## 1. Introduction

[2] The near horizon electromagnetic wave propagation is largely governed by the distribution of atmospheric refractive index which depends on the meteorological conditions. Knowledge of the refractivity information enables more precise assessment of the performance of both communications and radar systems. Conventional methods of the refractive index measurement consisting of detecting height dependence of temperature, pressure and humidity performed by radiosondes, microwave refractometers, or rocketsondes have some drawbacks, such as expensive and/or difficult deployment [*Halvey*, 1983]. Therefore, it is necessary to develop new methods for refractivity detection.

[3] *Richter* [1969] has pointed out temporal and spatial variations of radar echoes are related to temporal and spatial variations in the layers of the refractivity profile, which motivates the research of atmospheric refractivity estimation from radar clutter returns, i.e., refractivity from clutter (RFC). The advantage of RFC is that it provides a synoptic characterization of the refractive index structure over the spatial extent of the radar and it overcomes the necessity of additional data and/or sensing devices. In addition, RFC has the added advantage of being able to sense range-varying refractivity at a temporal sampling rate that can track changes in atmospheric conditions [*Vasudevan et al.*, 2007]. However, inferring the values of refractivity profile from radar clutter is a complex inverse problem because the relation between refractivity profile parameters and radar clutter is clearly nonlinear and ill-posed.

[4] In the last decade, many advances have been made in remotely sensing refractivity parameters from radar sea clutter [*Rogers et al.*, 2000, 2005; *Gerstoft et al.*, 2000, 2003a, 2003b, 2004; *Barrios*, 2004; *Weckwerth et al.*, 2005; *Yardim et al.*, 2006, 2007, 2008, 2009; *Vasudevan et al.*, 2007; *Douvenot et al.*, 2008, 2010; *Sheng and Huang*, 2009; *Sheng et al.*, 2009; *Huang et al.*, 2009]. Instead of determining the refractivity at each point over height at a given range, all these methods retrieve a few parameters to describe a probable characterization of the refractive index structure. The commonly used refractivity parameter models include bilinear model, trilinear model and five parameter model. Detailed discussions about these different RFC algorithms can be found in the works of *Vasudevan et al.* [2007], *Douvenot et al.* [2008, 2010], and *Huang et al.* [2009]. An important issue of these new techniques is how to evaluate their performance under real time. Operational applications necessitate short computation time, less than 10 min, to avoid error due to temporal evolution of refractivity. Through establishing many precomputed, modeled radar returns for different environments in a database, *Douvenot et al.* [2008, 2010] inverted real time profiles based on finding the optimal environment from the database. On the other hand, instead of using radar clutter returns, *Tabrikian and Krolik* [1999] proposed using point-to-point microwave measurements as a means of estimating tropospheric refractivity for the purposes of characterizing surface-based duct by the maximum a posteriori (MAP) method. *Valtr and Pechac* [2005a] used field measurements at a receiver site of a terrestrial point-to-point link in terms of angle-of-arrival spectra by matched field processing methods, and several refractivity models based on orthogonal function set were introduced to improve the estimation accuracy. Nevertheless, large numbers of forward model runs make them impractical for real time operational use.

[5] An adjoint model has particular relevance to inversion problems in which the unknown parameter space is much greater than the observation space. Methods based on adjoint models have been used in geophysical inversion [*Tarantola*, 1984], oceanography [*Huang et al.*, 2004; *Hursky et al.*, 2004], and atmospheric science [*Huang and Wu*, 2005]. In this paper we show how the technique can be used to invert for atmospheric refractivity in the electromagnetic wave propagation problems from field measurements at an array of radio receivers (see Figure 1). The derivation of the adjoint model begins with the parabolic wave equation for a smooth, perfectly conducting surface and horizontal polarization conditions. Through constructing the cost function and computing its gradient, the optimal solution of refractivity profile could be obtained by gradient-based iterations, which makes the computation real time. Other than estimating a few refractivity parameters, the optimal values of refractivity could be retrieved at each point over height, which is helpful to describe the vertical information of the refractivity in detail. For the convenience of inversion, the propagation environment is considered range independent in the numerical experiments.

[6] The reminder of this paper is organized as follows: Forward model, i.e., Fourier split-step parabolic wave equation model is introduced in section 2. Section 3 gives the detailed description of the implementation of the variational adjoint approach for atmospheric refractivity estimation. Finally, discussions of simulated runs and analysis are demonstrated in section 4.