In this study, TEC is calculated using GEONET data operated by the Geospatial Information Authority of Japan, and the following assumptions are made in processing the data . First, we approximate the F region, which contains many more electrons than other regions in the ionosphere, as a thin layer at an altitude of 300 km, because the two effects of the chemical reactions and diffusion are balanced, and the electron density is maximized at an altitude of 300 km. According to the results of the analysis of of ionogram information on the day of the 2011 Tohoku-Oki earthquake, the electron density peak of the ionosphere was at an altitude of 306 km in the data from Kokubunji, which is the closest to the epicenter at 440 km. This result suggests that the assumption of a hypothetical thin ionosphere at an altitude of 300 km in our analysis is reasonable. Note that the analysis of was conducted for the 11 March 2011 earthquake, and a similar analysis is needed to determine whether setting the ionosphere to 300 km is the most appropriate assumption for other earthquake cases. The point where the line connecting a GNSS satellite and a receiver intersects with this approximated thin layer is called the ionospheric point (IP). The footprint of the IP to the surface is called the sub-ionospheric point (SIP).

Two radio signals from the GNSS satellites, 1575.42 and 1222.60 MHz, are transmitted to the GNSS receivers, and the propagation time of the radio signals depends on the electron density in the atmosphere. Therefore, TEC between the GNSS satellites and the GNSS receivers can be estimated from the phase delay of these two types of radio signals. This TEC, which is in the pathway between a satellite and a receiver, is called the slant TEC, noting that in general the line of sight to the satellite is not vertical. The slant TEC at the time of the earthquake is used as the reference value for the time series slant TEC data. The time series slant TEC is defined as the difference between the slant TEC and the reference TEC value for each satellite receiver pair.

For each time series slant TEC data, a quadratic fitting is performed by the ordinary least-squares method for data points from 30 min before to 7 min after the time of the earthquake to be consistent with the previous study . These fitting curves are assumed to represent the time series slant TEC data, as it would have been in the absence of the effect of TEC depression caused by acoustic waves induced by the tsunami because it takes almost 7 min for acoustic waves to reach the ionosphere.

Then, we calculate the difference between the fitting curves and the time series slant TEC for each case. By multiplying the time series differences by the cosine of the angle θ between the vertical upward direction and the straight line between the satellite and the receiver, we obtain ΔvTEC, which is the variation of the vertical component of the slant TEC time series data. The conceptual diagram of the description of the data processing so far is drawn in Fig. .

To apply a low-pass filter to this ΔvTEC, we take a 300 s backward-moving average to obtain the low-frequency components of ΔvTEC. Since a TIH is a hole formed by the decrease of TEC, we want to focus on the decrease of TEC in our analysis. For this reason, in the following sections, we use the data of which the low-frequency ΔvTEC is less than 1. The low-frequency ΔvTEC is hereinafter referred as TEC for the sake of simplicity.

Since the time resolution of the available data is the 30 s interval, we set the time of the 2011 Tohoku-Oki earthquake occurrence to 05:46:30 (UTC) even though the exact occurrence time is 05:46:18 (UTC) according to the Japan Meteorological Agency.

In addition, the data include outliers due to broken receivers. We detected them using the k-nearest neighbor algorithm and removed these outliers from the data to be analyzed. The general principle of the k-nearest neighbor algorithm is that for a given data point, k-nearest neighbor data sets are identified, and labels are assigned to these k-nearest neighbor data sets and the given data point. Generally, this method requires the definition of the metric for measuring the distance (similarity) between data points, and the number k needs to be chosen as well. Here, simple but effective choices are made: the Euclidean distance is used to measure the similarity, and k is defined as the square root of the number of data points in the targeting range.

We analyze data over the area of 10∘ of latitude and 10∘ of longitude centered at 38.297∘ N, 142.373∘ E, the location of the epicenter of the 2011 Tohoku-Oki earthquake as reported by the USGS. Gaussian process (GP) regression is a method of regressing a function Y (here TEC is a function of horizontal coordinates) using a flexible nonlinear model based on a set of observed data. A GP is in fact a generalization of the multivariate normal distribution to infinite dimensions: any marginal distribution projected to finite dimensions is multivariate normal. The fitted GP here probabilistically represents all possible TEC surfaces that interpolate (up to a so-called nugget noise level) the observations. We employ here the Matérn kernel with an additional nugget that accounts for some noise about the observations: 1kν(xp,xq)=21-νΓ(ν)2νrlνKν2νrl,2cov(yp,yq)=kν(xp,xq)+σ2δp,q.

