def particle_depolarization(delta_m, delta_v, molecular_backscatter,
    particle_backscatter):
    """
    Calculate the linear particle depolarization ratio.
    
    Parameters
    ----------
    delta_m: vector
       The linear molecular depolarization ratio.
    delta_v: vector
       The linear volume depolarization ratio.
    molecular_backscatter: vector
       The molecular component of the total backscatter coefficient.
    particle_backscatter: vector
       The particle component of the total backscatter coefficient.
    
    Returns
    -------
    delta_p: vector
       The linear particle depolarization ratio.

    Notes
    -----
    The linear particle depolarization ratio is calculated by the formula:
    
    .. math::
       \\delta^p = \\frac{(1 + \\delta^m)\\delta^V \\mathbf{R} - (1 + \\delta^V)\\delta^m}
       {(1 + \\delta^m)\\mathbf{R} - (1 + \\delta^V)}
    
    References
    ----------
    Freudenthaler, V. et al. Depolarization ratio profiling at several wavelengths in pure
    Saharan dust during SAMUM 2006. Tellus, 61B, 165-179 (2008)
    """
    r = (molecular_backscatter + particle_backscatter) / molecular_backscatter
    delta_p = ((1 + delta_m) * delta_v * r - (1 + delta_v) * delta_m) / ((1 +
        delta_m) * r - (1 + delta_v))
    return delta_p
