Proceedings with Extended Abstracts (single PDF file) - Radio ...

where ˆδ nm = (δ l − δ m ) − 2z 0 (k l − k m ) represents the phase errors. As seen, these phaseerrors introduce a phase shift for each cosine function in the summation of the Fourierpower within a given range gate. If no errors are present, the cosine functions will bealigned at the imaged range z I and will add coherently.3 Phase calibration using genetic algorithmsThe Genetic algorithm (GA) is a robust optimization technique based on natural evolutionarymechanisms. Introduced by Holland [1975], the first algorithm was called thesimple genetic algorithm (SGA) with the goal of obtaining better estimates in global optimizationscenarios. SGA mimics nature’s evolutionary characteristics by manipulatinga given population of possible solutions (individuals), and searching for the best solutionto solve an optimization problem. Basically, SGA operates through the following steps:(1) creation of a population (possible solutions), (2) evaluation of each individual in thepopulation, (3) selection of the best individuals, and (4) genetic manipulation to create anew population. Therefore, after manipulation, a new population is produced with moreoptimal genetic characteristics. The SGA repeats the cycle until a certain condition issatisfied. This condition could be, for example, a predefined number of generations orsome desired fitness level.The proposed phase calibration algorithm, based on the SGA approach, is devisedas follows. First, an image of the total echo power is constructed as a function of timeand height. Then a predefined window is chosen as the region surrounding the highestsignal-to-noise ratio (SNR) pixel from the image. In the selected window, the SGA algorithmis applied. For our particular application with the EISCAT radar, five frequencieswill be used to implement RIM. Therefore, five unknown phase values φ l (l = 1, · · · , 5)must be estimated. The Fourier RIM power is considered as a fitness function to evaluatenew generations of potential solutions. When the fitness function is called for anevaluation, a unique phase calibration matrix denoted by Φ i for a specific individual i isformed with the elements Φ i lm = 〈φi l φim∗ 〉 = e −jδi lm where l and m represent the differentfrequency combinations (l, m = 1, · · · , 5). If δlm i ≈ ˆδ lm i , the Fourier RIM power will bemaximized and the term Φ i lm is chosen so as to cancel the original phase offsets. Then,the calibration operation is simply performed by an element-by-element multiplicationof the contaminated covariance matrix with calibration matrix ˆR lm = R lm · Φ i lm at everyevaluation. The selection algorithm assigns high fitness values to individuals that allowthe Fourier RIM power to be maximized. The SGA loop is repeated until a certainnumber of generations is reached. After the SGA process, the optimal phase calibrationmatrix Φ is formed and used to correct the entire original image.4 Experimental resultsFigure 1 shows a comparison of the corrupted Capon RIM power image with randomphase errors and the calibrated RIM image after the application of the proposed GAbasedmethod. The vertical white lines at approximately 10:35 UT emphasize the regionto which the calibration procedure was applied. The learning curves from the GA arealso provided in the bottom panels of the figure. Given the large vertical extent of thePMSE layers, any enhancement due to the calibration is difficult to observe.The effect of the GA-based calibration is more easily observed by scrutinizing a smallerregion of the data. Figure 2 provides the echo power, original RIM image, and calibratedRIM image for a 10-min period from 1035–1045 UT at an altitude of approximately85 km. Note the more natural transitions between range gates and finer detail in thecalibrated RIM image. Without calibration, the RIM power centers are distorted andunnaturally contained within each gate. Further, range weighting-function effects can beobserved similar to those reported by Chilson et al. [2003].139