Wavelet and Global Positioning System - Research Paper Example

WAVELET AND GPS By Student’s name Course code and name Professor’s name University name City, State Date of submission Abstract The use of GPS as a reliable tool for navigation is taking a different turn in these modern times. This has calls for improved levels of accuracy since the quality of this service cannot be compromised. This research paper looks at various types of GPS errors such as data inaccuracies, lack of precision in the utility instrumentation and medium of propagation. In order to check on these errors the research applies data collected for two days whose weathers differ. The residuals were analysed by applying the de-noising and threshold functions specifically found on the wavelet toolbox for MATLAB in order to come up with recommendations of how to treat the errors. This research showed that db3 is the best filter for the de-noising procedure for these residual values. In addition to this, the root mean square method established that filter db4 is best suited for decomposition purposes. It was also established that the Gaussian distributions for histograms can the best tools for observation when it comes to noise distribution as a soft curve out rightly indicates de-noised data. Contents Abstract 2 Contents 3 Introduction 4 Literature Review 6 GPS Errors 6 Wavelet Technique 9 Experimental Results 12 Data Acquisition 12 Data Processing 14 Conclusion 21 22 References List 23 Appendix A: Data Acquisition Process 25 Data Acquisition Day 1 25 25 Data Acquisition Day 2 26 Appendix B – Full GPS Data 27 Appendix C: Trial and Error 35 Appendix D – Corrected Distances 38 Introduction For many years man navigated to remote areas by use of ingenious methods such as angular measurements and guidance from the natural stars. The development of long range radio navigation, very high frequency omnidirectional radios and radio beacons in early 1940s inspired the need to revolutionize the way man navigates from one area to another especially during a time when the whole world was entangled in war. Modernization has seen the inception of the state-of-the-art Global Positioning System (GPS) based on satellites fundamentals by the department of defence of the United States with a purpose to fulfil military necessities. This vital innovation has now been made accessible for civilian use as a means of advancing lives as high usage has been noted in the mobile digital communications. Other areas in which the incorporation of GPS utility has been important include; marine seismic survey, offshore and onshore mapping, cadastral survey, transit systems, open-pit mining, vehicle navigation and precision engineering applications to list a few. Despite the fact that GPS is a satellite based application, the developers have strived to achieve required levels of accuracy through continuous positioning and timing. This is achieved through a constellation of twenty four satellites delegated towards the worldwide coverage weathers of all kinds. According to El-Rabbany (2002), these satellites are strategically placed in the orbital planes in groups of four so as to provide any GPS enabled gadget with information, location and positioning data at any given time. The GPS satellites have thus achieved a fully operational mode as a means of providing navigation data from over 26,000km above the earth surface. Basically, these signals are contaminated by errors ranging from atmospheric disturbance, multipath disturbance, orbital biases and receiver noise. This experiment thus aims at investigating the noise error characteristics of GPS receivers through the use of wavelet transform method in order to propose and implement corrections that may have a positive impact towards their elimination. The wavelet transforms applied in wave analysis is a mathematical means of finding out the variation in signal frequency with respect to time. According to MathWorks Incorporated (2013), wavelet transforms provide precise analysis for images and signals with a selective mode of study for each component under scrutiny. This is especially done by the allocation of various resolutions to the 1-D and 2-D graphics produced for this purpose. The special tool designed for this instance is the Wavelet Toolbox for MATLAB whose transformation capability is discrete, continuous and nondecimated. The data collected for investigation is fed directly into the data entry interface with an aim of de-noising for the purpose of finding out the intensities of rectifications that are required. This technique is hence used in decomposing the established GPS errors through low and high frequency classifications for correction to be carried out by use of Wavelet Toolbox for MATLAB. This study shall at the end achieve the data collection aspect coupled by GPS reading, data analysis and presentation. Literature Review GPS Errors The technological revolution in the global setup has seen various innovations come into existence in the 20th Century. The Global Positioning System is one of these important innovations owing to the massive benefits and applications that it poses to human life. Other than the conventional means of investigating positioning around the globe there are out of date technologies such as the magnetic compass, chronometer, sextant and radio-based gadgets. The unreliability of these equipment gave birth to the idea that global positioning could actually be carried out by use of satellite setups with ground controlled operations, satellite constellations orbiting around the earth and special receivers held by users. The need to make it simpler and cheaper for universal access of GPS data was encountered by its share of repercussions as the launch of satellites meant for this purpose