Here, cov( ) means a covariance function, r=|xp-xq|, Kν is a modified Bessel function of the second kind, Γ is the Gamma function, ν and l are positive parameters, σ2 is the variance of the noise (i.e., the nugget), and δp,q=1 if p=q and 0 otherwise. The Matérn kernel's smoothness ν generates a GP whose smoothness relates to ν and should thus be carefully chosen to match the smoothness of the function Y. By setting ν=5/2, we use a kernel function that is twice differentiable, which, in our analysis, conforms very well to the physical phenomena of TEC reduction.

After fitting our GP, the joint distribution of the estimates at any new locations are estimated (with uncertainty) even in areas where there is no measurement data. Here we predict the TEC surface over the area in increments of 0.01∘ in both latitude and longitude. However, using 1200 receivers, it takes more than 10 min to fit the full data due to costs of O(n3); that is, the computational cost is proportional to the cubic of the number of data points n.

A stochastic partial differential equation (SPDE) approach using the integrated nested Laplace approximation (INLA) can reduce the cost of fitting the GP. Such an approach is not only faster but also has demonstrated that spatial predictions are more accurate, less uncertain, and more robust than the standard covariance-based fitting of a GP, e.g., when mapping stratospheric ozone . In other words, the estimated values provided by the method are closer to the true values and have less dispersion in the spatial interpolation, and the method is robust against the absence of measurements. Exploiting Gauss–Markov random fields (GMRFs), the INLA-SPDE reduces costs to O(n32) for two dimensions. The crucial point is that a Gaussian spatial process with a Matérn covariance function is the stationary solution to a certain SPDE that can be solved using finite-element approaches and approximated using the INLA in the GMRF setting. Nevertheless, some effort must be put into creating a reasonable mesh that solves the SPDE using finite elements, shown in Fig. .

The number of elements in the mesh cannot be too large, as the computational burden would become too high, and not too small, as the fitting surface would not be a good approximation of the actual surface.

Using this INLA-SPDE method with about 5200 mesh elements, the average computational time to fit the full data and predict the surface in 30 s increments from 05:30:00 to 06:30:00 becomes less than 1 min, with a standard deviation of less than 5 s, whereas the average computational time based on the standard GP regression method is more than 10 min, with a standard deviation of more than 2 min. The exact values are described in Table .

For the two data points that are determined to be outliers by our method, we validate them as outliers as follows. Figure a is the 3D plot of the captured data by satellites at 06:00:00, and the two red dots are the points identified as outliers. It is clear that these two outliers have very different values from their neighboring measurement points, but such spikes do not correspond to any genuine physical variations. These points will move further away from the surrounding points over time, and eventually, the absolute values of the two outliers reach over 50 TECU, which can distort the fitting surface considerably and prevent proper TIH analysis.

In addition, to validate further, we create a semi-variogram cloud. In the semi-variogram cloud, half the value of the squared difference between feature values of two data points is plotted against the difference in geographical space between the two data points. Figure b shows the semi-variogram cloud of the data at 06:00:00 (UTC), where the red points are associated with points considered outliers by our method. We can see that these two points present extremely different values when compared to the other measurement points. This corresponds to a clear lack of correlation between these two points and the rest of the data set.

Our method identifies the receivers that correspond to the outliers. In this case, these outliers are observed by the receiver 960588 and 950175, respectively. According to the Geospatial Information Authority of Japan, either the antennas of the receiver or the receiver itself was replaced by a new one in the year following the 2011 Tohoku-Oki earthquake.

In the analysis, it is inappropriate to include observations measured by receivers that would have been broken. Therefore, the exclusion of these outliers is essential to the TIH analysis, and hence all the analyses in this study are implemented after removing the outliers.

Figure  shows that measured TEC data, 3D plot of its fitting surface, and 2D mapping of the fitting surface and confidence interval (CI) for both full data and sparse data at 06:08:30 (UTC). Although Fig. a shows that the observed TEC decreases near the epicenter, the position where it decreases the most and the range of the depression cannot be described in detail due to the limited number of data points. In addition, the degree of TEC decrease in Fig. a and the 3D plot of the TEC depression represented by blue dots in Fig. c show that the TEC values gradually change from 0 in the area where the TEC values continue to be stable to the minimum value in the region where they are decreasing the most. These facts indicate that, in the evaluation of a TIH, it is necessary to estimate the TEC values where no captured data exist, in order to use a measure that can take the overall variation into account rather than using a single data point.

