## 1. Introduction

[2] Accurate and efficient inversion of incoherent scatter radar measurements with the goal of estimating ionospheric parameters such as electron density, electron temperature, ion temperature, ion composition, etc., remains challenging despite the development of the exact forward theory [e.g., *Dougherty and Farely*, 1960, 1961; *Farley*, 1966; *Hagfors*, 1961, 1971] more than four decades ago. The reason lies in part in the inherent range smearing of information from one altitude over a number of altitudes, which is due to the length of transmitted pulse. Existing modulation techniques aim at elimination of range smearing from non-zero lags of autocorrelation function (ACF) measurements. Examples include double pulse and multiple pulse techniques [*Farley*, 1969, 1972], alternating codes [*Lehtinen and Haggstrom*, 1987] and random codes [*Sulzer*, 1986b] which have been successfully used in high resolution E-region measurements. For F-region measurements with high gain radars such as the Arecibo radar, however, these modulation techniques amplify the measurement noise in the form of uncorrelated clutter from unwanted ranges [*Holt et al.*, 1992]. In such cases, one can use the multiradar ACF technique [*Sulzer*, 1986a] which essentially divides the radar power into several independent un-coded long pulse transmissions at different frequencies to trade signal-to-noise ratio (SNR) ( the ratio of the power originating from one bit to the background and system noise power) for time resolution.

[3] Un-coded long pulse measurements, on the other hand, suffer from high correlation between ACF lag estimate errors. This correlation plays an important role in ISR error analysis, that is, in determining the uncertainty in estimated parameters. First, in order to provide an accurate error analysis one has to account for the complete error covariance matrix [*Huuskonen and Lehtinen*, 1996], whose computation and inversion is computationally prohibitive especially in the high SNR scenarios, where the error covariance matrix is far from diagonal. In addition, when SNR is high, lag/range integration does not work as effectively as it would for fully independent observations in terms of improvement in statistical accuracy [*Huuskonen et al.*, 1996; *Lehtinen et al.*, 1997]. Because the estimates are highly correlated in high SNR, the integration of signals from *N* adjacent altitudes does not yield an -fold increase in estimation accuracy as would be the case for fully independent observations.

[4] Since plasma ACF estimates from long-pulse measurements are not readily available from the measured signal ACF, any proper analysis technique is required to take the range smearing into account. *Nikoukar et al.* [2008] have developed a computationally efficient, hybrid methodology for analysis of long pulse data, where the smearing of information is removed from the data via a set of 1-D deconvolutions followed by nonlinear least squares optimization techniques applied to the deconvolved ACFs. The deconvolution allows for the retrieval of plasma ACF at individual altitudes separated by the distance determined by lag resolution (fine resolution grid). The results presented in the above work, however, show that in long pulse measurements and in the presence of noise in data, it is not possible to obtain estimated parameters in the same fine resolution grid. The reason is that in order to overcome the measurement noise, one is required to incorporate regularization techniques which introduce dependencies into adjacent estimated parameters, hence reducing the effective range resolution.

[5] The purpose of the work presented in this paper is to investigate the performance of amplitude modulated coding schemes in F-region ISR experiments in terms of statistical estimation error, range resolution, and signal-to-noise ratio. For this purpose, we consider several amplitude modulated codes with a fixed length with different duty cycles. Our goal is accomplished by formulating the inherent trade-off between estimation error and resolution as mathematical measures for model order selection [*Konishi and Kitagawa*, 2008; *Akaike*, 1973]. Subsequently, numerical simulations are performed to verify the improved performance of the amplitude modulated (AM) codes guided by our analytical framework rendering optimal range resolution in comparison with long pulse modulation.

[6] In section 2, we describe the structure of the error covariance matrix and show how it is affected by the shape of the modulation waveform. In section 3 we develop analytical approaches to determine the fundamental attainable resolution based on model order selection. We proceed by presenting numerical estimates of fundamental range resolution as a function of SNR for various AM waveforms with different on-off ratios (duty cycles) in section 4. The last section provides a summary and conclusion.