did not run smoothly. According to LaMarca and de Lara (2008), the errors observed in GPS readings during the early stages of invention of this technology were resoundingly high leading to its un-usability. GPS observations are contaminated by three categories of errors such as GPS satellites data inaccuracies, lack of precision in the utility instrumentation and medium of propagation errors. To begin with, the estimation of electromagnetic waves speed, receiver time, satellite transit time and position pose critical challenges to global positioning. According to Sen (2006) the parameters that determine GPS signals are mainly affected by the ephemeris and the satellite clock data. The ephemeris errors are defined as errors emanating from the orbital position of the satellites while the satellite clock data errors are according to Köhne and Wößner (2009) associated with synchronization problems due to frequency bias/ drift and phase bias. These errors have however been countered by the Kalman filter designed to approximate or predict these values for corrective measures. The Keplerian orbits followed by the GPS satellites are considered to be a major cause of the ephemeris data error to a tune of 2.1m. Rao (2010) states that the discovery of differential global positioning which depend on ground reference stations also played a huge role in error elimination. In this case, the levels of accuracy are mainly affected when the distance between the user and reference station increases. With the advanced calculations in the GPS industry, the location can now be calculated in terms of time a function which has now been incorporated into the differential GPS. Figure 1: Errors on GPS signals (Yasuda , 2006). User errors have been widely classified into two groups which are mainly; multipath and receiver noise. Multipath errors can be described as those errors occurring due to frequency interference on gadget antenna, signal amplifier, cables, multi-access noise and signal quantization noise. On the other side, multipath errors occur when the GPS signals reach the receiver from all directions causing confusion on the gadget. This form of confusion referred to as frequency bandwidth interference originates from buildings, the ground and any other hard surface that may reflect waves. Czerniak and Reilly, (1998) state that the reflected signal takes time to reach the gadget resulting to an error that may cost the accuracy by up to 5 meters. In order to minimize the errors emerging from multipath noise, the surfaces on which the signals reflect should be reduced. Further, the GPS receivers have been equipped with processing algorithms that are able to filter the multipath noise. Another special filter developed to mitigate the effects of pseuodorange measurements is the hatch filter which is good at receiver noises for GPS in motion. The receiving antenna may also be shaped for multipath reception for angles elevating above the horizon in order to filter multipath waves. The commercial electromagnet is also designed to receive a stgimulated response through a choke ring antenna (Kerkhoff, et al., 2010). The medium through which GPS waves are propagated may also affect the levels of desired accuracy. From the physics of waves, it is outright that whenever waves penetrate the vacuum, their speed is slightly tampered with. According to Townsend and Fenton (1994), the deviations caused by these errors are static since the GPS processors are programmed with certain parameters which remain static especially when it comes to speed of waves which is likened to that of light. At some point, it is also made clear that the larger errors which cause rays to bend by up to 10km while propagating through the transfer media such as the troposphere and the atmosphere, highly attribute to the accuracy of the GPS. Another important media worth this discussion is the ionosphere in which the deviations extend by up to 100km due to high electron charge. These kinds of delays in wave propagation are referred to as ionospheric delays (Doherty, et al., 1996). Wavelet Technique The wavelet technique has emerged as the most sought method for solving mathematical problems in engineering and physics. This method is responsible for the modern methods of analysing data compression, processing of images, propagating of waves, processing of signals recognizing of patterns in various fields of applied sciences, high precision medical imaging, object detection in wars, detecting of objects in the aviation industry and detecting of objects by marine vessels for research and accident alleviation. The decomposition of complex waveforms found in images, music, speech and other significant patterns is paramount for analysis – a property that is readily available with the wavelet transforms technique. Signal processing and transmission is done through a series of numbers that represent an advanced method of Fourier transforms. It is however advantageous in that it is applied in dynamic waves whereas the Fourier transforms are only for stationary waves (Sifuzzaman, et al., 2009). It has been found out by various researchers (The MathWorks 2013; Williams and Amaratunga 2011), that the wavelet transforms are developed under best basis for wave compression. This is approached through the generalized scaling functions that apply when it comes to the orthonormal functions which are deciphered as complex for analysis. This gives the wavelet transform method an advantageous ground over the rest of the methods that could have been utilized for the analysis of the GPS waves under study. Following the fact that signals are classified into continuous and discrete forms, the derivation of filter coefficients present a difficult problem to researchers. This has however been