# A Response Function Approach for Rapid Far-Field Tsunami Forecasting

## Abstract

Predicting tsunami impacts at remote coasts largely relies on tsunami en-route measurements in an open ocean. In this work, these measurements are used to generate instant tsunami predictions in deep water and near the coast. The predictions are generated as a response or a combination of responses to one or more tsunameters, with each response obtained as a convolution of real-time tsunameter measurements and a pre-computed pulse response function (PRF). Practical implementation of this method requires tables of PRFs in a 3D parameter space: earthquake location–tsunameter–forecasted site. Examples of hindcasting the 2010 Chilean and the 2011 Tohoku-Oki tsunamis along the US West Coast and beyond demonstrated high accuracy of the suggested technology in application to trans-Pacific seismically generated tsunamis.

### Keywords

Tsunami forecast DART station pulse response function source inversion boundary value problem## 1 Introduction

Tsunami forecasting, in particular in application to ocean-wide seismically generated tsunamis, relies on numerical modeling for predicting the tsunami wave height and currents before the actual tsunami arrives at the coasts. Conventionally, the modeling solves an initial value problem with an estimated tsunami source function. At the same time, deducing the tsunami source presents a major challenge. Existing instruments do not directly measure the tsunami source—an initial displacement of the sea surface in the earthquake (EQ) area. Instead, the source has to be inferred from related data.

The first-available land-based seismic and geodetic observations can provide for rapid warning for nearby coasts within a few minutes after the EQ, with more elaborated source products to follow (Melgar et al. 2016). The next available source estimate, produced by a technology termed W phase inversion of centroid moment tensor, uses measurements of long period (100–1000 s) seismic waves in the far field of the EQ. Since 2009, the Pacific Tsunami Warning Center (PTWC) routinely makes real-time tsunami forecasts taking the focal mechanism from the W phase inversion and assuming a uniform slip on a rectangular fault (Foster et al. 2012; Wang et al. 2012). Seismogeodesy with near-source real-time GPS measurements (Crowell et al. 2012; Melgar et al. 2013; Titov et al. 2016a) can improve characterization of the fault geometry, but its applicability is limited to EQs occurring in the vicinity of dense GPS arrays. This sequence of early to relatively early-available source models is used for estimating expected tsunami heights by coarsely defined ranges assigned to coastal zones (Melgar et al. 2016 and references therein).

Once a tsunami reaches open-ocean tsunameter stations, sea level measurements offer an attractive option to bypass the complexity of fault motions, and to reconstruct the tsunami source directly. Commonly, the source is assembled from pre-defined basis functions. It is assumed that hypothetical tsunamis radiated by these basis sources linearly superimpose in deep water. Then the basis sources/tsunamis are combined to achieve the best fit to the observations—a procedure termed source inversion. The resulting source is expected to further advance the accuracy of the forecast, which can now aim at predicting location-specific time-histories of wave elevation and currents.

The latter approach is implemented within a tsunami forecasting system termed SIFT (Short-term Inundation Forecasting for Tsunamis), developed at PMEL (Pacific Marine Environmental Laboratory, NOAA) and operationalized at both US Tsunami Warning Centers. The basis functions in SIFT are represented by “unit” EQs (Gica et al. 2008; Percival et al. 2011) with a fault length of 100 km, fault width of 50 km, and a pre-defined spatial orientation. This apparently coarse size of the basis functions and the prescribed EQ mechanism allow a user to obtain meaningful source inversions by fitting observations at as few as one or two deep-ocean stations. The observations are provided by the DART network comprising about 40 stations stretched along the Pacific Rim. The SIFT technology resulted in a sequence of successful forecasts and hindcasts (Tang et al. 2008, 2012, 2016; Wei et al. 2008). At the same time, the inverted tsunami source is sensitive to the subset of considered “unit” sources, choice of DARTs, and duration of the observations being fitted. As stated by Percival et al (2011), “proper selection of the unit sources and of the relevant data is vital for getting good results from the inversion algorithm” .

