Crustal deformation detection capability associated with a 1 fault slip on the interplate boundary in the GNSS-A seafloor 2 geodetic observation array (SGO-A), provided by Japan 3 Coast Guard 4

The GNSS-A technique is an observation method that can detect seafloor crustal deformations 25 with centimeter level accuracy. The GNSS-A seafloor geodetic observation array operated by 26 the Japan Coast Guard, called SGO-A, has been constructed near the Japan Islands along the 27 Nankai Trough and the Japan Trench. This observation array has detected several earthquakes’ 28 displacements and episodic slow crustal deformation. To compare the detection results of SGO- A with other observation networks and expand the SGO-A distribution, it is necessary to 30 correctly understand its detection capability. In this paper, the capabilities of current GNSS-A 31 (frequency: f = 4–6 times/year, position accuracy: σ (standard deviation) = 1.5 cm) to detect a 32 crustal deformation rate only, an event only, and crustal deformation rate and event together 33 were arranged by numerical simulations. Results suggested the following features: when it is 34 known that there is no event, the 95% confidence level (CL) for the estimation of crustal 35 deformation rate with 4-year observation is about 0.5–0.8 cm/year; when the deformation rate 36 is known, a signal of about 3.0 cm can be detected by observations of about 4 times before and 37 after the event. When the deformation rate and the event are detected together, to keep the false 38 positive low (about 0.05), the false negative becomes high (about 0.2–0.7 for detecting a signal 39 of 4.5–6.0 cm). The determined rate and event variations are approximately 1.8 cm/year 40 (95%CL) and 1.5 cm (standard deviation), respectively. We also examined the detection 41 capability for higher frequency and accuracy, to examine how the detection capability improves 42 by technological advancements in the future. Additionally, we calculated the spatial range of 43 event detectability using the determined values of detection sensitivity. Each seafloor site can 44 detect a slip event larger than 0.1 m scale within about 50 km radius. A subseafloor slip event 45 smaller than about 1 m at the distance of 100 km or more from the land can often be detected 46 only on the seafloor observation array. 47

3 about 100 km, while that of the GEONET is about 30 km, causes lower spatial resolution than 94 the onshore area. 95 Due to these limitations, it is necessary to collect enough data through long-term observations 96 to detect geodetic phenomena accurately by the GNSS-A. In addition, the magnitude of the 97 phenomena must be large enough to be detected. In this paper, we examined the detection 98 capability of the geodetic phenomena shown in Table 1 from sparse geodetic time series data  99 using statistical test and numerical experiments. Additionally, we examined the ability to detect 100 interplate boundary slip in the current SGO-A along the Nankai Trough and the Japan Trench. 101 102 2 Detection capability tests 103 Detection of the geophysical phenomena of seismic cycle as crustal deformation events from 104 the geodetic time series can be broadly divided into four types, as shown in Table 1. Steady 105 crustal deformation due to plate coupling or rigid block motions in the absence of events is 106 simply detected as linear steady trends (gray line in Table 1). The trend is estimated by 107 regression analysis (Fig. 1a). The effect of the amount and duration of the data on the estimation 108 of the trend was discussed in subsection 2.1. Regular earthquake or cumulative change of SSE 109 were detected as non-continuous step signal (yellow line in Table 1). The step is estimated by 110 taking the difference between the average positions before and after the event (Fig. 1b). The 111 effect of the amount of the data on the statistical test of the significance of the step was discussed 112 in subsection 2.2. In the actual cases, the trend and the step are often estimated simultaneously 113 (blue line in Table 1). Thus, in subsection 2.3, we discussed the methods for estimating these 114 two values simultaneously and their uncertainties (Fig. 1c). Post-seismic deformations are 115 detected as non-linear change that decreases with time (orange line in Table 1). To estimate the 116 change, various models have been considered such as logarithmic or exponential. In this paper, 117 although we do not consider non-linear change of time series due to the complexity of models, 118 we discussed it supplementarily in subsection 2.1. 119 In this section, we examined the ability of sparse geodetic data to detect these phenomena using 120 numerical pseudo datasets. In the following tests, we assumed that the time series data has a 121 Gaussian noise with standard deviation of σ and no systematic errors. 122 123