made easy with the inception of better algorithms thereby overcoming the limitations of Fourier transforms. The wavelet transforms also counter the lack of local information in the predecessors in order to emerge as the best analysis tool for various applications in the engineering industry. Figure 2: A typical comparison between the Fourier transforms and Wavelet transforms (Malegori & Ferrini, 2010). The wavelet technology has been applied in various fields based on the merit of the outcomes displayed during its usage. This is attached to the statistical ability that is attached to signal processing, data compression, de-noising capabilities waveform smoothing and signal processing. The wavelet technology has found its way into biological applications such as biometric verification, DNA analysis, body parts analysis such as heart and pressure, biology for recognition of cell membrane, fractural analysis, speech recognition and protein analysis. In other industries, the wavelet technology is used for; detection of values in the banking industry, traffic description for the internet, supervision of gear wheels in manufacturing industries and computer graphics analysis. In physics, the wavelet technology has been applied in astrophysics, molecular studies, optics, quantum physics, turbulence analysis, geophysical studies and fibre optics analysis (Sifuzzaman, et al., 2009). Experimental Results Data Acquisition In carrying out the data acquisition process, three hundred points were carefully chosen at Tothill Park in Plymouth to avoid any conflicting results. A handheld receiver was used in conjunction with mobile communication gadgets as a means of data verification. The data was gathered in a chosen span of thirty seconds to ensure that no discrepancies were recorded. This exercise was carried out from 20th to 21st March 2013 which made it two days of vigorous data collection. Figure 3: Data acquisition in progress. The GPS location determined for the constant point A was established as 50 22 40.68 N, 04 08 26.52 W as the longitude and latitude respectively. These were verified through use of mobile phone gadget in order to ensure that the handheld GPS receiver was in a plum working condition. The second position B was chosen and the distance from A measured as the original distance. Point B would however be varied from time to time as the main objective of the exercise. It was established that the initial distance from point A to B was 12 meters which kicked off the data collection exercise. The figure 3 shown below indicates the data acquisition process in progress. Graph 1: Error day 1 Graph 2: Error day 1 The choice of location was influenced by the fact that Tothill Park is open and lacks plants which could have caused a lot of errors. These errors are explained in the literature review section in depth but as a matter of clarification, the obstructions were avoided due to the multipath effect that they may have had on the whole procedure. The weather conditions of Tothill Park area during the first day were mild and sunny making it comfortable for the data collection exercise. During the second day, the conditions were however were severe in that it was rainy. This could also affect the data collected as a humid weather is a direct attribute to interference of the propagated GPS waves. The variance due this effect would be significant for this study as the errors calculated in the graphs shown above indicate. Data Processing The data collected for day one and two between point A and B was entered into the MATLAB user interface for analysis using the wavelet extension. The residuals were then imported into the wavelet tool box in a single dimensional manner and exposed to a range of filters. In order to extract the bias, de-noising was undertaken by applying filters db2, db6 and db8 thereby giving the results shown in the following graphs. Graph 3: Extraction of bias. The de-noised signal was imported and saved into the column vector for correction. This procedure was carried out for the data collected on both days and the graphs copied as shown in appendix E. However for illustration purposes the sample graphs of the corrected distances for db 4 day 1 and db6 day 2 are shown below. Graph 5: Db4 day 1 Graph 10: Db6 day 2 The root mean square for the corrected distances for day one and two was then produced as a table with the filter number tallied against the results as shown in the figure below. The correspondence of 12 which was the initial distance was notably maintained in the order of 0m since it was considered as the origin. Finally, the correction frequency that was considered as the most appropriate was considered for correction of the rest of the distance sets as highlighted in the root mean square table below. Considering the level of de-noising required, the filter db4 was chosen as the most productive for this purpose at 10.083.   RMSE Filter Day 1 Day 2 Db1 9.98 13.51 Db2 10.038 13.55 Db3 10.045 13.549 Db4 10.083 13.56 Db5 10.042 13.543 Db6 10.094 13.57 Db7 10.043 13.548 Db8 10.102 13.568 Table 1: Thresholds for db1 day 1 decomposition The corrected distances were subjected to the de-noising and threshold methods for single dimension analyses. The best level of de-noising was achieved at db3 in order to establish the three decompositions’ impacts on noise elimination. The correction distance achieved by the wavelet toolbox 1-D was done by carrying out adjustments on the noise or data adjustment on three levels of decomposition. The smoothest curve achieved indicated that the noise had been successfully eliminated – a point that could also be observed from the Gaussian distribution output. Trial and error was done for several datasets until the level of adjustability was assumed. The de-noised