Here, we present our method's success in surface fitting with uncertainty as expressed in Fig. c, e and g. In Fig. c, the fitting surface is depicted with red colors. The fitted surface in Fig. c is an almost perfect fit to the TEC data observed by the satellites. We can thus confirm that the TEC values on the surface do not change linearly from the stable area to its minimum. This fact indicates that the TEC oscillation, which is thought to be the effect of high-frequency components , remains even after implementing the data pre-processing, which is a low-pass filter. Figure e, which is the 2D projection of the fitting surface, shows that our method enables us to estimate the TEC values with a fine granularity even over the region where the data are not detected by the GNSS satellites and receivers. The estimated values are computed and displayed in increments of 0.01∘ in latitude and longitude. The result shows that our method can capture the TIH anisotropic structure . In addition, the 2D projection result shows that the epicenter of the earthquake and the place where the minimum TEC value is obtained are explicitly different. It is also found that the shape and region of the TIH region are almost the same as those of the initial tsunami shown by the simulation .

Unlike the case of full data displayed in Fig. a, it can be clearly seen from Fig. b that the number of data points in the sparse-data case is not sufficient to entirely analyze the TIH, where 95 % of the GNSS receivers are randomly removed from the observed data. This situation is highly possible in areas where a dense network of GNSS satellites and receivers has not been installed, and hence the appropriate analysis for the TIH is almost impossible from the captured data. Nevertheless, our new analysis method can overcome this sparse-data problem effectively.

In Fig. d, the fitting surface and almost 5 % of captured data are depicted by a red surface and blue dots, respectively. Despite the extremely small number of data points, our method is successful in surface fitting and uncertainty evaluation.

A 2D projection of the fitting surface with sparse data and its 99 % CI, in Fig. f and h, shows that the location and range of the TIH are adequately estimated and consistent with those of the full-data case shown in Fig. e and g. To be more specific, our method is able to estimate the anisotropic TIH, which is consistent with the location and shape of the initial tsunami. The uncertainty naturally increases by comparing Fig. g with h, since the number of data points is smaller than that of the original data. These results demonstrate that our new method based on GP regression overcomes the sparse-data problem by implementing surface fitting that adequately estimated TEC variation with uncertainty and captured TIH shape.

In this study, surface fitting for the sparse data is performed using the data received by the remaining 5 % of receivers after randomly excluding 95 % of the GNSS receivers. With such a small number of receivers, in theory it could happen that none of the 5 % of randomly chosen receivers would observe the data points that exist in the specific TIH region of great importance for the detection of the tsunami.

In our analysis, the minimum number of GNSS receivers within the target area detecting data from the satellite between 05:46:30 and 06:16:30, which is the period of analysis, is 832 (at 05:46:30), and 40 receivers were selected at random as a conservative estimate of 5 % of that number. Consider the aforementioned case where only receivers that do not measure the TIH are randomly selected to make up these 40 units. For example, at 06:12:00, there are 66 receivers receiving data with a TEC value of -4 or less and 25 receivers receiving data with a TEC value of -5 or less. At 06:12:00, there are 917 receivers detecting data from the satellite in the target area, and the probability that at least 1 of the 40 receivers randomly selected from these receivers contains 1 of the 66 receivers that receive -4 or less TEC value is around 95 %. Also, the probability that at least 1 of these 40 receivers contains 1 of the 25 receivers that receive -5 or less is approximately 70 %. Therefore, a random selection of receivers that cannot measure the TIH can happen, though the probability of that is very low. In this experiment, we analyze the usefulness of surface fitting in the sparse case, where there are receivers that measure the TIH.

The specific details of the situation of this experiment are described below: 10 experiments were conducted to randomly select 5 % of receivers, and the minimum observed values measured in each case are shown in Table . With a rare exception, the receiver number by which the minimum value is measured in each case is different. For example, at 06:12:00, in the case of random 9, the minimum value is about 0.8 TECU larger than the case with all data. Also, at 06:16:00, the minimum value of the full data is -5.19, but there is only one case, random 5, where the minimum value is less than -5.