The SIFT methodology relies on studies by PMEL scientists who emphasize that “only few source parameters are critical to characterize far-field tsunamis, mainly the location and the magnitude. Other parameters have secondary influence and can be predefined for forecast purposes” (Titov et al. 1999; Tang et al. 2016). In particular, “the waves are not very sensitive to details of the initial ocean surface deformation, as long as the [wave] energy and the general source area are determined correctly” (Tang et al. 2012). These findings imply that the method aims at constructing a source proxy, not necessarily the physical source, which would nevertheless approximate the tsunami waveform outside the near field.

Therefore, the next logical step is to by-pass the construction of the source proxy and use tsunami’s en-route measurements for recovering its wavefront and project its dynamics forward in time. To date, a few such attempts have been made. These studies were directed at forecasting local coasts with the measurements collected in their vicinity, and provided warning times under 1–2 h. Maeda et al. (2015) developed a data assimilation method to recover tsunami waves passing through a dense array of tsunameters. This methodology strongly focuses on using the Japan Trench cabled network with detectors spaced at 30–50 km. Thomson et al. (2011) forecasted the 2009 Samoa tsunami at the coast of Vancouver Island, BC, Canada, using the wave measurements at the NEPTUNE cabled observatory off-shore the island as a real-time plane-wave input into a regional tsunami forecast model. Elaborating upon physical assumptions similar to that used by Thomson et al. (2011), Power and Tolkova (2013) expressed their forecast with a response formalism. The Pulse Response Function, as introduced by Power and Tolkova (2013), is unique for each “detector-site” pair and for the directivity of the approaching tsunami, which is determined by the location of the tsunami source. The PRF does not depend on the details of the sea surface deformation in the tsunami origin. The response functions can be pre-computed for a set of anticipated approach directions corresponding to different tsunami origination areas, and then used in a tsunami event to generate instant real-time offshore and near-shore estimates.

In this paper, we introduce a new tsunami forecasting method which generates a prediction at a far-field forecast site as a combination of convolutions of the tsunami’s real-time measurements and pre-computed pulse response functions (PRFs). As will be detailed further, the proposed method provides an instant solution to the boundary value problem of numerically propagating the wave directly from the detectors in a linear propagation medium. This technology relies on mathematical formalism developed in Power and Tolkova (2013), adapted for using measurements at any number of detectors in the relative vicinity of the source area, outside the near field.

The new technology is described in Sect. 2, and is demonstrated with hindcasting the two largest trans-Pacific tsunamis of the twenty first century: the 2010/02/27 Chile (Sect. 3) and the 2011/03/11 Tohoku-Oki (Sect. 4) tsunamis. In both cases, tsunami measurements at open-ocean detector(s) in the EQ region are directly converted into the wave time-histories at remote offshore locations across the ocean, where the wave would arrive many hours later. Next, the PRFs derived for the Tohoku-oki tsunami were “blind-tested” with forecasting tsunamis generated by a hypothetical normal fault event and a hypothetical thrust event in the general area of the 2011 Tohoku-Oki EQ (Sect. 5). This exercise demonstrates that PRFs do not depend on the EQ mechanism and geometry, and are applicable to forecasting any tsunami originating in the same general area. In Sect. 6, the response-based forecast is extended to coastal sites. In particular, the response formalism is generalized for predicting wave heights in non-linear coastal environments such as a shallow harbor, and applied to hindcasting the above two tsunamis in the Crescent City Harbor, CA (Sects. 6.1–6.3). Forecast latency is briefly discussed in Sect. 7. Finally, Conclusions elaborate upon the potential of the suggested technology, its pros and cons compared to related conventional methods, and directions for further research.

## 2 Mathematical and Numerical Methods

The response formalism relies on an assumption that the azimuthal (across-the-beam) shape of the tsunami front is prescribed by the bathymetry and a location of the tsunami source; whereas the radial (along-the-beam) wave structure is determined by the source details. Then a tsunami beam within a certain azimuthal window can be reconstructed by combining a prescribed wavefront shape with the radial wave structure measured as the wave passes by a detector. To reconstruct an azimuthal structure of the departing wave within a larger angle, or even ocean-wide, the wave has to be assembled from several wave beams. The latter requires several detectors delimiting the source region.