signals differences for db1 day 1 and db1 day 2 are indicated in the graphs 11, 12, 13 and 14 below. Graph 11: Graph for db1 day1 decomposition Graph 12: Histogram for db1 day1 decomposition Graph 13: Thresholds for db1 day2 decomposition Graph 14: Histogram for db1 day2 decomposition The noise in the above histograms (Figures 12 and 14) has been eliminated by a huge margin as much as the signals are not purely Gaussian. The traces of noise that are observed in these histograms are called residual noises which are due to elimination of data which should have been useful for the achievement of a perfect Gaussian form. In order to counter this problem it was also noted that it could have been useful to adjust the coefficients for the thresholds as long possible in order to achieve the Coiflet-type wavelets. The de-noising procedure was carried out for a fixed number of filters contrary to the previous researches thereby yielding accidental errors. Conclusion The utilization of GPS has grown exponentially with new applications being discovered by the day. The advancement in navigation also calls for accuracy so as to avoid compromising of the quality of this vital service. The mentioned errors clearly show that researchers are keen to identify and eliminate them by incorporating modern technologies of different sophistication. The wavelet analysis tool in particular has been noted as one important tool for the analysis of such errors when it comes to these researches due to ease of usage and efficiency in data processing. This tool is especially helpful when it comes to the noise reduction function which was used in carrying out this research, decomposition of the data and root mean square calculation for error modulation. In a bid to minimize the errors to be dealt with in this research, the area chosen was void of obstacles such as tall buildings which could have deflected the satellite signal. The data was collected for two different days whose weather conflicted in order to establish the effect of weather as a minor objective of this research. It was established that the weather has a major effect on the accuracy levels as during the second day it was cold and rainy, leading to a larger error in comparison to day1. Importing this data on MATLAB wavelet toolbox for rectifications ranging from db1 to db8 was paramount for learning as the different levels of noise removal were observed during this process. The inbuilt root mean square feature used in this process was important in the correction of the distance variance since it was used for fine tuning of the noise level. In order to carry out this correction, the root mean square was applied in order to de-noise the achieved signal. This demonstrates exactly how the levels of accuracy are achieved in the GPS gadgets for advanced applications or industries. It was also noted that most histograms produced by wavelet toolbox took the Gaussian form translating to the fact that the noise had been successfully eliminated at all levels of trial. Although some discrepancies were noted on various levels of filtration, they are as a result of inbuilt software errors whose thresholds differ since they are meant for approximation purposes. References List Czerniak, R. J. & Reilly, J. P., 1998. Applications of GPS for Surveying and Other Positioning Needs in Departments of Transportation. Washington DC.: Transportation Research Board. Doherty, P. H., Raffi, E. & Klobuchar, J. A., 1996. Statistics of Time Rate of Change of Ionospheric Range Delay. The Mitre Corporation, pp. 1-10. El-Rabbany, A., 2002. Introduction to GPS: The Global Positioning System. Massachusetts: Artech House, Inc.. Kerkhoff, A., Harris, R. B., Petersen, C. P. & Pickard, A., 2010. Modifications to GPS Reference Station Antennas to Reduce Multipath. Texas, Applied Research Laboratories, The University of Texas at Austin, pp. 866-879. Köhne, A. & Wößner, M., 2009. Sources of Errors in GPS. [Online] Available at: http://www.kowoma.de/en/gps/errors.htm [Accessed 11 August 2013]. LaMarca, A. & de Lara, E., 2008. Location Systems: An Introduction to the Technology Behind Location Awareness. Pitsburg: Morgan & Claypool Publishers. Malegori, G. & Ferrini, G., 2010. Tip-sample interactions on graphite studied using the wavelet transform. Beilstein J. Nanotechnol. 2010, 1, p. 172–181. Rao, G. S., 2010. Global Navigation Satellite Systems. Delhi: Tata McGraw-Hill Education. Sen, S., 2006. GPS Filtering to Minimize Ionosphere Divergence Error for Safe Aircraft Landing, Michigan: ProQuest. Sifuzzaman, M., Islam, M. R. & Ali, M. Z., 2009. Application of Wavelet Transforma nd its Advantages compared to Fourier Transform. Journal of Physical Sciences, Vol. 13, pp. 121-134. The MathWorks, 2013. Continuous Wavelet Transform. [Online] Available at: http://www.mathworks.com/help/wavelet/gs/continuous-wavelet-transform.html [Accessed 12 August 2013]. Townsend, B. R. & Fenton, P. C., 1994. A Practical Approach to the Reduction of Pseudorange Multipath Errors in a Ll GPS Receiver. ION GPS-94, Salt Lake City September 20-23,, pp. 1-6. Williams, J. R. & Amaratunga, K., 2011. Introduction to wavelets for Engineering, Massachusetts: Massachusetts Institute of Technology. Yasuda , A., 2006. Errors on GPS Signal. [Online] Available at: http://www.soi.wide.ad.jp/class/20050026/slides/01/61.html [Accessed 11 August 2013]. Appendix A: Data Acquisition Process Data Acquisition Day 1 Photograph 1 – 4 Photograph 9 – 10 Data Acquisition Day 2 Photograph 1 – 4 Photograph 5- 6 Appendix B – Full GPS Data The table below consists of data collected at Tothil Park, Plymouth for two days from 20th march to 21st March 2013. The distance between A and B is measured by use of a GPS in order to obtain differential error in this study. The actual