Principle. CAT navigation measures the fluctuation width of the displacements in EAS particle arrival times between two arbitrary points located within the shower disk in order to calculate the distance between these two points, and then subsequently the two-dimensional coordinates (x, y) of these points are derived. Its principle is essentially the inverse of the CTS28,29,30 concept. The time difference in EAS particle arrival times between two arbitrary points (Point A and Point B) fluctuates around the average value of the time displacements. This fluctuation width, time displacement SD (standard deviation), increases as the distance between these points increases. This phenomenon has already been observed in prior CTS works, and it has been reported that time displacement SD is degraded as the distance between these points increases30. The CAT positioning procedure is described as follows.
Supposing that an EAS event occurs in which particles nearly simultaneously arrive at three points A (x0, y0), B (x1, y1), and C (x2, y2) which is a total of n times at the time ti (i = 1, 2, ... n) during the time interval T, the distance between A-B and the distance between A-C would be respectively:
DAB = [(x1-x0)2+(y1-y0)2]1/2, (1-1)
DAC = [(x2-x0)2+(y2-y0)2]1/2. (1-2)
On the other hand, since the fluctuation widths of the arrival time difference (time displacement SD) between A-B (sAB) and A-C (sAC) (CTS time fluctuations) are a function of D, DAB and DAC can be respectively expressed as:
DAB = f (sAB), (2-1)
DAC = f (sAC). (2-2)
sAB and sAC are given from the measurements. Therefore, if the positions of B and C are known, the position of A can be derived by substituting Eqs. (2-1) and (2-2) with Eqs. (1-1) and (1-2). Figure 1 illustrates the basic principle of CAT navigation. In this paper, the first goal is to experimentally determine the parameters of f (s) in Eq. (2).
In principle, a minimum of two detectors are required to take a coincidence and to identify an EAS event. However, since it is expected that the coincidence rate decreases as the distance between these two detectors increases, the relative fraction of the accidental coincidence (i.e., random timestamps) generated by open-sky muons increases. For example, when using two 100-cm2 detectors to take a dual coincidence with a coincidence time window of 100 ns, the average time interval at which an accidental coincidence occurs is 0.5´12´107 = 5,000,000 s (~2 months); if the size of the detectors increases to 1 m2, this average time interval is reduced to 500 s. However, even with 1-m2 detectors, by taking a triple coincidence measurement with a redundant detector, random timestamp generation errors will be reduced to a rate of once in 0.33´(10-2)3´(107)2 = 33,000,000 s (~1 year); hence this type of error will become negligible. On the other hand, the results from prior EAS MC simulation studies28 indicated that adding a redundant detector reduces the event rate by a factor of ~2. Therefore, there is a trade-off between the accidental rate and the positioning signal update rate. The decision of whether to use a redundant detector needs to be optimized based on the size of the receiver we use. In the current demonstration, considering that the detector size was relatively large (i.e., the open-sky muon rate is high), three detectors to identify an EAS event were used, and to reduce the generation of random timestamps. In the following subsections, we will discuss (1) f (s), the distance dependence of SD of CTS time fluctuations measured with the CTS experimental setup, and (2) the CAT navigation accuracy when paired with oven-controlled crystal oscillators (OCXO).