Trend estimation 124
First, we assessed the uncertainty of trend estimation (Fig. 1a). The steady trend corresponds to 125 4 the crustal movement velocity caused by plate coupling or rigid block motion. To detect the 126 heterogeneity of plate coupling in the focal region of the megathrust earthquake, it is necessary 127 to estimate the velocity with an accuracy of less than 1 cm/year. Here, we assessed how much 128 data are needed for this purpose. 129 The trend was estimated by linear regression with a linear function, + . The unbiased 130 variance of the trend b is represented as 131 is the time of i-th data, and ̅ is the average of . 133 To detect the heterogeneity of the crustal velocity field, the significance test of the differences 142 of the trend among adjacent sites is necessary. This test is also effective to detect the temporal 143 variation between two periods in the case of a non-linear trend such as post-seismic deformation. 144 We conducted a statistical hypothesis test to examine if the null hypothesis that the trends of 145 two time series are the same can be rejected at a significance level of 0.05. We tested 1,000 146 numerical experiments on two time series data with the same period, observation frequency, 147 and noise level. Fig. 2b shows the smallest difference for which the null hypothesis is rejected 148 at the 0.05 level. 149 150

Step detection without trend 151
When a seismic event occurs, a step-like signal appears in the time series. The step is detected 152 from the difference between the average position before and after the event. In this subsection, 153 we examined the effect of the amount of the data on the statistical test of the significance of the 154 step. For simplicity, we assumed that there is no trend in the time series of our statistical tests. 155 This means that either the trend has been removed by trend estimation, or the period of the data 156 5 is short enough to neglect the trend. Simultaneous estimation of both trend and step is discussed 157 in the next subsection. 158 We considered the case of detecting a signal in a time series as shown in Fig. 1b and conducted 159 a statistical test for the difference between the means value before and after the event. The 160 significance of the event is judged based on whether the null hypothesis that the mean value 161 before and after the event is equal is rejected or not. There are two types of errors in the 162 statistical test. A type I error (false positive) α is the rejection of a true null hypothesis and a 163 type II error (false negative) β is the non-rejection of a false null hypothesis. To detect an event 164 accurately, these two types of errors need to be small. In the statistical test, we first fixed α, and 165 then considered the condition to reduce β. The condition is determined by the amount of the 166 data (sample size) and the magnitude of the event (effect size). 167 The statistical power γ is defined as 168 Parameters in the equations are defined as follows: In the actual case, it is necessary to evaluate the detection as soon as possible. Fig. 3b shows γ 181 when the sample size after event (S2 in Fig. 1b) is set to 2. If the threshold of γ is set to 0.8, 182 required sample size before the event are 12 and 4 to detect events with size of 2σ and 2.7σ, 183 respectively. Fig. 3c shows the uncertainty of the detected step size with respect to the sample 184 size obtained by the law of propagation of errors. The uncertainty decreases according to 1/√ . 185 186

