Geomagnetic Disturbances During the Maule (2010) Tsunami Detected by Four Spatiotemporal Methods

Abstract

Separating tsunamigenic variations in geomagnetic field measurements in the presence of more dominant magnetic variations by magnetospheric and ionospheric currents is a challenging task. The purpose of this article is to survey the tsunamigenic variations in the vertical component (Z) and the horizontal component (H) of the geomagnetic field using four spatiotemporal methods. Spatiotemporal analysis has shown enormous potential and efficiency in retrieving tsunamigenic contributions from geomagnetic field measurements. We select the Maule (2010) tsunami event on the west coast of Chile and examine the geomagnetic measurements from 13 ground magnetometers scattered in the Pacific Ocean covering a wide area from Chile, crossing the Pacific Ocean to Japan. The tsunamigenic magnetic disturbances are possibly due to two types of contributions, one arising from direct ocean motion and the other from atmospheric motion, both associated with tsunami forcing. Moreover, even though the tsunami waves decrease considerably with increasing epicentral distance, the tsunamigenic contributions are retrieved from a magnetic observatory in Australia ($\sim$ 13,000 km radial distance from the epicenter). These results suggest that various types of tsunamigenic disturbances can be identified well from the integrated analysis framework presented in this work.

Appendix: Mathematical Tools

1.1 Continuous Wavelet Transform

A wavelet is an oscillating function of time which has finite energy, and is very suitable for analysis of nonstationary data. This class of functions is well localized in both time and frequency, which allows one to perform simultaneous time and frequency analysis (Morlet et al. 1982a, b; Grossmann and Morlet 1984). Mathematically, a wavelet is a function $\psi$ in the Hilbert Space $L^2({\mathbb {R}})$ that satisfies the admissibility condition, that is,

$$\begin{aligned} C_{\psi } = \int _{0}^{\infty } \dfrac{|{\hat{\psi }}(\omega )|^{2}}{\omega } {\text{d}}\omega < +\infty , \end{aligned}$$

(1)

where ${\hat{\psi }}$ denotes the Fourier transform of $\psi$. To guarantee that the integral in (1) is finite, $\psi$ must have zero average (${\hat{\psi }}(0) = 0$) and ${\hat{\psi }}$ must be a continuously differentiable function (Mallat 2008). The zero average requirement implies that a wavelet must oscillate in such a way that the net area below the curve is zero. There are several types of wavelets that are used for data analysis, and the choice of wavelet depends on the purpose of the analysis. Wavelets equal to the second derivative of a Gaussian occur frequently in applications and are called Mexican hats (Mallat 2008), defined as

$$\begin{aligned} \psi (t) = \dfrac{2}{\pi ^{1/4}\sqrt{3\sigma }}\left( \dfrac{t^2}{\sigma ^2} - 1\right) \exp {\left( \dfrac{-t^2}{2\sigma ^2}\right) }, \end{aligned}$$

(2)

where $\sigma > 0$. Another type of wavelet that appears frequently is the complex Morlet wavelet, commonly defined as

$$\begin{aligned} \psi (t) = \dfrac{1}{\pi ^{1/4}}e^{i\omega _{0}t}e^{-t^2/2}, \quad \omega _{0} = 2\pi f_{0}, \end{aligned}$$

(3)

where $f_{0}$ is the center frequency of the wavelet. As can be seen from Eq. (3), the Morlet wavelet consists of a complex wave within a Gaussian envelope that has unit standard deviation. The factor $\pi ^{-1/4}$ ensures that the wavelet has unitary energy.

There are two types of operations that one can do over wavelets: translation and dilation. For the first one, the central position of the wavelet along the time axis is shifted, whereas for the second, the wavelet is compressed or dilated. Strictly speaking, if $\psi$ denotes a wavelet, then it will be called the mother wavelet, and the family of functions $\psi _{a,b}$ spanned by those operations is known as daughter wavelets and is given by

$$\begin{aligned} \psi _{a,b}(t) = \dfrac{1}{\sqrt{a}}\psi \left( \dfrac{t - b}{a}\right) , \end{aligned}$$

