This paper puts forward possibilities of refractive index profile retrieval using field measurements at an array of radio receivers in terms of variational adjoint approach. The derivation of the adjoint model begins with the parabolic wave equation for a smooth, perfectly conducting surface and horizontal polarization conditions. To deal with the ill-posed difficulties of the inversion, the regularization ideas are introduced into the establishment of the cost function. Based on steepest descent iterations, the optimal value of refractivity could be retrieved quickly at each point over height. Numerical experiments demonstrate that the method works well for low-distance signals, while it is not accurate enough for long-distance propagations. Through curve fitting processing, however, giving a good initial refractivity profile could generally improve the inversions.
 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.
Richter  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.
 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. , Douvenot et al. [2008, 2010], and Huang et al. . 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  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.
 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.
 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.
where ∂x ≡ ∂/∂x and ∂z ≡ ∂/∂z, u = (x, z) represents a scalar component of the electric field, and x and z are corresponding to range and height, respectively. k0 is the free space wave number, m = m(x, z) is the modified index of refraction, takes into account the Earth's curvature and is defined by m = n + z/ae, n being the index of refraction and ae being the radius of the Earth. u(x, 0) = 0 represents the boundary condition at z = 0, and u(0, z) = ϕ(z) gives the initial field at the source range.
 The two most popular approaches to numerically solving PWEs are the use of implicit finite differences (IFD) and the Fourier split-step algorithm. Historically, IFD methods have had the advantage of allowing straightforward implementation of complex boundaries, while the split-step solution has proven to be significantly more stable numerically, thus permitting larger horizontal mesh increments and shorter computation times [Valtr and Pechac, 2005b]. In our calculations, the Fourier split-step algorithm will be adopted. The forward and inverse Fourier transforms are defined by,
where p is the transform variable often referred to as the vertical wave number or spatial frequency. Here, the transforms are written in continuous form, but with limits of integration placed upon z and p (since the discrete Fourier transform, by way of the FFT, is actually used). Z and P are determined by Nyquist's criteria, ZP = πN, where N is the transform size [Barrios, 1994]. It should be noted that an abrupt truncation of the field in height, however, will result in strong reflections from the nonphysical upper boundary. A practical approach to this problem is to add an “absorbing” region above the maximum altitude of interest where the field is attenuated smoothly to zero at height Z [Kuttler and Dockery, 1991]. In this paper, a cosine tapered (Tukey) filter array is used to filter the upper 1/4 of the field for this purpose.
 Let u(xk, z) be the electric field at range xk and height z. Then, the field at range xk+1 and height z, denoted by u(xk+1, z), is given by the Fourier split-step solution to the parabolic wave equation,
where δx is the range increment, given by δx = xk+1 − xk. Note that when calculating the exponential factor, m is allowed to vary in both x and z. As (3) shows, the field at range xk+1 is dependent on the field at the previous range step xk. Therefore, one must begin with an initial field ϕ(z) at range zero in order to propagate the field forward. Initial field is essentially the antenna aperture distribution, and that the far-field antenna pattern and its aperture distribution are a Fourier transform pair. The antenna pattern used in our calculations is a Gaussian antenna pattern. It is designated that the antenna is set at a height of 25 m and operates at a frequency of 8000 MHz with a beam width of 1 deg and an elevation angle of 0 deg.
3. Adjoint Model
3.1. Cost Function
 Note uobs(L, z) as the field measurement at the receiver of height z, where L is the horizontal distance between the transmitter and the receiver array (the relationship between the receiving power and uobs(L, z) is given by Gerstoft et al. [2003a]). The field predicted by the forward model is noted as u(L, z). Then, a convenient cost function formulation J is thus defined as,
In practice, the available measurements are often contaminated by noise. Previous studies of inverse problem have indicated that these problems are nonlinear and ill-posed [Huang et al., 2004; Huang and Wu, 2005]. With the aid of regularization techniques in inverse problem [Tikhonov and Arsenin, 1977; Huang and Wu, 2005], an additional stable function related to the field u(x, z) could be introduced to J in order to deal with the ill-posedness and make the calculation stable. Then, the improved cost function is defined as follows,
where ∣∂ zu∣2dxdz is a stable function and γ is a regularization parameter.
3.2. Tangent Linear Model
 In open ocean conditions, it was found that assuming range-independent environments leads to good propagation assessments 86% of the time [Hitney et al., 1985]. In order to simplify the computation, the refractive conditions are considered constant with the distance for the entire propagation path, that is, m = m(z).