Step detection with trend 187
In the actual case, it is necessary to detect the event from a short time series, in which the trend 188 6 and the timing of the event are unknown. If the trend is assumed to be unchanged before and 189 after the event, the time series is represented as a piece-wise line (Fig. 1c) as follows, 190 where t1 and t2 are the beginning and the ending time of the event, respectively. Because the 193 estimation of t1 and t2 is a non-linear regression problem, we used the numerical estimation 194 using grid search which is similar to the method proposed in Yokota and Ishikawa (2020). Once 195 t1 and t2 are fixed, fitting a piece-wise line is a linear regression problem. We searched the best 196 result which minimizes the c-AIC (Akaike, 1974;Sugiura, 1978), by varying t1 every 0.1 year. 197 The duration of the event (t2 -t1) is fixed to 1 year; the uncertainty of the estimation of duration 198 is discussed later. The c-AIC is defined as follows: 199 First, we examined the detection probability of the event. Fig. 4a shows the rate of false positive 212 of our method, when applying thresholds of 0, 5, and 10 for the -Δc-AIC. It shows the 213 probability that the piece-wise line is incorrectly determined to be more significant than the 214 straight line despite the absence of the event, i.e., D = 0. If the threshold is set to 0, the false 215 detection rate becomes larger than 0.6-0.7, suggesting that this threshold cannot be used 216 practically for detection of a step in a trend. The false detection rate can be improved by 217 increasing the threshold; in the case where the threshold is set to 10, the false detection rate is 7 about less than 0.05, which can be used practically. 219 Figs. 4b-d shows the probability of false negative of our method. These figures show the 220 probability that the piece-wise line is incorrectly determined to be less significant than the 221 straight line, despite the existence of an event. It is improved by increasing annual frequency 222 and event size. Contrary to the case of the false positive, the false negative rate deteriorates by 223 increasing the threshold of -Δc-AIC. In the actual case, it is necessary to set appropriate 224 thresholds according to the purpose, due to the trade-off relation between false positive and 225 false negative. For example, accepting a high false negative rate will decrease the false positive 226 rate due to the trade-off relation. However, a high false negative rate indicates that we are failing 227 to detect many of the events that are actually occurring. Thus, an exceedingly high false 228 negative rate has an adverse effect on earthquake disaster prevention, so it will be necessary to 229 accept false positive to some degree. On the other hand, an exceedingly high false positive rate 230 indicates that we are detecting events that are not actually occurring, which might lead to false 231 findings for research of the physical earthquake process; thus, it is also necessary to reduce the 232 false positive rate. 233 Next, we discuss the accuracy of trend and event determinations. Since the determination 234 accuracy cannot be discussed for thresholds with high false positive rates, we evaluated the case These results suggest that the occurrence time can be determined approximately ± 0.5 years 241 with a 90% probability in cases where the annual observation frequency is larger than 4 242 times/year and the event size is larger than 3σ. 243 The duration of the event was fixed to 1 year in this verification. The event duration is estimated 244 from two unknown parameters, the occurrence time and the end time of the event. Therefore, 245 the detection accuracy of the duration is always worse than that of the occurrence time. 246 247

Summary of detection capability 248
As concrete cases, we compared the cases of campaign GNSS-A and daily-GNSS (Table 2). 249 The annual observation frequency (f) is set to 4-6 times/year. The standard deviation of 250 8 horizontal positioning (σ) is set to 1.5 cm. In the case of daily-GNSS, the standard deviation of 251 positioning is better than 0.5 cm in the horizontal component (e.g., Nakagawa et al., 2009;Suito, 252 2016 On the other hand, GNSS-A needs longer observation period to detect the phenomena accurately. 255 Because the crustal deformation due to the actual interplate coupling is in the order of 256 centimeters, it is desirable to determine the crustal deformation rate with an accuracy of less 257 than 1 cm/year (95%CL). According to Fig. 2a (Fig. 6b). From these results, crustal deformations due to subseafloor slips of 299 about 1 m or less (corresponds to Mw 6.5 or less) in a range of about 100 km or more away from 300 the land area can be detected only on the seafloor observation array. When planning the future 301 expansion of the observation array, it is effective to consider these results and to install sites in 302 locations where the detection capability to detect smaller slip events can be improved. An array 303 with higher spatial density will allow us to analyze the location and physical process of slip 304 events more accurately. 305 306 4 Summary 307 We examined the signal and event detection capability of the current GNSS-A time series data 308 using statistical method. We arranged the detection capability of crustal velocity, size and timing 309 of seismic event in Table 2. In addition to the detection capability of the current low-frequency 310 and low-accuracy time series data, we also examined the detection capability for higher 311 frequency and accuracy, which may be realized in the future. By constructing a seafloor  One-side sample size S1 (in Fig. 1b) with respect to Δ and γ when S2 (in Fig. 1b) Table 1 Signals related to the earthquake cycle. 436 Table 2 Summary of the signal detection and determination capabilities of GNSS-A and daily-437 GNSS in 2021. 438