As described above, we can see that the minimum value and distribution of TEC are different when we randomly choose receivers. And the following Fig.  shows that our fitting method works for such different sparse distributions of TEC. In the left half of Fig. , three sets of randomly chosen sparse data are drawn, indicating that the data points are distributed differently. On the right-hand side of Fig. , the 2D projections of the fitting surface for these sparse data are displayed, and it can be seen that the shape of the TIH is almost the same, though not exactly the same. Specifically, Fig. b2 shows a slightly wider TIH form than Fig. a2, and Fig. c2 has a TIH that is evenly spread in the east–west direction compared to Fig. a2 and b2. These results show that if we can measure a few observation points with reduced TEC, we can apply our fitting method to capture a TIH even if there are far fewer data points relative to the original data.

How the calculation of the TIH volume, i.e., the volume of the area where the value of TEC is less than 0 in the target area, is affected in this situation is analyzed in Fig. . The volume of the TIH is used as a measure to determine the effect. It can be seen from Fig.  that in each case, there are very few measurement points with a value of -4 or lower, and even the number of measurement points with a value of -3 or less is fewer than 10. However, when compared to the volumes calculated using all the data represented by the horizontally lines, it can be seen that across all cases the values are very close to one another, at each time of the three selected times. Furthermore, although the number of data points below -2 varies, the computed volumes are almost similar. Obviously, if there are no data points in the TIH area, volume calculation itself is impossible but is highly unlikely as we explained above. However, if data points shown in the figure are measured in the TIH, volume calculation is possible. Since a single receiver can receive data from multiple satellites, the number of receivers needed to receive data in the TIH is extremely small.

Since we are able to estimate all the TEC variations in the target area, we analyze the shape of the TIH in detail using the observed data. Figure  displays the distribution of TEC below different levels at different times. Specifically, in Fig. a1 to a3, the shapes of the TIH limited to TEC values of -2 or less are drawn at 06:08:00, 06:12:00, and 06:16:00 (UTC). Similarly, in Fig. b1 to b3 and c1 to c3, the shapes of TIHs with TEC values of -3 or less and -4 or less, respectively, are drawn. The tsunami source was set to 38∘ N, 143.4∘ E, which was calculated in a previous study as the average of tsunami sources obtained from previous research papers. According to , the initial tsunami reproduces the TIH, and the TIH is spread out over the initial tsunami. Then, use the TEC values around the tsunami source area to derive the relationship between the TIH and the initial tsunami. Based on these previous studies, using the observed data, the expansion of the TIH around the tsunami source was analyzed as follows.

As shown in Fig. a1 to a3, the region with TEC less than -2 can be seen to expand less to the northern direction than the other directions. In addition, a TEC decrease isolated from the TIH directly above the tsunami can be seen in the southwestern direction. As an overall trend, it can be said that the expansion of the TIH in the eastern and southwestern directions is more pronounced than in the other directions.

However, when we observed the behavior of Fig.  b1 to b3, where the value of TEC is less than -3, we found that the expansion of the TIH is different from that of the TIH with a TEC value less than -2 described in Fig. a1 to a3. First of all, the area of the TIH has not expanded as much as that of the TIH with TEC less than -2. In addition, the southwestward expansion is not as large as that of the TIH in Fig. a1 to a3. For example, the eastward expansion appears to be the most pronounced, and also the northward expansion is less pronounced than in other directions, which are similar characteristics.

In the case of the TIH with TEC less than -4, shown in Fig. c1 to c3, the westward expansion is smaller than the expansion in the other directions. In addition, the TIH center does not remain directly above the vicinity of the tsunami source but appears to be moving to the southeast. It can also be seen that the region where TEC is less than or equal to -4 does not expand significantly during the period of 20 to 30 min after the earthquake, but it does not shrink significantly either. In other words, when we focus on the TEC values below -4, the shape of TEC is almost stable.

From Fig. , it can be inferred that the time evolution of the shape of the region with small TEC variation and that with large TEC variation do not necessarily coincide. To investigate this point in more detail, we analyze the graphs with the distance in kilometers plotted by time for the most expanded points in the eight directions from the tsunami source. Here, the distance is computed by Hubeny's distance formula .

Here, the location of the tsunami source is the same value used in , and its coordinates are calculated by referring to , , , and . The eight directions are evenly divided into north, northwest, west, southwest, south, southeast, east, and northeast from its tsunami center position.

In Fig. , the distance between the most expanded point in the eight directions and the tsunami source when the TEC value is -2 or less (Fig. a), -3 or less (Fig. b), and -4 or less (Fig. c) are drawn, respectively. The eight directions and the color coding for each are the same as in Fig. .