distance and the measured distances are denoted by D0 and D1 respectively. Day 1 Day 2   Point A   Point B     Point A Point B Time Latitude Longitude Latitude longitude D1 D0 GPS error Time Latitude Longitude Latitude longitude D1 D0 GPS error 0 50.377967 4.140700 50.377883 4.140667 9 12 -3 0 50.376511 4.140014 50.376533 4.140167 11.112 12 -0.888 30 50.377939 4.140706 50.377883 4.140667 6 12 -6 30 50.376472 4.140025 50.376600 4.140133 16.152 12 4.152 60 50.377969 4.140683 50.377883 4.140667 9 12 -3 60 50.376497 4.140117 50.376600 4.140133 11.490 12 -0.510 90 50.377986 4.140664 50.377883 4.140667 11 12 -1 90 50.376456 4.140028 50.376583 4.140150 16.643 12 4.643 120 50.377986 4.140664 50.377883 4.140667 11 12 -1 120 50.376453 4.139989 50.376583 4.140133 17.767 12 5.767 150 50.377989 4.140681 50.377883 4.140667 11 12 -1 150 50.376483 4.140106 50.376600 4.140117 12.997 12 0.997 180 50.377989 4.140681 50.377883 4.140667 11 12 -1 180 50.376489 4.140114 50.376600 4.140117 12.357 12 0.357 210 50.377978 4.140681 50.377883 4.140667 10 12 -2 210 50.376533 4.140117 50.376617 4.140133 9.341 12 -2.659 240 50.377981 4.140686 50.377883 4.140667 10 12 -2 240 50.376489 4.140131 50.376600 4.140117 12.394 12 0.394 270 50.377981 4.140686 50.377883 4.140667 10 12 -2 270 50.376461 4.140075 50.376600 4.140133 15.988 12 3.988 300 50.377986 4.140742 50.377883 4.140667 12 12 0 300 50.376467 4.140061 50.376583 4.140150 14.423 12 2.423 330 50.377981 4.140750 50.377883 4.140667 12 12 0 330 50.376481 4.140078 50.376583 4.140150 12.523 12 0.523 360 50.377956 4.140775 50.377883 4.140667 11 12 -1 360 50.376517 4.140075 50.376583 4.140150 9.123 12 -2.877 390 50.377953 4.140753 50.377883 4.140667 9 12 -3 390 50.376492 4.140217 50.376583 4.140133 11.782 12 -0.218 420 50.377950 4.140769 50.377883 4.140667 10 12 -2 420 50.376503 4.140000 50.376583 4.140133 13.024 12 1.024 450 50.378003 4.140817 50.377883 4.140667 17 12 5 450 50.376514 4.140000 50.376583 4.140133 12.208 12 0.208 480 50.377997 4.140794 50.377883 4.140667 15 12 3 480 50.376508 4.140153 50.376567 4.140133 6.632 12 -5.368 510 50.377983 4.140772 50.377883 4.140667 13 12 1 510 50.376481 4.140014 50.376550 4.140133 11.420 12 -0.580 540 50.377964 4.140789 50.377883 4.140667 12 12 0 540 50.376489 4.142981 50.376550 4.140133 7.757 12 -4.243 570 50.377958 4.140792 50.377883 4.140667 12 12 0 570 50.376511 4.140078 50.376550 4.140117 5.129 12 -6.871 600 50.377972 4.140708 50.377883 4.140667 10 12 -2 600 50.376469 4.140019 50.376550 4.140117 11.304 12 -0.696 630 50.377975 4.140711 50.377883 4.140667 10 12 -2 630 50.376472 4.140056 50.376550 4.140100 9.204 12 -2.796 660 50.377978 4.140731 50.377883 4.140667 11 12 -1 660 50.376464 4.139994 50.376550 4.140100 12.154 12 0.154 690 50.377972 4.140767 50.377883 4.140667 12 12 0 690 50.376481 4.140008 50.376567 4.140083 10.953 12 -1.047 720 50.377983 4.140761 50.377883 4.140667 12 12 0 720 50.376497 4.140069 50.376567 4.140100 8.021 12 -3.979 750 50.377983 4.140761 50.377883 4.140667 12 12 0 750 50.376486 4.140014 50.376583 4.140100 12.416 12 0.416 780 50.377978 4.140747 50.377883 4.140667 11 12 -1 780 50.376514 4.140083 50.376583 4.140117 8.076 12 -3.924 810 50.377969 4.140728 50.377883 4.140667 10 12 -2 810 50.376497 4.140039 50.376583 4.140117 11.050 12 -0.950 840 50.377978 4.140722 50.377883 4.140667 11 12 -1 840 50.376497 4.140092 50.376583 4.140100 9.593 12 -2.407 870 50.377986 4.140753 50.377883 4.140667 12 12 0 870 50.376514 4.140044 50.376583 4.140100 8.669 12 -3.331 900 50.378000 4.140728 50.377883 4.140667 13 12 1 900 50.376486 4.140019 50.376600 4.140100 13.893 12 1.893 930 50.378006 4.140736 50.377883 4.140667 14 12 2 930 50.376500 4.140033 50.376600 4.140100 12.083 12 0.083 960 50.377981 4.140708 50.377883 4.140667 11 12 -1 960 50.376456 4.139997 50.376600 4.140100 17.638 12 5.638 990 50.377994 4.140731 50.377883 4.140667 13 12 1 990 50.376478 4.140014 50.376600 4.140100 14.900 12 2.900 1020 50.377975 4.140700 50.377883 4.140667 10 12 -2 1020 50.376483 4.140033 50.376600 4.140100 13.807 12 1.807 1050 50.377956 4.140681 50.377883 4.140667 8 12 -4 1050 50.376486 4.140028 50.376600 4.140083 13.263 12 1.263 1080 50.377964 4.140678 50.377883 4.140667 8 12 -4 1080 50.376475 4.139917 50.376600 4.140100 19.032 12 7.032 1110 50.377969 4.140636 50.377883 4.140667 9 12 -3 1110 50.376478 4.139722 50.376600 4.140100 30.040 12 18.040 1140 50.377953 4.140711 50.377883 4.140667 8 12 -4 1140 50.376492 4.140008 50.376600 4.140117 14.287 12 2.287 1170 50.377961 4.140703 50.377883 4.140667 9 12 -3 1170 50.376494 4.139981 50.376617 4.140117 16.669 12 4.669 1200 50.377964 4.140694 50.377883 4.140667 9 12 -3 1200 50.376494 4.139906 50.376600 4.140117 19.023 12 7.023 1230 50.378031 4.140719 50.377883 4.140667 16 12 4 1230 50.376458 4.139825 50.376600 4.140100 25.069 12 13.069 1260 50.378031 4.140697 50.377883 4.140667 16 12 4 1260 50.376464 4.140022 50.376600 4.140083 15.743 12 3.743 1290 50.378003 4.140697 50.377883 4.140667 13 12 1 1290 50.376472 4.140083 50.376583 4.140100 12.411 12 0.411 1320 50.378011 4.140733 50.377883 4.140667 14 12 2 1320 50.376467 4.139981 50.376583 4.140100 15.493 12 3.493 1350 50.377983 4.140736 50.377883 4.140667 12 12 0 1350 50.376492 4.140014 50.376583 4.140117 12.530 12 0.530 1380 50.377931 4.140742 50.377883 4.140667 7 12 -5 1380 50.376461 4.140011 50.376583 4.140117 15.515 12 3.515 1410 50.377983 4.140808 50.377883 4.140667 14 12 2 1410 50.376528 4.140061 50.376583 4.140133 8.024 12 -3.976 1440 50.378008 4.140825 50.377883 4.140667 17 12 5 1440 50.376519 4.140036 50.376583 4.140150 10.756 12 -1.244 1470 50.377953 4.140806 50.377883 4.140667 12 12 0 1470 50.376489 4.139986 50.376600 4.140150 16.962 12 4.962 1500 50.377958 4.140808 50.377883 4.140667 13 12 1 1500 50.376481 4.140006 50.376600 4.140150 16.773 12 4.773 1530 50.377997 