### 2.1 Response Formalism with a Single Detector

*h*(

*t*) satisfies a series of conditions: \(h(0)=1\) and \(h(i\Delta t_0)=0\) for all integer \(i \ne 0\); \(\sum { h(t-i\Delta t_0)} =1\); \(\int {h(t){\rm{d}}t}=\Delta t_0\); and its spectrum is localized below \(1/2\Delta t_0\). Impulse

*h*(

*t*) acts over the series

*q*(

*t*) as a low-pass filter with a cut-off at the Nyquist frequency. A function

*h*(

*t*) with such properties is known as an interpolating pulse (Sweldens and Schroder 2000); its particular example adopted for this work and shown in Fig. 1 is described in (Power and Tolkova, 2013). The sampling interval, selected hereafter at \(\Delta t_0=2\) min, determines the period \(2\Delta t_0\) of the shortest wave component registered at the detector.

*p*(

*t*) is the pulse response function (PRF). It represents the tsunami signal at the site, divided by

*h*(0), in a hypothetical wave which propagates in the tsunami’s direction and yields the surface elevation time-history

*h*(

*t*) at the detector (the interpolating pulse

*h*(

*t*) in Eq. (2) is dimensionless, but the surface elevation

*h*(

*t*) is measured in meters. Hereafter, the dimensionality will be implied by the context and not shown explicitly). In the half-space down the tsunami path from the detector, this hypothetical wave can be obtained as a solution to the SWE initialized in either of two ways:

- 1.
by varying the surface elevation as \(2 \cdot h(t)\) with zero flow velocities on a boundary \(\Gamma\) drawn through the detector and coinciding with the tsunami wave front; or

- 2.
by a specific initial deformation centered on the detector. The deformation has an elongated shape containing the assumed boundary \(\Gamma\) on its centerline, with the shape’s profile across \(\Gamma\) being \(\eta =2 \cdot h(r/c)\), where

*c*is the long wave celerity offshore, and*r*is a distance from the shape’s centerline (Fig. 2). In this work, the same*c*value is applied at all points on \(\Gamma\).

In a linear medium (deep water), either procedure produces two waves, each with the time-history *h*(*t*) at the detector. One wave travels toward the forecasted site, and the other—back to the tsunami source. If variations of the local wave celerity *c* on \(\Gamma\) are negligible, the above two options yield identical results. Then expression (3) provides a solution at the site to an equivalent boundary value problem in an initially still domain, separated from the source area by curve \(\Gamma\) and forced by an incoming wave with an elevation time-history \(\tilde{q}(t)\) on \(\Gamma\).

The solution to the above problem should also yield a time-history \(2 \cdot h(t)\), \(t \ge 0\), at the detector—the result of superposition of the left half on the initial disturbance propagating to the right and the right half propagating to the left. Then an offshore wave celerity

*c*might need to be adjusted (and the initial condition re-derived), to ensure that the first zero surface displacement at the detector occurs at exactly \(\Delta t_0\). The latter condition defines the value of*c*regardless of its initial estimate (for instance, with the depth at the detector).Half of the initial disturbance tends to focus back on the source. For the PRF computations, the bathymetry/topography in the source area might need to be modified, to allow the back-traveling pulse to exit the domain without being reflected onto the detector (particular bathymetry modifications used in this study are described in the Supplementary Appendix).

The implication of linearity of the propagation holds when non-linear effects in the SWE solutions are negligible, which is justified when the forecasted site is located in deep water or the tsunami height remains small.

An actual tsunami wave is unlikely to be uniform along its entire wavefront. Therefore expression (3) can be physically meaningful only within a certain azimuthal window enclosing the detector, and the resulting prediction applies only to sites located in the path of the wave beam within this window.

### 2.2 Response Formalism with Multiple Detectors: Combining Wave Beams

*K*individual detectors:

*k*-th wavefront segment, as described in Sect. 2.1.