In Fig. a, it can be seen that the expansion in the northeastern direction progresses earlier than other directions. A little later, the eastward expansion continues to progress, and finally the eastward expansion progresses more than the northeastward expansion. The expansion to the south, after increasing for some reason, begins to decrease and then continues to expand again. Eventually, it will be as large as the eastward expansion. The westward expansion is a discontinuous movement due to the definition of distance and direction in this analysis. Specifically, as depicted in Fig. a2 and a3, the region with TEC less than -2 has a special shape in the western direction, and the distance in this direction tends to be more discontinuous than the distances in other directions. What is noteworthy is the expansion of the distance in the northern and northwestern directions. The progress of the distances in these directions is clearly smaller than those in other directions. For example, compared to the expansion in the eastern direction, the distance in the northern and northwestern directions is less than half. Also, it is worth mentioning the TIH withdrawal (05:59:30–06:01:00) in the southward direction. Due to the influence of the geomagnetic field, the plasma tends to move more to the south. In addition, the backward-moving average frequency filter applied in this study does not completely exclude the high-frequency component. For these reasons, we detected a decrease in the electron density as a result of the recombination caused by the oscillation of the plasma due to the high-frequency component on the southern side. This temporary decrease in electron density is caused by a different mechanism than the TIH formation, which is caused by the low-frequency component.

In Fig. b, the time evolution of the distances in all directions appears to be continuous. In this case, too, the expansion toward the northeast progresses in the initial stage, but then the expansion toward the east progresses significantly. The expansion in the southern direction can be said to be slower than that of the two directions mentioned above, but it eventually expands by the same distance as that in the eastern direction. The southeastern direction shows a trend similar to the expansion in the southern direction, and although the initial expansion is not so large, the final value is large. It can be seen that the northern and southwestern directions do not expand as much even at the beginning and then remain in a stable state with almost the same value over time. The northwestern direction, as with Fig. a, is one of the directions that does not expand the most. The most significant difference between Fig. b and a is the expansion in the western direction. In Fig. a, the distance in the western direction expands discontinuously and finally reaches the highest value. On the other hand, in Fig. b, the distance in the western direction is the smallest in almost all time periods.

In Fig. c, the time at which the image begins to expand is greatly delayed because the threshold is set even smaller than in Fig. a and b. Initially, the expansion in the northwestern and southern directions is significant, and as time passes, the expansion in the eastern direction becomes the largest. Eventually, the distance in the southeastern direction becomes larger than the distances in the other directions at the end of the period, but its maximum value is smaller than the value recorded by the eastern direction around 06:10:30 (UTC). The expansions in the northern and southwestern directions show almost the same development. For the northwestern and western directions, as in Fig. b, only the smallest expansion is shown during the period.

Unlike the high-frequency component of the TEC variation , the low-frequency component of the TEC variation displays large drops in its values and remains fixed within a region, as shown in Fig. . Since this TEC reduction is caused by the initial tsunami, staying in the same location is theoretically correct. As shown in Fig. , the initial tsunami estimated by using inversion analysis mostly overlaps with the TIH where the TEC reduction is large. Although there have been studies to estimate the tsunami source using TEC fluctuation data, these studies estimate the tsunami source as a point . We show here for the first time that our method of estimating the entire TIH by GP regression can be used to estimate the initial-tsunami region as shown in Fig. . In the case of TEC values less than -3 shown in Fig. a1–a3, the TIH almost overlaps the initial-tsunami areas; in other words, the TIH with TEC values less than -3 is located in the region which is almost the same as the initial-tsunami region, while Fig. b1–b3 with TEC values less than -4 shows that the TIH stays within the initial-tsunami region.

Figure  shows the uncertainty in estimating the values of TEC using the full data. In this figure, the uncertainty is defined as 3 times the standard deviation. In general, interpolation between data points can be performed with a small uncertainty, while extrapolation has a larger uncertainty. Even in the case of interpolation, if the data points are sparse, the uncertainty will be large. Therefore, it can be seen from Fig.  that the uncertainty is larger in areas where the measurement points are sparse or where extrapolation is performed, as can be seen by comparing with Fig. .