CAT reference data taking. CAT reference data were taken to determine the parameters of f (s). The experimental setup was the following: one detector (Receiver A), consisting of a 1500-cm2 plastic scintillator (ELJEN200), and a photomultiplier tube (PMT; Hamamatsu R7724), and other four 1-m2 detectors (Receivers B-E) with the same configuration as Receiver A. All of the detectors were installed indoors on the ground floor of a building. Then, the triple coincidence was taken between Receivers A-B-C, Receivers B-C-D, and Receivers B-C-E to measure the time displacement SD. The distance between Receiver A and Receiver B was 10 m, the distance between Receiver A and Receiver C was 50 m, the distance between Receiver B and Receiver D was 35 m, and the distance between Receiver B and Receiver E was 180 m. All of the detectors except for Receiver E were wired with an RG coaxial cable to get rid of the local clock drift effect. (The positioning results using an independent local clock will be described in a later section.) For Receiver E, cabling was not possible since multiple third-party buildings were located in between Receiver B and Receiver E, so GPS-time-synchronized results acquired in the prior works were utilized30. The electronic circuit that is used for data collection was the same as previous experiments which have already been described in many references20,21,28,29, so its properties are only briefly mentioned here. The electronic circuit consists of a discriminator, 4-channel TDC, OCXO, and CPLD, and the PMT signals output from Detectors A-E are first digitized by the discriminator and input to each stop channel of the TDC. In parallel, the 10MHz TTL pulse output from the OCXO is fed to the TDC start channel, and the difference (Dt) between the output timing of the OCXO's 10MHz pulse and the PMT signal is measured. A cosmic-ray arrival timestamp (CAT stamp) is generated by adding Dt to (the number of OCXO 10MHz pulses) ´ (100ns). CAT stamp lists generated from these detectors are used for finding coincidence events in a given time window, and the temporal displacements between the coincided CAT stamps (called single signals) were subsequently calculated. The averaged triple coincidence rate for the combination including the 50-m baseline (the distance between detectors) was (68 s)-1, and the triple coincidence rate for the combination including the 180-m baseline was (360 s)-1. According to the EAS-MC results based on the prior work28, the average triple coincidence time intervals are respectively 3-4s and 40-50s for the 50-m and 180-m baselines with 1-m2 detectors. This discrepancy comes from two factors: (A) the efficiency of Receivers B-E is about 50%30, and (B) the area of Receiver B is 1,500 cm2. Therefore, the detection rate has dropped to ~1/20 and ~1/8, respectively in comparison to the estimation. In this work, a positioning accuracy of 2 to 3 m9 was targeted. The distance dependence of the CTS time fluctuations measured with this setup will be described in the next subsection.
Distance dependence of the CTS time fluctuations. The distance dependence of the time displacement SD measured with the current setup was almost linear, but a better fit is made with the following quadratic function (Figure 2).
s = 0.00171D2 + 0.49178237D + 4.8652112, or (3-1)
D = [-0.49178237 - {0.491782372 - 0.00684 ´ (4.8652112 - s)}1/2]/0.00342 (3-2)
The fitting errors are shown in Table 1.
Accuracy of CAT positioning and the positioning information update rate. The accuracy of CAT positioning depends on the granularity of the time displacement SD. Therefore, a certain number of EAS events are required. From this section onwards, the triple coincidence event rate will be referred to as the single signal update rate, and the rate at which a cluster (single signal cluster) containing a certain number of single signals is generated is called the positioning signal update rate. Here, the cluster size refers to the number of single signals in the single signal cluster. As the cluster size increases, the SD granularity improves, but the positioning signal update rate is degraded.
Figures 3A-3C show time-series graphs of the time displacement SD determined within each cluster by dividing the single signals obtained for D = 50 m into clusters of different sizes. As the cluster size increases, the temporal fluctuations in SD are reduced, but the positioning signal update rate decreases. Figures 3D-3F show positioning signals obtained by converting SD shown in Figures 3A-3C into the distance D using Eq. (3-2). The resultant distancing errors are respectively 9.4 m, 6.1 m and 3.8 m for a cluster size of 10, 20 and 50. In order to achieve the targeted accuracy, a cluster size of at least 50 is required. However, when considering the time required for the positioning signal update, a cluster size of 10 is ideal, and as is explained later, positioning errors can be improved by rejecting imaginary solutions of Eq. (1). Therefore, a cluster size of 10 is employed in the following discussions.
Positioning with local clocks. In the previous subsections, all the detectors were connected via wires, so the CTS time fluctuations only originated from the EAS time structure. However, in an actual CAT operation scenario, positioning is wirelessly performed using independent local clocks. Consequently, time fluctuations due to clock drift are added to the EAS time structure.