4.140567 50.377883 4.140667 14 12 2 1530 50.376469 4.139958 50.376583 4.140150 18.577 12 6.577 1560 50.377986 4.140703 50.377883 4.140667 11 12 -1 1560 50.376472 4.139978 50.376600 4.140100 16.643 12 4.643 1590 50.378014 4.140675 50.377883 4.140667 14 12 2 1590 50.376497 4.140042 50.376600 4.140117 12.605 12 0.605 1620 50.378053 4.140761 50.377883 4.140667 19 12 7 1620 50.376497 4.139986 50.376617 4.140150 17.649 12 5.649 1650 50.377939 4.140644 50.377883 4.140667 6 12 -6 1650 50.376467 4.140069 50.376617 4.140150 17.630 12 5.630 1680 50.377892 4.140533 50.377883 4.140667 9 12 -3 1680 50.376508 4.140056 50.376617 4.140150 13.783 12 1.783 1710 50.377969 4.140639 50.377883 4.140667 9 12 -3 1710 50.376517 4.140044 50.376600 4.140133 11.207 12 -0.793 1740 50.378006 4.140689 50.377883 4.140667 13 12 1 1740 50.376486 4.140111 50.376600 4.140117 12.670 12 0.670 1770 50.377894 4.140703 50.377883 4.140667 2 12 -10 1770 50.376519 4.140072 50.376600 4.140117 9.495 12 -2.505 1800 50.377906 4.140700 50.377883 4.140667 3 12 -9 1800 50.376542 4.140064 50.376600 4.140150 8.909 12 -3.091 1830 50.377892 4.140736 50.377883 4.140667 5 12 -7 1830 50.376464 4.140053 50.376583 4.140150 14.964 12 2.964 1860 50.377922 4.140714 50.377883 4.140667 5 12 -7 1860 50.376439 4.140001 50.376583 4.140133 18.588 12 6.588 1890 50.377950 4.140828 50.377883 4.140667 13 12 1 1890 50.376492 4.140022 50.376583 4.140117 12.196 12 0.196 1920 50.377950 4.140794 50.377883 4.140667 11 12 -1 1920 50.376453 4.139986 50.376583 4.140117 17.218 12 5.218 1950 50.377953 4.140836 50.377883 4.140667 14 12 2 1950 50.376450 4.140083 50.376583 4.140100 14.873 12 2.873 1980 50.378006 4.140844 50.377883 4.140667 18 12 6 1980 50.376525 4.140022 50.376583 4.140067 7.211 12 -4.789 2010 50.377997 4.140811 50.377883 4.140667 16 12 4 2010 50.376492 4.140097 50.376583 4.140083 10.240 12 -1.760 2040 50.377986 4.140817 50.377883 4.140667 15 12 3 2040 50.376492 4.140089 50.376583 4.140100 10.223 12 -1.777 2070 50.377997 4.140819 50.377883 4.140667 16 12 4 2070 50.376492 4.139967 50.376583 4.140100 13.903 12 1.903 2100 50.378003 4.140800 50.377883 4.140667 16 12 4 2100 50.376456 4.139772 50.376583 4.140100 27.242 12 15.242 2130 50.378028 4.140800 50.377883 4.140667 18 12 6 2130 50.376458 4.139922 50.376567 4.140100 17.437 12 5.437 2160 50.378036 4.140658 50.377883 4.140667 16 12 4 2160 50.376464 4.139775 50.376583 4.140100 26.600 12 14.600 2190 50.378031 4.140633 50.377883 4.140667 16 12 4 2190 50.376500 4.139767 50.376567 4.140117 25.903 12 13.903 2220 50.378036 4.140650 50.377883 4.140667 17 12 5 2220 50.376511 4.140014 50.376567 4.140100 8.686 12 -3.314 2250 50.378000 4.140658 50.377883 4.140667 12 12 0 2250 50.376497 4.140078 50.376567 4.140100 7.881 12 -4.119 2280 50.377919 4.140725 50.377883 4.140667 5 12 -7 2280 50.376519 4.139958 50.376550 4.140083 9.493 12 -2.507 2310 50.377911 4.140694 50.377883 4.140667 3 12 -9 2310 50.376469 4.140003 50.376550 4.140100 11.304 12 -0.696 2340 50.377911 4.140694 50.377883 4.140667 3 12 -9 2340 50.376503 4.140056 50.376550 4.140117 6.809 12 -5.191 2370 50.377911 4.140694 50.377883 4.140667 3 12 -9 2370 50.376450 4.140014 50.376533 4.140100 11.097 12 -0.903 2400 50.377925 4.140700 50.377883 4.140667 5 12 -7 2400 50.376494 4.140006 50.376550 4.140100 9.111 12 -2.889 2430 50.377933 4.140700 50.377883 4.140667 6 12 -6 2430 50.376506 4.139992 50.376550 4.140100 9.135 12 -2.865 2460 50.377958 4.140694 50.377883 4.140667 8 12 -4 2460 50.376533 4.139997 50.376550 4.140100 7.520 12 -4.480 2490 50.377989 4.140633 50.377883 4.140667 11 12 -1 2490 50.376497 4.139972 50.376550 4.140117 11.805 12 -0.195 2520 50.377972 4.140650 50.377883 4.140667 9 12 -3 2520 50.376469 4.140061 50.376550 4.140117 9.786 12 -2.214 2550 50.377972 4.140650 50.377883 4.140667 9 12 -3 2550 50.376522 4.139908 50.376583 4.140117 16.261 12 4.261 2580 50.377922 4.140689 50.377883 4.140667 4 12 -8 2580 50.376453 4.139981 50.376583 4.140133 18.114 12 6.114 2610 50.377903 4.140764 50.377883 4.140667 7 12 -5 2610 50.376522 4.140072 50.376600 4.140100 8.870 12 -3.130 2640 50.377942 4.140728 50.377883 4.140667 7 12 -5 2640 50.376489 4.140017 50.376617 4.140117 15.880 12 3.880 2670 50.377997 4.140653 50.377883 4.140667 12 12 0 2670 50.376531 4.140133 50.376633 4.140117 11.489 12 -0.511 2700 50.377992 4.140675 50.377883 4.140667 12 12 0 2700 50.376522 4.140133 50.376633 4.140117 12.411 12 0.411 2730 50.377992 4.140675 50.377883 4.140667 12 12 0 2730 50.376522 4.140133 50.376633 4.140133 12.355 12 0.355 2760 50.377986 4.140692 50.377883 4.140667 11 12 -1 2760 50.376522 4.140133 50.376633 4.140150 12.411 12 0.411 2790 50.378011 4.140608 50.377883 4.140667 14 12 2 2790 50.376522 4.140036 50.376633 4.140150 14.760 12 2.760 2820 50.378047 4.140564 50.377883 4.140667 19 12 7 2820 50.376467 4.140097 50.376633 4.140150 18.906 12 6.906 2850 50.378056 4.140589 50.377883 4.140667 19 12 7 2850 50.376544 4.139964 50.376633 4.140150 16.488 12 4.488 2880 50.378056 4.140589 50.377883 4.140667 19 12 7 2880 50.376497 4.140031 50.376617 4.140167 16.419 12 4.419 2910 50.378044 4.140614 50.377883 4.140667 18 12 6 2910 50.376517 4.140014 50.376617 4.140150 14.724 12 2.724 2940 50.378044 4.140614 50.377883 4.140667 18 12 6 2940 50.376528 4.140078 50.376600 4.140133 8.945 12 -3.055 2970 50.378044 4.140614 50.377883 4.140667 18 12 6 2970 50.376500 4.140122 50.376600 4.140167 11.557 12 -0.443 3000 50.378033 4.140639 50.377883 4.140667 16 12 4 3000 50.376575 4.140025 50.376617 4.140133 8.971 12 -3.029 3030 50.377933 4.140700 50.377883 4.140667 6 12 -6 3030 50.376478 4.140064 50.376617 4.140133 16.210 12 4.210 3060 50.377900 4.140750 50.377883 4.140667 6 12 -6 3060 50.376514 4.139983 50.376617 4.140133 15.613 12 3.613 3090 50.377950 4.140739 50.377883 4.140667 9 12 -3 3090 50.376453 4.140014 50.376633 4.140133 