In Fig. , the initial-tsunami and TIH regions overlap, but at 06:12:00 and 06:16:00, the TIH region extends more toward the eastern direction of the Pacific Ocean than the initial-tsunami region. Specifically, this tendency is observed in the area east of the initial-tsunami area and west of the dotted line at 145.37∘ E. In addition, the TIH region at 06:12:00 appears to be wider eastward of the Japan Trench than the TIH region at 06:16:00. In this regard, for example, looking at the uncertainty values for each region listed in Fig. , at 06:12:00, the uncertainty for the region east of the initial-tsunami region and west of the dotted line at 145.37∘ E is slightly larger. However, it is unlikely that this was the reason for capturing the non-overlapping phenomenon of the initial-tsunami and TIH regions.

show that acoustic waves propagate upward, which are gradually refracted, and their effects propagate horizontally in the ionosphere. According to this principle, the TIH is expected to spread evenly in the east–west direction of the tsunami generation area. , who analyzed the measured data, showed that slant TEC decreased in the area east of the tsunami generation area after the 2011 Tohoku-Oki earthquake. The reason why the TIH calculated by our method appears to be extended to the east of the Japan Trench when compared to the initial-tsunami area is that the initial-tsunami height is highest on the Japan Trench. The acoustic waves from the highest region on the trench propagate into the atmosphere and affect the neutral atmosphere evenly in the east–west direction, resulting in the recombination of ions and electrons, causing the TIH to spread in the east–west direction around the trench. Therefore, we believe that the estimation by our method correctly captures the variation of the electron density. In future reverse calculations of the initial-tsunami area and height based on TIH information, it is necessary to take into account that the area with the highest initial tsunami has a large impact on TIH formation.

Also, the initial tsunamis estimated by other researchers show that the initial-tsunami regions overlap the TIH region as in Fig. , where these initial tsunamis are roughly estimated by looking at them. In , they used an algorithm developed by to compute the initial sea surface displacement based on their fault-determination procedure. On the other hand, in , they investigated the effect of the rupture process on a tsunami source inversion. Estimated sea surface elevation of a tsunami source by their inversion method based on the assumption of finite rupture velocity of 2 kms-1 is shown. These results indicate that our TIH estimates overlap not only with the initial tsunami estimated by but also with the initial tsunami determined by other researchers.

In this section, the relationship between the TIH and the initial tsunami is analyzed in more detail with the help of data provided by Tatsuhiko Saito, the first author of . Based on the assumption that the TIH is triggered by tsunamis, it is expected that the locations of high tsunamis and high TEC reduction in the TIH are in close proximity to each other.

Figure a1 to a3 show the contour lines of the initial-tsunami height and the overlaid TIH that is plotted at three different time points. From these figures, it can be seen that the region with the largest decrease in TEC and the region with the highest initial-tsunami wave height are very close to each other. Furthermore, the decrease in TEC appears to correspond with the increase in the initial-tsunami height.

To make a more precise comparison in this regard, graphs comparing the initial tsunami and TIH at the latitude and longitude where the initial-tsunami height is the highest are illustrated in the second and third rows of Fig. , respectively. From the figures shown in the second line, it can be seen that the longitude with the largest decrease in TEC and the longitude with the highest initial-tsunami height roughly match. On the western side of the location where the initial tsunami is highest, the area where the tsunami height starts to increase and the area where TEC starts to decrease appear to be almost the same, but this is not necessarily the case on the eastern side.

From the figures shown in the third row, the latitudes with the largest decrease in TEC and the highest initial-tsunami height roughly matched, and unlike the east–west asymmetry when the latitude is fixed, the TEC decrease is generally symmetrical in the north–south direction. This symmetry seems to correspond to the shape of the initial tsunami.

Furthermore, the time series variation of TEC from 05:46:30 to 06:16:30 (UTC) is shown in Fig. , when fixed at the latitude and longitude with the highest initial-tsunami height, respectively. Figure a shows that the TEC decrease is the largest in the region where the initial-tsunami height is the highest when we observe the time series variation of TEC for 30 min after the earthquake. However, in the process of TEC decrease, it is also observed that there is a time period when TEC decreases the most in a slightly western region. In more detail, we can see that the area where TEC is decreasing the most is slightly moving from the west to the east and expanding with a larger decrease. In addition, there is an asymmetry of TEC decrease in the east–west direction.

On the other hand, from Fig. b, the TEC decrease appears to be symmetrical in the north–south direction when the longitude is fixed at the position where the initial tsunami is highest. However, by checking the details of the time series change of TEC decrease, we can see that the position where the TEC decreases the most is moving from north to south, and the decrease is expanding.