 Set perturbations to m as (z) = m(z) + α(z), taking u and , respectively, as the solution of the model (1) corresponding to m and . Note (x, z) is the Gâteaux differential of u(x, z),
Then satisfies the following tangent linear model (TLM),
 Define two inner products as follows,
where is the conjugate function of h.
 The Gâteaux differential of J[m] with respect to at point m is,
where Re[u] is the real component of a complex number u.
 On the other hand, from the definition of J′[m; ],
Combining (10) with (11), the following equation could be obtained,
3.3. Adjoint Model
 Multiply (7a) by w(x, z), and integrate it on the domain [0, L] × [0, Z],
Considering the initial boundary condition (7b–7c), (13) could be computed separately,
It should be noted that in the above computation, the conjugate boundary conditions of adjoint field w(x, z) are set as w(x, 0) = 0, w(x, Z) = 0 and ∂zw(x, Z) = 0. Substitute the above four terms into (13) to obtain,
 Combining (12) with (15), the following adjoint equation and initial boundary conditions could be obtained,
So the gradient of the cost function (5) at point m is obtained,
 With this gradient, then the steepest descent iteration formulas could be adopted as follows,
where ρi is iteration step for mi, and the determination of ρi could refer to [Byrd et al., 1995].
 The solution of the adjoint model (16) can be also computed by Fourier split-step algorithm,
where δx is the range increment, given by δx = xk+1 − xk, which should be identical to δx in (3), and G(x, p) = γ2p2U(x, p). However, being different from the solution of (3), w(xk, z) is obtained by integrating the adjoint model in reverse direction.
 The optimal solution of m can be obtained by gradient-based iterations. The whole iteration process is expressed as a flowchart in Figure 2, where ɛ is a given small positive real number.
4. Numerical Experiments and Analysis
 The approach described above is novel in that it introduces the variational optimal control methods in combination with the regularization techniques into refractivity estimation. Since the measured data were not available, numerical experiments were performed to test the theoretical results above.
 Owing to m is very close to unity, for environmental inputs, modified refractivity M, defined by M = 106 × (m − 1), is used to describe the information of the atmospheric environment. In our simulations, a surface-based duct capping over an evaporation duct is used to model the observed modified refractivity profile. The initial profile is given with the gradient of 0.118 M-units/m, which is consistent with the mean over much of the world. In the implementation of Fourier split-step transform, the Fourier transform size N is set to the order of 512 to satisfy N = 29, the vertical increment δz is 1.0736 m determined by δz = π/P, the range increment δx is set to be 1 m and the effective propagation range L is set to be 100 m. Because the available measurements are often contaminated by noise, here, 10% additive white Gaussian noise is added to the true field measurements. Figure 3a gives the observed modified refractivity profile (Mobs, real line), the initial profile (Minit, dashed line), and the values of retrieved modified refractivity at each point over height (Minv, dot) without considering the additional stable function, i.e., γ = 0. The stable function is taken into account in the cost function and the prior parameter γ is set to be γ = 0.001, then the corresponding retrieved values are shown in Figure 3b. The iteration numbers and the values of cost function J at each iteration step of the two different cost functions are given in Figure 3c.
 From the inversion results given in Figure 3, it is clear that in both experiments, the cost function could decrease to zero and the retrieved profile could approach to the true values. With the additional stable function in the cost function, however, the iteration number is less than without it, and the inverted values are converging closer to the observed profile, which indicates that with a proper regularization parameter, the regularization ideas introduced into the inverse problem is helpful to deal with the ill-posedness of the problem. The regularization parameter is chosen as a priori information which can be exactly determined by the discrepancy principle [Huang and Wu, 2005]. From Figure 3c, it could be seen that the decreasing of the cost function is very fast at the beginning of the iteration process. When the inversion accuracy achieves some extent, the cost function will decrease smoothly. The iteration number of γ = 0 is 44 and the computation time is 7.9 s. The iteration number of γ = 0.001 is 38 and the computation time is 6.8 s. Our computation source is ThinkPad R400 equipping with dual CPUs (P8600, 2.40GHz) and 2 GB EMS memory bank.
 In practical operations, the transmitter and the receiver array could not be disposed thus adjacently. On the other hand, using parabolic wave equation to simulate the electric field propagations in the medium does not obtain correct results for very short ranges, and parabolic wave equation models the propagating waves inside a cone with a very small angle, which does not include high altitudes at very short ranges. Therefore, the distance between the transmitter and the receiver array should be much longer than 100 m. Referring to previous works with real data in RFC researches, the range increment δx is set to be 600 m and the effective propagation range L is set to be 60 km. Other settings are kept the same as above. The inversion results are shown in Figure 4, where the dash-dotted curve is the fitting data of the retrieved values with the least square method.
