A variational regularization of Abel transform for GPS radio occultation
Abstract. In the Global Positioning System (GPS) radio occultation
(RO) technique, the inverse Abel transform of measured bending angle (Abel
inversion, hereafter AI) is the standard means of deriving the refractivity. While concise
and straightforward to apply, the AI accumulates and
propagates the measurement error downward. The measurement error propagation
is detrimental to the refractivity in lower altitudes. In particular, it
builds up negative refractivity bias in the tropical lower troposphere. An
alternative to AI is the numerical inversion of the forward Abel transform,
which does not incur the integration of error-possessing measurement and
thus precludes the error propagation. The variational regularization (VR)
proposed in this study approximates the inversion of the forward Abel
transform by an optimization problem in which the regularized solution
describes the measurement as closely as possible within the measurement's
considered accuracy. The optimization problem is then solved iteratively by
means of the adjoint technique. VR is formulated with error covariance
matrices, which permit a rigorous incorporation of prior information on
measurement error characteristics and the solution's desired behavior into
the regularization. VR holds the control variable in the measurement space
to take advantage of the posterior height determination and to negate the
measurement error due to the mismodeling of the refractional radius. The
advantages of having the solution and the measurement in the same space are
elaborated using a purposely corrupted synthetic sounding with a known true
solution. The competency of VR relative to AI is validated with a large
number of actual RO soundings. The comparison to nearby radiosonde
observations shows that VR attains considerably smaller random and
systematic errors compared to AI. A noteworthy finding is that in the
heights and areas that the measurement bias is supposedly small, VR follows
AI very closely in the mean refractivity deserting the first guess. In the
lowest few kilometers that AI produces large negative refractivity bias, VR
reduces the refractivity bias substantially with the aid of the background,
which in this study is the operational forecasts of the European Centre for
Medium-Range Weather Forecasts (ECMWF). It is concluded based on the results
presented in this study that VR offers a definite advantage over AI in the
quality of refractivity.