Figure  shows the time series of the TIH volume, which is computed by the trapezoidal quadrature method for the region where TEC estimated by surface fitting has a negative value. In other words, the volume between the flat surface, where the TEC values are equal to 0, and the fitting surface, which has a negative value, is calculated.

The main effect by acoustic waves induced by the initial tsunami is the reduction of TEC in the ionosphere by moving the plasma along the magnetic field and causing recombination. More specifically, although there are regions where TEC increases due to complex physical mechanisms, the magnitude of the initial tsunami can be assessed by focusing on the decrease in TEC. Therefore, the volume of the region with a negative TEC value is considered to be related to the magnitude of the initial tsunami.

In Fig. a, surface fitting is implemented with simple GP regression, while the INLA-SPDE method is applied to the GP regression process in Fig. b. The solid lines in Fig. a and b display the TIH volumes computed for the full data, and the dashed lines are for the sparse data. In the case of the sparse data, we repeat 10 times the random selection of 5 % of the receivers and the GP regression to fit the surfaces and then calculate the average value using each computed volume. This iterative process is intended to exclude a possible influence of the random seed used in the sparse-data selection on the results. As shown in Fig. , the measurement points are moving and not uniformly distributed in the targeting range. Still, the time series of the computed TIH volumes looks continuous, as shown in Fig. a and b.

The volume of the TIH begins to increase almost 10 min after the earthquake occurrence and continues to increase until about 28 min after the earthquake shown in Fig. a and b. More specifically, the reason the volume of the TIH continues to increase until 28 min after the earthquake is that the TIH is formed by the extremely low-frequency component of acoustic waves that travel from the sea surface to the ionosphere. The recombination of plasma and electrons caused by this low-frequency component is the direct cause of the TIH, and this recombination takes time, so the volume continues to increase over a long period of time. Simulations of TIH formation using a physical model presented by also show that the TIH forms over a similar time period. The past study of actual post-earthquake TEC variation data has confirmed that the percentages of TEC depressions for huge simulated tsunamis reach almost 40 times as large as those for smaller simulated tsunamis . In this context, a huge simulated tsunami means a tsunami with a simulated initial-tsunami height of around 5 m, and a small simulated tsunami means a tsunami with a simulated initial-tsunami height of 1 m or less in their study. Although it is impossible to understand the time evolution of TIH volume in advance, since no study has considered and analyzed the entire TEC variation in the targeting range under the extremely complex TIH mechanism and our new method is applied to only one tsunami case, it is a sufficiently conservative estimate that the threshold is set to about 10 %–20 % of the maximum TIH volume according to the aforementioned previous study. Therefore, if the volume exceeds this threshold, it is reasonable to draw the conclusion that a huge tsunami is being generated. Though it is impossible to describe what the definition of a huge tsunami in an explicit and strict manner is, since there is no general definition of a huge tsunami, a huge tsunami in this context means a tsunami with the same magnitude as the 2011 Tohoku-Oki earthquake. By convention, tsunamis triggered by a magnitude of about nine earthquakes are considered to be huge tsunamis. Also, in this analysis, a provisional threshold is set to 200 000 TECUkm2.

In the case of the full data, both panels show that the volumes calculated from the fitting surface (but not accounting for uncertainties in the approximation) reach the threshold 1 and 2 min earlier, respectively, than the volumes of 80 % and 99 % CI. Similarly, in the case of the sparse data, the time difference is about 2 and 4 min, respectively, to reach the threshold for the volumes computed from the fitting surface and both CIs. It means that thanks to our uncertainty computations, making sure that a warning is at a high level of confidence, based on data, of either 80 % or 99 % results in delays for advisories of only, respectively, 1–2 or 2–4 min.

The warning system based on this method is highly feasible because surface fitting and the estimation of the TEC values for the full data can be processed in less than 1 min based on the INLA-SPDE method. However, in the case of the sparse-data fitting, our implementation of the INLA-SPDE method sometimes fails due to the geometric meshing optimized for larger data sets, where the benefit of this method is. Nevertheless, the robustness and feasibility of this method never deteriorate because it is possible to compute the surface and estimated values in less than 10 s for the sparse-data case based on the standard GP regression method. Our method is the first to demonstrate that we can calculate the volume of TIHs accurately in real time and use it as a measure of TIHs even when only a limited number of measurement points are available.