 The results given in Figure 4 show that the accuracy with 60 km propagation distances is less than the accuracy with 100 m. In low propagation ranges, the inversion values fluctuate surrounding the observed profile. While in long propagation ranges, the inversion values fluctuate surrounding the initial profile but follow the synoptic structure of the observed profile. The reason of this phenomenon is likely to be that with the increment of the propagation ranges, the fading of the receiving signal power will become larger and larger, which will enhance the uncertainty of the inversion. Comparing Figure 4a with Figure 4b, it is clear seen that the structure of the retrieved profile in Figure 4b is better than the structure in Figure 4a. This is because the additional stable function is propitious to improve the accuracy of the retrieved results. The computation time corresponding to Figure 4a is 11.5 s and 9.0 s corresponding to Figure 4b, respectively.
 On the other hand, the inversion accuracy of adjoint approach is dependent on the space of observations. If the observations are extraordinarily insufficient, it is difficult to obtain the qualified inversions. In the above simulations, the vertical resolution of the receiver array is set to be identical to the FFT bin spacing. How the inversion would be for other vertical spacing array elements? Figure 5 shows the estimations for sparse vertical resolutions of the receiver array, where the propagation range L is set to be 60 km and the regularization parameter γ is set to be γ = 0.001.
 From the above numerical experiments, we could see that in the long distance the method is not accurate enough and the inversion accuracy is decreasing with reduction of observations. However, the final goal of the refractivity estimation is not to give the exact refractivity profile, but to propose a potential structure that could be able to render an approximation of the real atmospheric condition to predict microwave propagation for assessing the performance of both communications and radar systems. Figure 6 gives the modeled raypaths diagrams computed by ray-tracing approach and the coverage diagrams (dB) of the modeled propagation loss computed by Fourier split-step method to parabolic wave equation. The entire area of the diagrams is 0–200 km in range and 0–400 m in height. Figures 6a and 6b show the raypaths for the observed refractivity profile and the retrieved refractivity profile (fitted) in Figure 4b, respectively. The next two plots show the modeled propagation loss for the two different environments. The absolute difference of propagation loss values of Figures 6c and 6d is shown in Figure 6e. Figure 6 gives an indication of how well the inverted profile is able to predict the propagation characteristics. Because of the ducting propagation, the “radar holes” and the “skip areas” agree fairly well for the two different environments.
 Although the modeled raypaths have a good match between the observed and inverted environments, however, the maximum value of the absolute difference of propagation loss is up to 60 dB. This is quite substantial and great efforts should be made to reduce the difference for illustrating the goodness of the technique. The adjoint process is a local optimization method and the initial guess has an impact on the inversion accuracy. If the initial guess is too far from the correct solution, there is no mechanism for escaping a spurious local minimum [Hursky et al., 2004]. Generally, giving a good initial guess could improve the inversion results to some extent. The inversion results of different initial profiles are shown in Figure 7.
 A novel approach to estimate atmospheric refractivity at each point over height from field measurements at an array of radio receivers has been presented. This scheme introduces the variational optimal control methods in combination with the regularization techniques into refractivity inversion. The optimal solution could be obtained by gradient-based iterations, which makes the computation real time. Numerical experiments show that in low propagation ranges, the retrieved results could converge to the observed profile with high accuracy. By virtue of the fading of the receiving signal power, the inversions are gradually deviating from the true values in long propagation range cases.
 The adjoint process is a local optimization method and the inversion accuracy is dependent on the space of observations and initial guess. Generally, a good initial guess could ameliorate the inversions. In practical operations, we could use the historical observations as the initial guess to replace the standard refractivity profile with the gradient of 0.118 M-units/m to improve the retrieved profiles.
 Inverse problems typically lead to ill-posed mathematical models. Especially, their solutions are unstable under data perturbations, so special numerical methods should be developed to cope with these instabilities. Here, the regularization ideas incorporating additional information about the desired solution were introduced into the establishment of the cost function to stabilize the ill-posedness of the problem. Yet, the simulations demonstrate that regularization is helpful to deal with the ill-posedness of the inversion but cannot solve it.
 In this paper, only smooth, perfectly conducting surface and horizontally homogeneous atmospheric conditions were discussed with simulations. Future work is required to evaluate the performance of the method with real measured data, as well as more complex environment conditions.
 The paper was much improved by the suggestions from the anonymous reviewers. This work has been supported by the National Natural Science Foundation of China (grant 40775023).