21.790 12 9.790 3120 50.377950 4.140761 50.377883 4.140667 9 12 -3 3120 50.376464 4.139997 50.376633 4.140133 21.170 12 9.170 3150 50.377950 4.140769 50.377883 4.140667 10 12 -2 3150 50.376475 4.140050 50.376633 4.140133 18.571 12 6.571 3180 50.377900 4.140739 50.377883 4.140667 5 12 -7 3180 50.376497 4.140011 50.376633 4.140133 17.441 12 5.441 3210 50.377900 4.140719 50.377883 4.140667 4 12 -8 3210 50.376489 4.140022 50.376633 4.140150 18.441 12 6.441 3240 50.377981 4.140689 50.377883 4.140667 10 12 -2 3240 50.376469 4.139992 50.376633 4.140133 20.809 12 8.809 3270 50.377911 4.140731 50.377883 4.140667 5 12 -7 3270 50.376489 4.140061 50.376633 4.140150 17.254 12 5.254 3300 50.377889 4.140731 50.377883 4.140667 4 12 -8 3300 50.376481 4.140056 50.376633 4.140167 18.726 12 6.726 3330 50.377919 4.140769 50.377883 4.140667 8 12 -4 3330 50.376497 4.140014 50.376633 4.140150 17.951 12 5.951 3360 50.377900 4.140681 50.377883 4.140667 2 12 -10 3360 50.376489 4.140236 50.376633 4.140167 16.800 12 4.800 3390 50.377900 4.140739 50.377883 4.140667 5 12 -7 3390 50.376500 4.140011 50.376633 4.140167 18.479 12 6.479 3420 50.377919 4.140781 50.377883 4.140667 9 12 -3 3420 50.376525 4.140081 50.376633 4.140167 13.506 12 1.506 3450 50.377869 4.140731 50.377883 4.140667 4 12 -8 3450 50.376525 4.140000 50.376633 4.140167 16.876 12 4.876 3480 50.377889 4.140761 50.377883 4.140667 6 12 -6 3480 50.376522 4.140047 50.376617 4.140167 13.492 12 1.492 3510 50.377939 4.140761 50.377883 4.140667 9 12 -3 3510 50.376508 4.139961 50.376600 4.140133 15.907 12 3.907 3540 50.378000 4.140769 50.377883 4.140667 14 12 2 3540 50.376486 4.139931 50.376600 4.140133 19.161 12 7.161 3570 50.377981 4.140681 50.377883 4.140667 10 12 -2 3570 50.376464 4.139914 50.376600 4.140133 21.708 12 9.708 3600 50.377881 4.140739 50.377883 4.140667 5 12 -7 3600 50.376444 4.140050 50.376600 4.140133 18.279 12 6.279 3630 50.377919 4.140769 50.377883 4.140667 8 12 -4 3630 50.376492 4.140025 50.376600 4.140117 13.688 12 1.688 3660 50.377850 4.140739 50.377883 4.140667 6 12 -6 3660 50.376494 4.139897 50.376600 4.140117 19.492 12 7.492 3690 50.377889 4.140706 50.377883 4.140667 2 12 -10 3690 50.376456 4.139889 50.376600 4.140133 23.632 12 11.632 3720 50.377900 4.140761 50.377883 4.140667 6 12 -6 3720 50.376450 4.140025 50.376617 4.140133 20.062 12 8.062 3750 50.377939 4.140750 50.377883 4.140667 8 12 -4 3750 50.376514 4.140061 50.376600 4.140150 11.464 12 -0.536 3780 50.377969 4.140750 50.377883 4.140667 11 12 -1 3780 50.376542 4.139958 50.376600 4.140150 15.060 12 3.060 3810 50.377969 4.140689 50.377883 4.140667 9 12 -3 3810 50.376553 4.139964 50.376617 4.140150 14.988 12 2.988 3840 50.377969 4.140681 50.377883 4.140667 9 12 -3 3840 50.376517 4.140072 50.376617 4.140133 11.934 12 -0.066 3870 50.377931 4.140719 50.377883 4.140667 6 12 -6 3870 50.376547 4.140028 50.376600 4.140117 8.612 12 -3.388 3900 50.377939 4.140739 50.377883 4.140667 8 12 -4 3900 50.376561 4.139944 50.376600 4.140117 12.956 12 0.956 3930 50.377961 4.140739 50.377883 4.140667 10 12 -2 3930 50.376494 4.139953 50.376600 4.140117 16.518 12 4.518 3960 50.377950 4.140706 50.377883 4.140667 7 12 -5 3960 50.376575 4.139997 50.376600 4.140117 8.915 12 -3.085 3990 50.377919 4.140650 50.377883 4.140667 4 12 -8 3990 50.376547 4.140011 50.376600 4.140117 9.512 12 -2.488 4020 50.378011 4.140661 50.377883 4.140667 14 12 2 4020 50.376483 4.139911 50.376583 4.140100 17.408 12 5.408 4050 50.377931 4.140661 50.377883 4.140667 5 12 -7 4050 50.376572 4.139975 50.376600 4.140117 10.510 12 -1.490 4080 50.377931 4.140661 50.377883 4.140667 5 12 -7 4080 50.376600 4.140069 50.376583 4.140117 3.828 12 -8.172 4110 50.377931 4.140661 50.377883 4.140667 5 12 -7 4110 50.376586 4.140078 50.376583 4.140117 2.776 12 -9.224 4140 50.377981 4.140789 50.377883 4.140667 13 12 1 4140 50.376533 4.140089 50.376583 4.140117 5.897 12 -6.103 4170 50.377969 4.140789 50.377883 4.140667 12 12 0 4170 50.376542 4.140125 50.376567 4.140117 2.842 12 -9.158 4200 50.377889 4.140750 50.377883 4.140667 5 12 -7 4200 50.376553 4.140028 50.376567 4.140117 6.490 12 -5.510 4230 50.377911 4.140761 50.377883 4.140667 7 12 -5 4230 50.376564 4.140042 50.376550 4.140133 6.681 12 -5.319 4260 50.377881 4.140731 50.377883 4.140667 4 12 -8 4260 50.376550 4.140097 50.376567 4.140133 3.160 12 -8.840 4290 50.377881 4.140750 50.377883 4.140667 5 12 -7 4290 50.376533 4.140083 50.376567 4.140133 5.129 12 -6.871 4320 50.377889 4.140731 50.377883 4.140667 4 12 -8 4320 50.376522 4.140028 50.376567 4.140117 8.010 12 -3.990 4350 50.377969 4.140661 50.377883 4.140667 9 12 -3 4350 50.376503 4.140017 50.376550 4.140117 8.824 12 -3.176 4380 50.378019 4.140650 50.377883 4.140667 15 12 3 4380 50.376494 4.139997 50.376550 4.140117 10.483 12 -1.517 4410 50.378000 4.140650 50.377883 4.140667 13 12 1 4410 50.376475 4.139947 50.376550 4.140133 15.612 12 3.612 4440 50.377969 4.140761 50.377883 4.140667 11 12 -1 4440 50.376458 4.139981 50.376567 4.140117 15.437 12 3.437 4470 50.377969 4.140661 50.377883 4.140667 9 12 -3 4470 50.376458 4.140031 50.376550 4.140117 11.882 12 -0.118 4500 50.377950 4.140661 50.377883 4.140667 7 12 -5 4500 50.376472 4.140042 50.376567 4.140117 11.772 12 -0.228 4530 50.377931 4.140639 50.377883 4.140667 5 12 -7 4530 50.376481 4.140014 50.376567 4.140117 12.034 12 0.034 4560 50.377961 4.140719 50.377883 4.140667 9 12 -3 4560 50.376483 4.140003 50.376567 4.140100 11.550 12 -0.450 4590 50.377950 4.140700 50.377883 4.140667 7 12 -5 4590 50.376494 4.139997 50.376567 4.140100 10.845 12 -1.155 4620 50.377931 4.140703 50.377883 4.140667 5 12 -7 4620 50.376500 4.139983 50.376567 4.140117 12.014 12 0.014 4650 50.377900 4.140761 50.377883 4.140667 6 12 -6 4650 50.376503 4.139969 50.376567 4.140117 12.628 12 0.628 4680 50.377886 4.140756 50.377883 4.140667 6 12 -6 4680 50.376508 