We develop a new method to analyze the total electron content (TEC) depression in the ionosphere after a tsunami occurrence. We employ Gaussian process regression to accurately estimate the TEC disturbance every 30

The damage caused by tsunamis can be devastating. For example, almost 20 000 people died in the tsunami following the 2011 Tohoku-Oki earthquake in Japan. One reason for such levels of casualties is that current tsunami height predictions are relatively unreliable, even following an identified earthquake event, and so early warning systems are not as effective as required. Initial sea surface deformations are typically indirectly determined from seismological inversions of the earthquake source. However, some of these early estimates are sometimes much lower than expected: for instance the initially estimated value for the 2011 Tohoku-Oki earthquake of

Research on tsunami warnings has been conducted for a long time and has undergone remarkable technical evolution with the development of various
technologies

Furthermore, the initial-tsunami wave cannot be precisely inferred from seismic information alone due to the complexity of the relationship between
the earthquake source and the initial wave. For example so-called tsunami earthquakes generate much larger tsunamis than expected from the seismic
source, e.g., the 2010 Mentawai tsunami

A path towards accurate warnings is to estimate the tsunami ionospheric holes (TIHs) generated in the ionosphere after the initial-tsunami occurrence

In Japan, the GPS Earth Observation Network System (GEONET), which is a network of more than 1200 receivers, enables us to observe the behavior of TEC in the ionosphere with a large number of data points. The most prominent case of the TEC changes in the ionosphere observed by GEONET is the tsunami following the 2011 Tohoku-Oki earthquake. By focusing on the changes in the ionosphere after the earthquake and observing the high-frequency component of the TEC fluctuations,

Furthermore,

To overcome these problems, we implement below a statistical method for the analysis of TEC using satellite data, which allows us to estimate TEC values even over areas with no measurements and to evaluate the whole TIH even without a dense measurement network such as GEONET in Japan. Our approach does not make any assumptions about the nature of the source of the tsunami. This method enables us to calculate the volume (with uncertainty) of the hole as an assessment of the scale of the TIH, and we propose to use its volume as a measure of the TIH. We believe that estimating the TIH provides a new and important tool for tsunami early warning systems that is independent of seismology.

In Sect.

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

Two radio signals from the GNSS satellites, 1575.42 and 1222.60

For each time series slant TEC data, a quadratic fitting is performed by the ordinary least-squares method for data points from 30

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

The schematic image of TEC depression detected by a satellite and a receiver.

To apply a low-pass filter to this

Since the time resolution of the available data is the 30

In addition, the data include outliers due to broken receivers. We detected them using the

Panel

We analyze data over the area of 10

Here, cov( ) means a covariance function,

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

A stochastic partial differential equation (SPDE) approach using the integrated nested Laplace approximation (INLA)

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

Computational time for the TEC surface fitting and the TEC value estimation.

The red star is the epicenter, and the two large black circles are the outliers. Panels

Figure

For the two data points that are determined to be outliers by our method, we validate them as outliers as follows. Figure

Panel

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

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.

Left-hand side is for the full data; right-hand side is for the sparse data using only 5 % of the GEONET receivers. Panels

Figure

Here, we present our method's success in surface fitting with uncertainty as expressed in Fig.

Unlike the case of full data displayed in Fig.

In Fig.

A 2D projection of the fitting surface with sparse data and its 99 % CI, in Fig.

Table of the minimum TEC values and the receiver numbers that observed them at three different times.

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

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

Three different sparse-data distributions, random 0, random 3, and random 8, and these fitting surfaces mapped in two dimensions at 06:12:00 (UTC). The black dashed line indicates the location of the axis of the Japan Trench.

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.

Comparison of the volume of the TIH calculated by sparse data (40 receivers) and the volume calculated using all data. Random choices are independently implemented 10 times. Points with square marks indicate the number of data points with a TEC value of

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.

Expansion of the tsunami ionospheric hole. Panels

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

As shown in Fig.

However, when we observed the behavior of Fig.

In the case of the TIH with TEC less than

From Fig.

Here, the location of the tsunami source is the same value used in

Expansion of the tsunami ionospheric hole. Panel

In Fig.

In Fig.

In Fig.

In Fig.

The initial tsunami and TIHs with TEC values less than

Unlike the high-frequency component of the TEC variation

Uncertainty of the estimated TEC values at 06:08:00, 06:12:00, and 06:16:00. The uncertainty in this case is defined as 3 times the standard deviation. The area surrounded by the blue line is the simulated initial tsunami by inversion analysis with 130 small basis functions implemented by

Figure

In Fig.

TIH and the initial-tsunami comparison. In the first row, the initial tsunami estimated by

Also, the initial tsunamis estimated 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

Overlap of initial tsunami and TEC. Panels

Initial-tsunami height and time series variation of TEC from 05:46:30 to 06:16:30 (UTC). Panel

Figure

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.

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.

The time series of TIH volume for full data and sparse data with a one-sided CI. The red solid line is the volume calculated with a fitting surface for full data. The solid yellow line and the solid blue line are a one-sided 80 % CI and 99 % CI, respectively. The dashed lines are for sparse data with only 5 % of receivers. The horizontal black line is a provisional threshold. Panel

Furthermore, the time series variation of TEC from 05:46:30 to 06:16:30 (UTC) is shown in Fig.