In conventional muPS that utilizes the muon’s time of flight, distances are determined on an event-by-event basis, so if the clock drifts while acquiring four tracks, this clock drift directly affects positioning accuracy. For example, a drift of 10 ns leads to a positioning error of 3 m. Moreover, this time offset generally changes non-linearly. On the other hand, CAT navigation uses SD as positioning information. Therefore, a time offset between clocks does not affect positioning accuracy. Furthermore, clock drifts can be linearly approximated since its single signal update rate is much higher than conventional muPS. By tracing the time series of a single signal with a straight line and subtracting this fitted straight line from each data point, it is possible to cancel out this drift effect. In summary, the positioning procedure using the clock is as follows.
(1) Record CAT stamps in each receiver using the local clock associated with each detector.
(2) The generated CAT stamps are sent to the central processing unit via Wi-Fi.
(3) The cross-detector time displacements are computed from these CAT stamps at the central processing unit.
(4) If the time displacements are smaller than the given time window, it is regarded as a single signal, and it is added to the single signal cluster at the central processing unit.
(5) Once a certain number of single signals have been accumulated, a linear function is fitted to the time-sequential cross-detector time displacement data points to remove the local clock’s relative drift effect.
(6) The time displacement SD of the data points is calculated at the central processing unit.
(7) The distance between the detectors is derived with Eq. (3-2) at the central processing unit.
(8) The position of the receiver is derived with Eq. (1) at the central processing unit.
(9) The receiver’s position information is sent back to the receiver via Wi-Fi.
In this demonstration, the above procedure was implemented in tandem with OCXOs (wireless) and cables (wired), so that the OCXO effect can be separated from the EAS time structure effect. Figure 4 shows an example of the time displacement data between the detectors, and the fitted linear function for D = 50 m. Tables 2 and 3 respectively show the SD of the time displacement data points after subtraction of the fitted function and deriving the distance with Eq. (3-2). 10 independent runs are shown as an example. It can be seen that there is a large deviation in the value in Run #6 for D = 180m. This error is caused by the nonlinear clock drift (see a green dashed line in Figure 5) since the time scale required for positioning signal update was longer for D = 180m. Figure 5 shows the OCXO drift. The run numbers shown in Figure 5 correspond to the run numbers shown in Tables 2 and 3. Besides this irregularity, the error (1SD) due to the clock drift had a value of 3-4 meters regardless of the length of D.
At the end of this section, the 2-dimensional wireless positioning results with OCXO will be shown. The geometric configuration and positioning results within the framework of WGS84 are shown in Figure 6A. If an imaginary solution was derived in Eq. 1, that dataset was discarded. Since imaginary solutions indicate that the circular functions in Eq. (1) do not intersect, low quality positioning signals are automatically discarded in this process. Additionally, regarding the conjugate solutions, the values closer to the initial position were taken. The resultant positioning accuracy in this geometric configuration was 3.4 m (SD) in the x direction and 4.8 m (SD) in the y direction. This x-y asymmetry comes from this geometric configuration. Figures 6B and 6C show the x- and y-direction slices of the current positioning results. The slices in the y direction are symmetrical in positive and negative directions, but the slices in the x direction are asymmetrical. This asymmetry comes from the current solution selection criteria.
The current results were obtained with 1-m2 detectors and a 1,500-cm2 detector. A 1-m2 detector weighs more than 10 kg and cannot be used as a portable receiver, whereas a 1,500-cm2 detector weighs less than 2 kg and is the size of a large laptop (e.g. the size of Dell Alienware m18 is 410.3×319.9×25.1mm3 and 4.23 kg) so it is more portable. Additionally, by improving the buffer capacity of the data acquisition electronic circuit, efficiency can be improved to nearly 100%. With this improvement, it is expected that the time scale required for positioning signal update can be improved to 170 seconds (for a cluster size of 10) for a range of 50 m.