(4)

where $a > 0$ and $b \in {\mathbb {R}}$ are called dilation and translation parameters, respectively. It is important to note that the factor $1/\sqrt{a}$ ensures that all daughter wavelets have the same energy. Thus, relative to every wavelet $\psi$, the continuous wavelet transform (CWT) of a signal f for an instant of time b and scale a is defined by

$$\begin{aligned} \begin{aligned} Wf(a,b)&= \int _{-\infty }^{\infty }f(t)\dfrac{1}{\sqrt{a}}\psi ^{*}\left( \dfrac{t-b}{a}\right) {\text{d}}t\\&= \int _{-\infty }^{\infty }f(t)\psi _{a,b}^{*}(t){\text{d}}t \\&= \langle f, \psi _{a,b}\rangle \end{aligned} \end{aligned}$$

(5)

where $\langle \cdot , \cdot \rangle$ denotes the usual inner product of $L^{2}({\mathbb {R}})$. Equation (5) can be thought of as an analysis equation, since it represents the projection Wf(a, b) of f over a daughter wavelet centered at b with scale a. Since the projections are computed to every value of b and a, at the end of the process one obtains a two-dimensional transform plane with all the projections or wavelet coefficients, which represents the decomposition of the signal f over a family of daughter wavelets. Each of these coefficients has an energy which is given by

$$\begin{aligned} E(a, b) = |Wf(a,b)|^2, \end{aligned}$$

(6)

and the plot of E(a, b) is called a scalogram. The CWT preserves the total energy of a signal f and can be recovered by integrating (6) for all values of a and b:

$$\begin{aligned} E = \dfrac{1}{C_{\psi }}\int _{0}^{\infty }\int _{-\infty }^{\infty }|Wf(a,b)|^2{\text{d}}b\dfrac{{\text{d}}a}{a^2}. \end{aligned}$$

(7)

The main feature of the CWT is time–frequency localization (Soman et al. 2004). So, to obtain a time–frequency plane it is necessary to convert the scales to so-called pseudo-frequencies, which can be done using the equation

$$\begin{aligned} f = \dfrac{f_{c}}{a}, \end{aligned}$$

(8)

where $f_{c}$ is the center frequency of the mother wavelet ($a = 1$) defined by

$$\begin{aligned} f_{c} = \dfrac{1}{2\pi }\sqrt{\dfrac{\int _{0}^{\infty }\omega ^{2}|{\hat{\psi }}(\omega )|^{2}{\text{d}}\omega }{\int _{0}^{\infty }|{\hat{\psi }}(\omega )|^2{\text{d}}\omega }}, \end{aligned}$$

(9)

as discussed in Addison (2002). Specifically, the CWT measures the local matching between a signal f and the daughter wavelet at a particular instant of time b and scale a. A large coefficient value means that f and $\psi _{a,b}$ correlate well in the vicinity of b, whereas a low value indicates the opposite. Similarly to the Fourier transform, from the wavelet coefficients one can recover the original signal f with the synthesis equation defined as

$$\begin{aligned} f(t) = \dfrac{1}{C_{\psi }}\int _{0}^{\infty }\int _{-\infty }^{\infty }Wf(a,b)\psi _{a,b}(t){\text{d}}b \dfrac{{\text{d}}a}{a^{2}}, \end{aligned}$$

(10)

which is also called inverse continuous wavelet transform (ICWT) (Mallat 2008).

1.2 Discrete Wavelet Transform

In this work, we use the Daubechies (db2) wavelet function of order 2, and therefore the nonzero low filter values are $h \approxeq \left[ \frac{1+\sqrt{3}}{4\sqrt{2}}, \frac{3+\sqrt{3}}{4\sqrt{2}}, \frac{3-\sqrt{3}}{4\sqrt{2}}, \frac{1-\sqrt{3}}{4\sqrt{2}} \right]$, and the high-pass are $g=(-1)^{k+1}h(1-k)$. Also, we only study the discrete scales $j=1,2,\; {\text{and}}, \,3$ which are associated with the center frequencies of 3, 6, and 12 min, respectively. These center frequency values are obtained due to our analysis wavelet choice and to the data sample rate of 1-min time resolution; see additional details in Klausner et al. (2014b, 2016a, b, 2017).

