The atmospheric boundary layer (ABL) is the closest layer to the Earth's surface. Characterized by significant diurnal variations and turbulent motions, it is of great importance to the vertical transport of heat, momentum, water vapor and chemical composition, the evolution of large-scale weather processes and the study of climatology (Garratt, 1994). The ABL height is one of the important parameters of the ABL, which is usually used in the parameterization of the physical processes related to the ABL. An accurate ABL height has great influence on improving the precision of numerical weather forecasting models and general circulation models (Medeiros et al., 2005). Therefore, estimating the ABL height has been a challenging topic and one of the frontier research fields in atmospheric science (Hong et al., 1998; Xu et al., 2002; Li et al., 2006; Mao et al., 2006; Seidel et al., 2010; Dai et al., 2014).
As a new method of atmospheric remote sensing, the GPS radio occultation (RO) technique has the advantages of high precision, high vertical resolution, global quasi-uniform coverage, all-weather conditions, and long-term stability, among others, and has received extensive attention. In particular, the occultation project named the Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC), jointly implemented by Taiwan and the United States, was able to obtain nearly 2000 profiles a day during the five years of its mission (Liou et al., 2007). These occultation profiles have important applications in the study of global climate change and numerical weather forecasting.
In view of the high vertical resolution characteristics of occultation data, many scholars use them to determine ABL heights. (von Engeln et al., 2005) proposed the determination of ABL height by using the truncation height in full spectrum inversion. (Sokolovskiy et al., 2006) estimated ABL depths (i.e., ABL top heights) by determining the maximum lapse (breakpoint) of the refractivity vertical gradient. Subsequently, in a follow-up study (Sokolovskiy et al., 2007), they proposed that the ABL depth can also be determined by the bend angle profile. (Basha and Ratnam, 2009) used the high vertical resolution radiosonde data from the tropical station Gandaki to determine the ABL height by calculating the minimum value of the refractivity vertical gradient, and compared their results with those determined using COSMIC RO data, showing good consistency. (Guo et al., 2011) applied the breakpoint method proposed by (Sokolovskiy et al., 2006) to the refractivity data from COSMIC RO, and obtained the global distribution and seasonal variation of the marine ABL height. (Ao et al., 2012) defined the minimum vertical gradient of the profiles of refractivity and vapor pressure as the ABL top, and introduced the relative minimum gradient (or sharpness parameter) to characterize the quality of the ABL height. (Chan and Wood, 2013) improved the breakpoint method proposed by (Sokolovskiy et al., 2006) through fulfilling more constraints to ensure the quality of the results, and presented the seasonal cycle characteristics of the global ABL height. (Liao et al., 2015) and (Liu et al., 2016) also used a similar method to obtain and evaluate the seasonal and diurnal variations of the global marine ABL height.
The work mentioned above mainly used refractivity profile data, with bending angle (BA) profile data rarely employed. However, the BA, as an intermediate product in the occultation data processing chain, has some unique advantages. For example, the BA is a rawer product than refractivity, and thus contains less noise in data processing. Furthermore, because the BA is calculated directly from the observational data (and not by the Abel transform), there is no limitation on atmospheric ducting. In addition, since refractivity is obtained through integrating the BA according to the Abel inversion formula, refractivity profiles are smoother than BA profiles, which means that BA profiles contain more information than refractivity profiles (Fig. 1). This was also pointed out by (Rieder and Kirchengast, 2001), who showed in their Fig. 1 that the BA profile exhibited the highest signal-to-noise ratio. Therefore, BA profile data are more suitable for detecting ABL heights.
On the other hand, most of the methods mentioned above for determining ABL height are reduced to calculating the vertical derivatives of refractivity or BA profiles and carried out usually by the finite difference method. This can be addressed in the framework of inverse problems. From the viewpoint of the theory of inverse problems, calculating derivatives of functions using limited and discrete observational data (often known as numerical differentiation), belongs to inverse problems. Plus, numerical differentiation is generally ill-posed as an inverse problem, which is characterized mainly by numerical instability (Tikhonov and Arsenin, 1977). Hence, if conventional methods, such as the finite difference method, are applied to solving numerical differentiation problems, noise in the observational data will be amplified, resulting in derivatives far from true values that are sometimes even completely useless (Hanke and Scherzer, 2001; Ramm and Smirnova, 2001). To overcome the ill-posedness (mainly numerical instability) in numerical differentiation, some regularization strategies, such as Tikhonov regularization, have been introduced, and some stable numerical differentiation methods, such as the cubic spline interpolation method, mollification method, and variational regularization method, have been developed to obtain stable approximate derivatives (Ramm and Smirnova, 2001; Cheng et al., 2003).
In the above context, the present paper aims at determining ABL heights from BA profiles. Unlike in previous work, the present paper adopts another approach, i.e., combining the finite difference algorithm with the Tikhonov regularization technique, to obtain the vertical derivatives of BA profiles and then determine the ABL heights.
This paper is structured as follows: Section 2 introduces the numerical differential method and data used in this paper. Section 3 is the validation of the method by comparing with the ABL heights from COSMIC products, and results obtained by the finite difference method upon refractivity. Finally, a summary and discussion are presented in section 4.