## 1. Introduction

[2] Incoherent scatter radar (ISR) is the most comprehensive ground-based technique for studying the Earth's ionosphere. The incoherent scatter echo is the result of the scattering of electromagnetic energy, radiated from the radar, by electron density fluctuations in the ionospheric plasma, which in the most common case are influenced by much slower, massive positive ions. The frequency spectrum of the received signal provides information about electron and ion temperatures, ion composition and velocity. The analytical relationship between the spectrum and these parameters has been well established in the literature [e.g., *Dougherty and Farely*, 1960, 1961; *Farley*, 1966; *Hagfors*, 1961, 1971]. With the estimation of these parameters from incoherent scatter (IS) measurements, one can deduce many further ionospheric parameters such as electric field strength, conductivity and current, neutral air temperature, and wind speed.

[3] Although the exact forward theory of incoherent scatter was established more than four decades ago, inversion, i.e., the extraction of parameters from incoherent scatter spectra, has remained an open problem primarily because of two major factors. The first complication stems from the fact that variation of different plasma parameters may give rise to similar changes in the IS spectrum [*Vallinkoski*, 1988]. For example, the distinction between changes of the spectra due to ion composition or temperature ratio is very difficult. The same is true for ion mass and ion temperature for fixed temperature ratio. The second factor is the range smearing of information from one altitude over a number of altitudes, which is due to the length of transmitted waveform. An example is shown in Figure 1 where different lags of the autocorrelation function (ACF) are plotted prior to and after imposing the range ambiguity for an uncoded modulation of length 280 *μ*s with a 430 MHz radar (thin and thick curves, respectively). The estimation of ionospheric parameters from the range-smeared ACF would result in a greater ion temperature than electron temperature, which is not physical.

[4] Traditionally, incoherent scatter analysis has been based on performing nonlinear least squares fitting at individual altitudes (height-by-height or gated analysis) [*Lehtinen et al.*, 1996], where in the most common case the spacing between the altitudes is determined by the pulse length. The important assumption in this technique is that plasma parameters are considered constant in all regions which have influence on an individual set of spectral or ACF measurements. The effects of the transmitted waveform are then accounted for in the form of various corrections to ACF lag estimates. This method, although simple and fast, is based on an unrealistic assumption (constant parameter profiles for each range-gate) and suffers from coarse resolution of the parameters, as well as a bias which is introduced into the profiles [*Holt et al.*, 1992; *Lehtinen et al.*, 1996].

[5] Another method is the full-profile analysis [*Holt et al.*, 1992; *Lehtinen et al.*, 1996], which attempts to determine the complete altitude profiles of parameters simultaneously. The basic estimation procedure is to first calculate theoretical spectra on the basis of an initial parameter grid (followed by compensating for the smearing of information across the range, if applicable). Subsequently, the difference between the measured and predicted ACFs is used to update the plasma parameter values iteratively. The technique is optimal in the sense that all information including the range smearing effect as well as the covariance between the lag-estimate errors, can be incorporated into the analysis. The main limitation of the method, on the other hand, is its significant computational cost. Attempts have been made to reduce the cost by exploiting interpolation techniques from a coarse parameter grid to a fine range grid [*Holt et al.*, 1992; *Lehtinen et al.*, 1996]. The accuracy of the method, however, is highly dependent on the resolution of the parameter grid. A very fine spatial resolution, which is required to approximate fine details of the ambiguity function or to capture sharp gradients of parameter profiles, increases the number of parameters of the optimization search space and thus makes the technique more computationally expensive. One basic reason is due to the requirements of any nonlinear optimization technique, one of which is the computation of the derivative of the minimization function with respect to the search variables (parameters in the coarse grid). Even though analytical expressions of the derivatives of the lags of the theoretical autocorrelation function with respect to the ionospheric parameters are available, they cannot be exploited in the optimization procedures of the full-profile techniques. Therefore, forward difference, which slows down the speed of computation, is the only method that can be used for derivative calculation.

[6] In this paper, we develop the theory of a new hybrid inversion technique which aims at obtaining estimates that are close to optimal at a fraction of computational cost. The technique is based on a correction to the effect of the transmitted waveform on the ACF lag profiles through a deconvolution process, and subsequent estimation of parameters from the plasma ACF at individual altitudes.

[7] Note that some coding schemes such as alternating codes [*Lehtinen and Haggstrom*, 1987] can be utilized to eliminate the range smearing from the non-zero lags of ACF measurements analytically (provided that the medium coherence is maintained over code transmission cycles). These modulation techniques, however, increase the noise in the form of uncorrelated clutter from unwanted ranges and as such are not suitable for high-gain radars such as Arecibo radar. In such cases, one can use the coding technique developed by *Sulzer* [1986] which essentially divides the radar power into several independent uncoded long-pulse transmissions at different frequencies. In this paper we focus on such situations and investigate the performance of the proposed approach on simulated data using the long-pulse modulation. The performance comparison of different analysis techniques on ISR experiments is the subject of a future publication.

[8] The paper is organized as follows: section 2 describes background information on the incoherent scatter radar equation as well as the concepts of range smearing and ambiguity. In section 3, we present a detailed description of our proposed method and its justification using the radar equation. Section 4 investigates the performance of the new method on simulated data and compares the corresponding estimation results with those of common methods currently used. The last section provides a summary and conclusion.