# Kalman Filter and Wavelet Cross-Correlation for VHF Broadband Interferometer Lightning Mapping
## 1 Introduction  
Lightning is a natural phenomenon that occurs in the atmosphere. When electrical discharges are generated, electromagnetic radiations over different ranges of frequencies are produced, usually extending from the Ultra-Low Frequency (ULF) through the Ultra-High Frequency (UHF) [1]. Electromagnetic fields are emitted either from cloud-to-ground (CG) strikes, intra-cloud (IC) lightning,and some in-cloud processes [2; 3; 4; 5; 6]. The physics behind lightning initiation suggested that lightning is produced when a considerable amount of warm air is up-drafted to a height where an extremely low temperature can cause particles of frozen ice to produce electric charge disconnections [7]. Lightning strikes can cause harm to human beings and extend effects to systems operating in the surrounding environment. These effects can be minimized if reliable lightning mapping and localization systems are deployed to enable accurate prediction and detection of lightning strikes [4; 8; 9; 10; 11]. Lightning mapping provides sufficient information on detected CG flashes at a particular range of frequencies. The very high frequency (VHF) band has been important for lightning mapping and for studying the lightning process over the past four decades [12]. The VHF lightning mapping is carried out in two different ways: time of arrival (TOA) and direction of arrival (DOA) or what is generally known as interferometry (ITF). The VHF lightning mapping has been traditionally performed using the TOA approach and narrow-band ITF systems to locate the radiation sources [13; 14; 15]. The TOA uses the time difference of arrival (TDOA) technique which requires at least four to five antennae [6]. On the other hand, the narrowband ITF system uses closely spaced multiple baseline antennae (in order of n wavelength, n\(\lambda\)) to resolve the radiation measurement ambiguities and obtain the desired angular resolution [9]. The use of broadband ITF for lightning localization was first introduced by [2], and has been developed rapidly during the past few decades [16; 17; 18; 19; 20]. Unlike the narrowband ITF, the broadband ITF uses all frequency components which are then used to produce a better resolution of lightning discharges. In the latest development of ITF systems, an array of closely spaced antennae (three perpendicular antennae) are used to capture and map the lightning radiation into the two-dimensional mapping of sources [9]. Despite the significant improvements made to map lightning using ITF systems, the mapping accuracy remains a challenge. In this paper, we exploit the effects of different parameters of well-established methods to estimate the lightning maps accurately. We propose a modified cross-correlation method to solve problems in the DOA calculation and to improve the lightning location accuracy. We examine various noise filtering to reduce the low and high noise frequency contents. Noisy lightning signals impact the cross-correlation performance and lead to incorrect estimation of phase difference. Different generalized cross-correlation techniques are used to estimate the TDOA in a segment-wise manner. These techniques include conventional time and frequency domain cross-correlation. However, these generalized cross-correlation methods are more sensitive to the embedded noise in the signal, which results in wrong TDOA estimations. Therefore, we extend the proposed framework to include the wavelet-based cross-correlation, which reduces the noise and improves the system location accuracy. We then compare the proposed technique to the conventional cross-correlation techniques. We also investigate the effect of interpolating the cross-correlation function in determining the time difference. To assess the performance of the proposed lightning mapping approach, we introduce a simulation technique to generate a broadband ITF lightning.