Moreover, we use the EWC method to narrow the period band by choosing the first three wavelet decomposition levels according to the physical process in which we intend to highlight. Here, the EWC corresponds to the weighted geometric mean of the square wavelet coefficients per 8 min, as shown in the following equation

$$\begin{aligned} E W C=\frac{1}{7}\left( \sum _{k}^{N}\left| d_{k}^{j=1}\right| ^{2}+2 \sum _{k}^{N}\left| d_{k}^{j=2}\right| ^{2}+\sum _{k}^{N}\left| d_{k}^{j=3}\right| ^{2}\right) \end{aligned}$$

(11)

where N is equal to 8.

1.3 Hilbert–Huang Transform

The Hilbert–Huang transform (HHT) is an empirical method for analyzing nonstationary data that come from nonlinear processes (Huang and Shen 2005). This transform consists of two main parts: empirical mode decomposition (EMD) and Hilbert spectral analysis (HSA). Despite the importance of HSA, only EMD will be discussed in this article. It is important to note that the main difference between HHT and the conventional transforms is that for the former, the signal under analysis is expanded in terms of an adaptive basis (a basis that comes from the data), whereas for the others the expansion occurs in terms of a prior established basis (Huang and Shen 2005), e.g, sines and cosines for Fourier transform and various translated and dilated versions of the mother wavelet for CWT. Moreover, it can be said that HHT is more an algorithm than a transform, due to the absence of an analytical foundation that will require significant time and effort to accomplish (Huang and Shen 2005).

The EMD method assumes that any data consist of intrinsic modes of oscillations, and those intrinsic modes are obtained by an algorithm called the sifting process, which decomposes a given signal in a set of intrinsic mode functions (IMF), which are functions that satisfy the following conditions (Huang and Shen 2005):

• The number of maxima and minima points and the number of roots must either be equal or differ at most by 1.

• At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero.

To perform the sifting process over a given signal f, the first step is to find its extrema points (minimum or maximum) and connect them through a cubic spline, conceiving an upper and a lower envelope for the signal. The second step is to calculate an average envelope from those obtained in the first step and subtract it from the original signal. After these two steps, a test is applied to the resulting signal to determine whether it satisfies the IMF conditions or a stoppage criterion. If it does not, then these steps are applied to the resulting signal until an IMF is obtained. Finally, this IMF is subtracted from f, producing a residue. At this point, the steps are applied to the residue until another IMF is obtained, yielding another residue. This process of obtaining successive residues stops when the last one becomes a monotonic function from which no further IMFs can be extracted.

Suppose that the sifting process stops at the n-th iteration. So, for a residue $r_{j}$ with $0 \le j \le n-1$, an IMF is extracted by the following recurrence relation

$$\begin{aligned} {\left\{ \begin{array}{ll} h_{0} &{}= r_{j}\\ h_{i-1} - m_{i} &{}= h_{i} \end{array}\right. }, \quad i \ge 1, \end{aligned}$$

(12)

where $m_{i}$ is the average envelope computed from the upper and lower envelope of $h_{i - 1}$ and $r_{0} = f(t)$. Considering that $h_{i}$ converges to an IMF for some value of i, then the IMF associated with the residue j will be denoted by $c_{j}$, and the following equation holds

$$\begin{aligned} r_{j} = r_{j -1} - c_{j}, \quad 1 \le j \le n. \end{aligned}$$

(13)

Therefore, from Eq. (13), one can write the original signal f in terms of n IMFs obtained at the end of the sifting process, as follows

$$\begin{aligned} \begin{aligned} r_{1}&= f(t) - c_{1}\\ r_{2}&= r_{1} - c_{2}\\&\vdots \\ r_{n}&= r_{n-1} - c_{n} \end{aligned} \implies f(t) = \sum \limits _{i = 1}^{n}c_{i} + r_{n}. \end{aligned}$$

(14)

1.4 Mean Absolute Percentage Error (MAPE)

The MAPE is a statistical measure, given in percentage terms, of the accuracy of a forecasting model and is defined by

$$\begin{aligned} M = \dfrac{1}{n}\sum _{i = 1}^{n}\left| \dfrac{A_{i} - F_{i}}{A_{i}}\right| , \end{aligned}$$

(15)

where $A_{i}$ is the actual value and $F_{i}$ is the forecast value. Because of the simplicity of Eq. (15), the MAPE is widely used to indicate the accuracy of a forecasting model, but it is scale-sensitive and should not be used with low-volume data.