def brunt_index_correct(alpha):
    """
    Apply empirical corrections from Heyer & Brunt

    Using the empirical correction from Brunt & Heyer 2002a, where

    .. math::
        lpha = (0.33 \\pm 0.04)eta -(0.05 \\pm 0.08)

    :math:`eta` is the index of the integrated velocity spectrum. This is
    Equation 4.1. The relation to the velocity structure index is:

    .. math::
        eta = 2\\gamma + 1

    Then the conversion to :math:`\\gamma` is:

    .. math::
        \\gamma = (1.5 \\pm 0.18) lpha - (0.19 \\pm 0.20)

    These values are based off a broken linear fit in Section 3.3.1 from
    Chris Brunt's thesis. These are based off calibrating against uniform
    density field's with different indices.
    """
    return 1.52 * alpha - 0.19
