Geophysical Monitoring for Geologic Carbon Storage. Группа авторов

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Geophysical Monitoring for Geologic Carbon Storage - Группа авторов


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href="#fb3_img_img_a8ca9777-0a18-562e-a451-1d62ca97c893.png" alt="upper T left-parenthesis omega right-parenthesis equals exp left-brace minus left-bracket delta left-parenthesis omega right-parenthesis plus i phi left-parenthesis omega right-parenthesis right-bracket right-brace"/>

      where δ accounts for the absorption, ϕ accounts for the dispersion, and ω is the instantaneous frequency, which depends on ω o , ω m , M, and t. As defined, δ embodies the product of path length, extinction coefficient, and concentration. The wave exiting the sample is then

      (3.4)upper E left-parenthesis t right-parenthesis equals upper T left-parenthesis omega right-parenthesis dot upper E 0 exp left-brace i left-parenthesis omega 0 t plus upper M sine left-parenthesis omega Subscript m Baseline t right-parenthesis right-brace

      where the transmission function serves as a susceptibility. The intensity of the wave leaving the sample, I(t), is given by the electric field E(t) times its complex conjugate. This leads to (Supplee et al., 1994)

      (3.5)StartLayout 1st Row StartLayout 1st Row upper I left-parenthesis t right-parenthesis equals upper I 0 e Superscript minus 2 delta 0 Baseline left-brace 1 plus left-parenthesis 2 sigma-summation Underscript n equals 0 Overscript infinity Endscripts upper J Subscript n Baseline left-parenthesis upper M right-parenthesis dot upper J Subscript n plus 1 Baseline left-parenthesis upper M right-parenthesis 2nd Row Subscript Sub Subscript Sub Sub Subscript Sub Sub Superscript Sub Subscript Sub Superscript Subscript Superscript Super Subscript Super Sub Subscript Super Sub Superscript Super Subscript Super Superscript Superscript Baseline dot left-bracket delta Subscript negative n minus 1 Baseline minus delta Subscript n plus 1 Baseline plus delta Subscript negative n Baseline minus delta Subscript n Baseline right-bracket dot cosine left-parenthesis omega Subscript m Baseline t right-parenthesis right-parenthesis EndLayout 2nd Row StartLayout 1st Row plus left-parenthesis 2 sigma-summation Underscript n equals 0 Overscript infinity Endscripts upper J Subscript n Baseline left-parenthesis upper M right-parenthesis dot upper J Subscript n plus 1 Baseline left-parenthesis upper M right-parenthesis 2nd Row Subscript Sub Subscript Sub Sub Subscript Sub Sub Superscript Sub Subscript Sub Superscript Subscript Superscript Super Subscript Super Sub Subscript Super Sub Superscript Super Subscript Super Superscript Superscript Baseline dot left-bracket phi Subscript negative n minus 1 Baseline minus phi Subscript negative n Baseline plus phi Subscript n plus 1 Baseline minus phi Subscript n Baseline right-bracket dot sine left-parenthesis omega Subscript m Baseline t right-parenthesis right-parenthesis right-brace EndLayout EndLayout

      FMS is often implemented using a small modulation index (i.e., small M) and where only the n = 0 sidebands are important. Under such conditions, the equation for the FMS signal intensity can be simplified considerably to give the often‐used working equation

      (3.6)StartLayout 1st Row upper I left-parenthesis t right-parenthesis equivalent-to upper I 0 e Superscript minus 2 delta 0 Baseline dot left-bracket 1 plus upper M dot left-parenthesis delta Subscript negative 1 Baseline minus delta 1 right-parenthesis dot cosine left-parenthesis theta right-parenthesis 2nd Row plus upper M dot left-parenthesis phi Subscript negative 1 Baseline plus phi 1 minus 2 phi 0 right-parenthesis dot sine left-parenthesis theta right-parenthesis right-bracket EndLayout

      Note, for completeness, that when phase matching is employed (such that θ .=0), and the M goes to zero limit is taken, one arrives at

      (3.7)upper A b s o r b a n c e identical-to minus ln left-parenthesis StartFraction upper I Over upper I Subscript o Baseline EndFraction right-parenthesis equals 2 delta 0

Schematic illustration of an example of (a) Voigt absorption profile and (b) the corresponding FMS signal plotted as functions of the carrier frequency.

      which is essentially Beer's law for absorption at the carrier frequency ω o (the factor of two here results from taking the product of the transmission function and its complex conjugate).

Schematic illustration of (a) a relatively sharp Voigt absorption profile and (b) the corresponding FMS signal plotted as functions of the carrier frequency. Schematic illustration of least-squares fitting of FMS spectra.

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