒌 −𝑫 𝒌−𝟏 )𝒔𝒊𝒏𝑰 𝒌 𝒄𝒐𝒔𝑨 𝒌 𝟎 ]
𝒅∆𝒓 𝒌 𝒅𝒑 𝒌
𝟏 𝟐
=
Similarly, for the interval between stations k and k+1 we can write:
𝒅∆𝒓 𝒌+𝟏 𝒅𝑫 𝒌
𝒅∆𝒓 𝒌+𝟏 𝒅𝑰 𝒌
𝒅∆𝒓 𝒌+𝟏 𝒅𝑨 𝒌
𝒅∆𝒓 𝒌+𝟏 𝒅𝒑 𝒌
=[
]
Substituting j = k+1 and again differentiating equation (A.3) we get:
[ −𝒔𝒊𝒏𝑰 𝒌 𝒄𝒐𝒔𝑨 𝒌 −𝒔𝒊𝒏𝑰 𝒌+𝟏 𝒄𝒐𝒔𝑨 𝒌+𝟏 −𝒔𝒊𝒏𝑰 𝒌 𝒔𝒊𝒏𝑨 𝒌 −𝒔𝒊𝒏𝑰 𝒌+𝟏 𝒔𝒊𝒏𝑨 𝒌+𝟏 −𝒄𝒐𝒔𝑰 𝒌 −𝒄𝒐𝒔𝑰 𝒌+𝟏 ]
𝒅∆𝒓 𝒌+𝟏 𝒅𝑫 𝒌
𝟏 𝟐
=
[ (𝑫 𝒌+𝟏 −𝑫 𝒌 )𝒄𝒐𝒔𝑰 𝒌+𝟏 𝒄𝒐𝒔𝑨 𝒌+𝟏 (𝑫 𝒌+𝟏 −𝑫 𝒌 )𝒄𝒐𝒔𝑰 𝒌+𝟏 𝒔𝒊𝒏𝑨 𝒌+𝟏 −(𝑫 𝒌+𝟏 −𝑫 𝒌 )𝒔𝒊𝒏𝑰 𝒌+𝟏 ]
𝒅∆𝒓 𝒌+𝟏 𝒅𝑰 𝒌
𝟏 𝟐
=
[ −(𝑫 𝒌+𝟏 −𝑫 𝒌 )𝒔𝒊𝒏𝑰 𝒌+𝟏 𝒔𝒊𝒏𝑨 𝒌+𝟏 (𝑫 𝒌+𝟏 −𝑫 𝒌 )𝒔𝒊𝒏𝑰 𝒌+𝟏 𝒄𝒐𝒔𝑨 𝒌+𝟏 𝟎 ]
𝒅∆𝒓 𝒌+𝟏 𝒅𝑨 𝒌
𝟏 𝟐
=
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