latitude derivations

  1. latitude from polygon

    symbol

    description

    unit

    variable name

    \(\lambda\)

    longitude

    \(degE\)

    longitude {:}

    \(\lambda^{B}(i)\)

    longitude

    \(degE\)

    longitude_bounds {:,N}

    \(\phi\)

    latitude

    \(degN\)

    latitude {:}

    \(\phi^{B}(i)\)

    latitude

    \(degN\)

    latitude_bounds {:,N}

    Convert all polygon corner coordinates defined by \(\phi^{B}(i)\) and \(\lambda^{B}(i)\) into unit sphere points \(\mathbf{p}(i) = [x_{i}, y_{i}, z_{i}]\)

    \(x_{min} = min(x_{i}), y_{min} = min(y_{i}), z_{min} = min(z_{i})\)

    \(x_{max} = max(x_{i}), y_{max} = max(y_{i}), z_{max} = max(z_{i})\)

    \(\mathbf{p}_{center} = [\frac{x_{min} + x_{max}}{2}, \frac{y_{min} + y_{max}}{2}, \frac{z_{min} + z_{max}}{2}]\)

    The vector \(\mathbf{p}_{center}\) is converted back to \(\phi\) and \(\lambda\)

  2. latitude from range

    symbol

    description

    unit

    variable name

    \(\phi\)

    latitude

    \(degN\)

    latitude {:}

    \(\phi^{B}(l)\)

    latitude boundaries (\(l \in \{1,2\}\))

    \(degN\)

    latitude_bounds {:,2}

    The pattern : for the dimensions can represent {latitude}, or {time,latitude}.

    \[\phi = \frac{\phi^{B}(2) + \phi^{B}(1)}{2}\]
  3. latitude from vertical profile latitudes

    symbol

    description

    unit

    variable name

    \(\phi\)

    single latitude for the full profile

    \(degN\)

    latitude {:}

    \(\phi(i)\)

    latitude for each profile point

    \(degN\)

    latitude {:,vertical}

    \(N\)

    number of profile points

    The pattern : for the dimensions can represent {time}, or no dimensions at all.

    \[\begin{split}\begin{eqnarray} N & = & max(i, \phi(i) \neq NaN) \\ \phi & = & \phi(N/2) \end{eqnarray}\end{split}\]
  4. latitude from sensor latitude

    symbol

    description

    unit

    variable name

    \(\phi\)

    latitude

    \(degN\)

    latitude {:}

    \(\phi_{instr}\)

    latitude of the sensor

    \(degN\)

    sensor_latitude {:}

    The pattern : for the dimensions can represent {time}, or no dimensions at all.

    \[\phi = \phi_{instr}\]