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Understanding superposition for a layer model

  1. Apr 6, 2015 #1
    Greetings all,

    This question can cover many sub-forums here, but I'll put it in General Math since I believe it deals with superposition.

    Now, I am studying a layered-earth model for programming purposes, and what I mean by that is, for example, we say air is one layer, grass is another, and the dirt underneath is the final layer. So in essence this would be a three layer system. What I'm trying to understand is an equation that defines N-layers to something called apparent conductivity. But I'm more concerned about the right hand side of the equation.

    Now, while I present pseudo-code here to demonstrate, if anyone needs me to go in to more detail or write in Latex, let me know.

    So, for the three layered system, the equation for apparent conductivity looks like:

    sigma_a = sigma1*(1-R(z1))+sigma2*(R(z1)-R(z2))+sigma3*R(z2)

    where sigma 1 is the conductivity of the first layer, sigma 2 is the conductivity of the second layer, etc., and R(z1) and R(z2) are the relative contributions to the apparent conductivity from all material below a certain layer.

    Now, say we have a 4 layer system. Now the equation expands to:

    sigma_a = sigma1*(1-R(z1))+sigma2*(R(z1)-R(z2))+sigma3*R(z2)-R(z3))+sigma4*R(z3)

    Now the first part (sigma1*(1-R(z1))) will always stay the same. The last part (sigma4*R(z3)) stays the same and can be written as sigma_N*R(z(N-1)), where N is the total amount of layers. But the middle information is always going to increase depending on the total number of layers.

    This is where I think it has to do with superposition in some way but I am having a difficult time seeing it. Any assistance would be appreciated.

  2. jcsd
  3. Apr 8, 2015 #2

    Simon Bridge

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    Homework Helper

    Define "superposition".
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