On the other hand, from Fig.

Figure

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.

The volume of the TIH begins to increase almost 10

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

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

In this paper, we compute the volume of the ionospheric depression generated by a tsunami, in real time and with enough confidence to issue warnings. The surface fits the TEC data using a Gaussian process regression after removing outliers. It enables us to estimate the TEC values over the entire target area. Furthermore, uncertainty can be properly evaluated for the estimated values of TEC according to the density of observations.

The TIH captured by our method is located east of the epicenter. This is consistent with the initial tsunami estimated by other research groups being
east of the epicenter

As shown in our results, this new method is robust, as it works in situations where measurements are not uniformly distributed and moving and TIHs display
anisotropy and even if the number of observed data points is sparse. Since our estimates of the shape of the anisotropic TIHs reflect the signature
of the initial-tsunami wave, we demonstrate that using one specific data point such as the minimum observed value as a scale of a TIH

In addition, although there have been papers referring to a TIH based on observational data

As for the high-frequency component of the TEC variability, past studies

We also believe that our method can estimate the initial tsunami independently from previous methods. Previous methods include, for example, the method of estimating a tsunami from the source and magnitude of an earthquake. By detecting seismic waves at multiple observation points, it is possible to calculate the approximate location of the earthquake and to some extent the exact size and magnitude of the earthquake within 2 min. Based on this information, the initial tsunami can be estimated.

In addition, a method has been developed to calculate the initial shape of a tsunami in reverse by calculating the shift of vessel speed due to the passage of a tsunami from the data of the automatic identification system (AIS), which is required to be installed on ships sailing in the area. This method is expected to be used to predict the initial tsunami around the world. However, some problems exist, such as the fact that the exact magnitude of a huge earthquake with a magnitude greater than 8 is not immediately known, the fault displacement (estimated from seismic waves) does not always match the initial sea level change, and the initial sea level change cannot be known.

In addition to the above methods, Japan has developed a system called the REal-time GEONET Analysis system for Rapid Deformation monitoring (REGARD),
which analyzes GEONET data in real time and extracts crustal deformation during earthquakes to automatically estimate fault models and earthquake
scale within 3

Therefore, in addition to the various existing initial-tsunami estimation methods mentioned above, if the initial-tsunami shape and height estimation
based on our developed TIH estimation can be realized in the future, it is expected that the combination of these methods will enable us to realize
an even more accurate estimation of initial-tsunami height and range. From this point of view, even if it takes about 20

Furthermore, the existence of the second and third waves can be estimated by looking at the time variation of the TIH. Since the presence of large second and third waves affects the shape of the TIH, it can be used to determine whether the tsunami warning should be maintained or canceled after it is issued. In fact, after the 2011 earthquake in Japan, it took 1.5 d for the tsunami warning to be lifted. This is an advantage of our method, even if it takes time to obtain useful information.

As for the method to obtain the initial-tsunami information by calculating an inverse from the TIH information, we expect that the combination of the
TIH estimated by our method and the acoustic wave propagation model may be able to estimate the initial-tsunami area. At present, although

GPS data were provided by the Geospatial Information Authority of Japan. Currently, the data can be purchased from the Japan Association of Surveyors at

The supplement related to this article is available online at:

RK conceptualized the project and methodology, curated the data, performed the formal analysis and investigation, created the visualizations, and wrote the original draft of the paper. MK acquired project funding, curated the data, and worked with the software. TN acquired project funding, curated the data, worked with the software, and reviewed and edited the paper. AS supervised the project, conceptualized the methodology, and reviewed and edited the paper. SG conceptualized and supervised the project, acquired project funding, performed the investigation, and reviewed and edited the paper.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors acknowledge the use of the UCL Myriad High Performance Computing Facility (Myriad@UCL), as well as associated support services, in the completion of this work. Ryuichi Kanai is supported by the Japan Student Services Organization. This research was partly supported by the Ministry of Education, Culture, Sports, Science and Technology through a Grant-in-Aid for Scientific Research (B, no. 17H02058, 2017–2020, Masashi Kamogawa) and Earthquake Research Institute (University of Tokyo) cooperative research program (Masashi Kamogawa and Ryuichi Kanai). Serge Guillas was supported by the projects “Uncertainty Quantification of complex computer models. Applications to tsunami and climate” (EPSRC grant no. EP/N510129/1) and “Real-time Advanced Data assimilation for Digital Simulation of Numerical Twins on HPC” (EPSRC grant no. EP/T001569/1) of The Alan Turing Institute. Masashi Kamogawa was supported by Adaptable and Seamless Technology transfer Program through Target-driven R & D (A-STEP) from Japan Science and Technology Agency (JST) (grant no. JPMJTM19YG). The authors are grateful to Tatsuhiko Saito, the first author of

This research has been supported by the Ministry of Education, Culture, Sports, Science and Technology (grant no. 17H02058); the Engineering and Physical Sciences Research Council (grant nos. EP/N510129/1 and EP/T001569/1); Tokai University (grant no. IORD2018-01); and the Adaptable and Seamless Technology Transfer Program through Target-driven R&D (grant no. JPMJTM19YG).

This paper was edited by Maria Ana Baptista and reviewed by two anonymous referees.