4.140003 50.376567 4.140117 10.359 12 -1.641 4710 50.377950 4.140711 50.377883 4.140667 8 12 -4 4710 50.376522 4.140036 50.376583 4.140133 9.680 12 -2.320 4740 50.377961 4.140675 50.377883 4.140667 8 12 -4 4740 50.376528 4.140078 50.376583 4.140133 7.327 12 -4.673 4770 50.377933 4.140689 50.377883 4.140667 5 12 -7 4770 50.376531 4.140089 50.376583 4.140133 6.660 12 -5.340 4800 50.377931 4.140647 50.377883 4.140667 5 12 -7 4800 50.376533 4.140114 50.376583 4.140117 5.562 12 -6.438 4830 50.377922 4.140675 50.377883 4.140667 4 12 -8 4830 50.376547 4.140056 50.376583 4.140117 5.907 12 -6.093 4860 50.377936 4.140689 50.377883 4.140667 6 12 -6 4860 50.376544 4.140008 50.376583 4.140117 8.816 12 -3.184 4890 50.377939 4.140644 50.377883 4.140667 6 12 -6 4890 50.376539 4.139989 50.376583 4.140117 10.321 12 -1.679 4920 50.377931 4.140703 50.377883 4.140667 5 12 -7 4920 50.376528 4.139925 50.376583 4.140133 16.013 12 4.013 4950 50.377931 4.140703 50.377883 4.140667 5 12 -7 4950 50.376508 4.139964 50.376583 4.140133 14.626 12 2.626 4980 50.377922 4.140742 50.377883 4.140667 6 12 -6 4980 50.376500 4.140000 50.376567 4.140133 12.014 12 0.014 5010 50.377939 4.140722 50.377883 4.140667 7 12 -5 5010 50.376497 4.140014 50.376567 4.140117 10.619 12 -1.381 5040 50.377908 4.140756 50.377883 4.140667 6 12 -6 5040 50.376497 4.140050 50.376600 4.140117 12.368 12 0.368 5070 50.377919 4.140769 50.377883 4.140667 8 12 -4 5070 50.376483 4.140097 50.376600 4.140117 13.046 12 1.046 5100 50.377931 4.140700 50.377883 4.140667 5 12 -7 5100 50.376475 4.140111 50.376600 4.140117 13.905 12 1.905 5130 50.377933 4.140719 50.377883 4.140667 6 12 -6 5130 50.376489 4.140089 50.376600 4.140117 12.511 12 0.511 5160 50.377958 4.140719 50.377883 4.140667 9 12 -3 5160 50.376497 4.140025 50.376600 4.140117 13.148 12 1.148 5190 50.377958 4.140761 50.377883 4.140667 10 12 -2 5190 50.376503 4.140042 50.376600 4.140117 12.048 12 0.048 5220 50.377975 4.140792 50.377883 4.140667 13 12 1 5220 50.376494 4.139967 50.376600 4.140133 16.657 12 4.657 5250 50.377981 4.140703 50.377883 4.140667 11 12 -1 5250 50.376483 4.139978 50.376583 4.140133 15.662 12 3.662 5280 50.377992 4.140672 50.377883 4.140667 12 12 0 5280 50.376475 4.139997 50.376600 4.140133 16.922 12 4.922 5310 50.377958 4.140633 50.377883 4.140667 8 12 -4 5310 50.376467 4.140036 50.376600 4.140117 15.889 12 3.889 5340 50.378022 4.140681 50.377883 4.140667 15 12 3 5340 50.376453 4.140042 50.376583 4.140117 15.460 12 3.460 5370 50.377942 4.140678 50.377883 4.140667 6 12 -6 5370 50.376447 4.140000 50.376583 4.140117 17.248 12 5.248 5400 50.377928 4.140664 50.377883 4.140667 4 12 -8 5400 50.376447 4.140056 50.376583 4.140133 16.108 12 4.108 5430 50.377917 4.140700 50.377883 4.140667 4 12 -8 5430 50.376444 4.140064 50.376600 4.140133 17.985 12 5.985 5460 50.377936 4.140717 50.377883 4.140667 6 12 -6 5460 50.376442 4.140081 50.376600 4.140117 17.791 12 5.791 5490 50.377933 4.140731 50.377883 4.140667 7 12 -5 5490 50.376439 4.140097 50.376583 4.140117 16.120 12 4.120 5520 50.377961 4.140731 50.377883 4.140667 9 12 -3 5520 50.376439 4.140133 50.376583 4.140133 16.061 12 4.061 5550 50.377906 4.140753 50.377883 4.140667 6 12 -6 5550 50.376447 4.140167 50.376583 4.140133 15.318 12 3.318 5580 50.377894 4.140725 50.377883 4.140667 4 12 -8 5580 50.376456 4.140197 50.376583 4.140117 15.313 12 3.313 5610 50.377958 4.140703 50.377883 4.140667 8 12 -4 5610 50.376461 4.140139 50.376583 4.140117 13.681 12 1.681 5640 50.377967 4.140700 50.377883 4.140667 9 12 -3 5640 50.376469 4.140106 50.376583 4.140100 12.670 12 0.670 5670 50.377992 4.140681 50.377883 4.140667 12 12 0 5670 50.376478 4.140053 50.376600 4.140100 13.997 12 1.997 5700 50.377936 4.140714 50.377883 4.140667 6 12 -6 5700 50.376481 4.140019 50.376583 4.140117 13.346 12 1.346 5730 50.377922 4.140758 50.377883 4.140667 7 12 -5 5730 50.376489 4.140006 50.376583 4.140117 13.129 12 1.129 5760 50.377961 4.140733 50.377883 4.140667 9 12 -3 5760 50.376500 4.139961 50.376583 4.140133 15.330 12 3.330 5790 50.377986 4.140789 50.377883 4.140667 14 12 2 5790 50.376503 4.139919 50.376583 4.140117 16.608 12 4.608 5820 50.378019 4.140731 50.377883 4.140667 15 12 3 5820 50.376517 4.140003 50.376583 4.140133 11.860 12 -0.140 5850 50.378019 4.140719 50.377883 4.140667 15 12 3 5850 50.376528 4.140028 50.376567 4.140133 8.645 12 -3.355 5880 50.378019 4.140719 50.377883 4.140667 15 12 3 5880 50.376544 4.140042 50.376567 4.140133 6.954 12 -5.046 5910 50.377986 4.140772 50.377883 4.140667 13 12 1 5910 50.376547 4.140069 50.376567 4.140117 3.986 12 -8.014 5940 50.378000 4.140742 50.377883 4.140667 14 12 2 5940 50.376531 4.140086 50.376567 4.140117 4.563 12 -7.437 5970 50.377958 4.140669 50.377883 4.140667 8 12 -4 5970 50.376528 4.140061 50.376583 4.140133 8.024 12 -3.976 6000 50.377958 4.140708 50.377883 4.140667 8 12 -4 6000 50.376519 4.140044 50.376583 4.140117 8.758 12 -3.242 6030 50.377919 4.140733 50.377883 4.140667 6 12 -6 6030 50.376506 4.139989 50.376583 4.140133 13.406 12 1.406 6060 50.377939 4.140731 50.377883 4.140667 7 12 -5 6060 50.376500 4.139947 50.376583 4.140117 15.174 12 3.174 6090 50.377981 4.140761 50.377883 4.140667 12 12 0 6090 50.376506 4.140000 50.376600 4.140117 13.369 12 1.369 6120 50.377939 4.140756 50.377883 4.140667 8 12 -4 6120 50.376522 4.140017 50.376583 4.140117 9.821 12 -2.179 Appendix C: Trial and Error Figure 1: Threshold for day 1 db 1 Figure 2: Histogram for day 1 db 1 Figure 3: Thresholds for db6 day 1 decomposition Figure 4: Histogram for db6 day 1 decomposition Figure 5: Threshold for db8 day2 decomposition Figure 6: Histogram for db8 day2 decomposition Appendix D – Corrected Distances Graph 1: Db2 day 1 Graph 2: Db2 day 2 Graph 3: Db3 day 1 Graph 4: Db3 day 2 Graph 5: Db4 day 2 Graph 6: Db5 day 1 Graph 7: Db5 day 2 Graph 9: Db6 day 1 Read More