*K*wave beams and making it continuous in the azimuthal direction, one might want to make the beams overlapping and tapered across with a function \(\tilde{h}(\rho )\), where \(\rho\) is a distance along \(\Gamma _k\). As a result, an initial deformation \(\eta _k(P)\) at point

*P*for the

*k*-th PRF computations is given by:

*r*and \(\rho\) are distances of the point

*P*from the

*k*-th detector counted across and along the wavefront \(\Gamma _k\), as shown in Figs. 2 and 3; other notations are the same as in Sect. 2.1. Azimuthal interpolator \(\tilde{h}_k(\rho )\) describes a spatial span of the

*k*-th wavefront segment and an optional azimuthal tapering factor at the

*k*-th detector. Anticipated spacial spans can vary from non-overlapping segments (as shown in Fig. 3) to wider segments delimited at the

*k*-th detector by the rays propagating to the \((k+1)\)-th and \((k-1)\)-th detectors. The latter option was adopted in this work. If no wavefront singularity occurs between the

*k*-th and \((k+1)\)-th detectors, the linear interpolation is applied: \(\tilde{h}_k(\rho )=1-\rho /\rho _{k+1}\), \(0<\rho <\rho _{k+1}\), where \(\rho _{k+1}\) is a distance at which \((k+1)\)-th ray passes by the

*k*-th detector. If both

*k*-th and \((k+1)\) detectors are located in deep water with no local bathymetric peculiarities, \(\rho _{k+1}\) can be approximated by the distance of the \((k+1)\)-th detector’s projection onto the

*k*-th wavefront (see Fig. 3). With \(\tilde{h}_{k+1}(\rho )\) selected analogously, the

*k*-th and the \((k+1)\)-th wave beams continuously transition into each other. On the contrary, if there is a ridge or another topographic feature breaking a wavefront at a distance \(\rho _{\rm{ridge}}\) from the

*k*-th detector, the initial deformation extends uniformly to the ridge and ends there: \(\tilde{h}_k (\rho <\rho _{\rm{ridge}})=1\), \(\tilde{h}_k (\rho >\rho _{\rm{ridge}})=0\).

### 2.3 Numerical Models

Three numerical models solving the shallow-water equations (SWE) on a sphere are used in this study for PRF computations (Cliffs, Alaska GI-T) and for forecasting with simulated events (RIFT).

The Cliffs model is an open-source relative of the Method of Splitting Tsunamis (MOST) model used in the SIFT system. As MOST, Cliffs solves the fully nonlinear SWE in a characteristics form with an explicit finite-difference scheme developed at the Novosibirsk Computing Center of the Siberian Division of the Russian Academy of Sciences in 1984–1989 (Titov et al. 2016). The friction term is based on Manning formulation. The primary difference between Cliffs and MOST algorithms is the treatment of the land–water boundary (Tolkova 2014, 2016b).

The Alaska GI-T model stems from the TUNAMI model by Imamura (1996), which solves the fully nonlinear SWE with a leap-frog numerical scheme; friction in Manning formulation. The primary modification in the Alaska GI-T model is the method used for wetting/drying (Nicolsky et al. 2011). Currently, the Alaska GI-T model is used to predict propagation and run-up of hypothetical tsunamis along the Alaska shore (Suleimani et al. 2013; Nicolsky et al. 2015).

The RIFT model is a real-time tsunami forecast model, developed in the US Pacific Tsunami Warning Center (PTWC) for rapid ocean-scale simulations (Wang et al. 2012). The model solves the linearized SWE with a finite-difference scheme similar to one used in Imamura (1996), without friction.

## 3 Case Study: Forecasting the 2010 Chile Tsunami Along North America from a DART off the Coast of Peru

All simulations in this section were carried out with Cliffs using ETOPO bathymetry at 2 arc-min resolution. To compute the wavefront shape \(\Gamma\), a small initial arc drawn around the EQ epicenter^{1} was propagated numerically over realistic bathymetry until its hump reached the DART; the wavefront \(\Gamma\) was defined by the coordinates under the maximal sea level elevation along the arc, as shown in Fig. 4. As long as this initial arc is reasonably small, its radius and profile can be fairly arbitrary, since the wavefront shape at the detector is not sensitive to the details of the source function (the main premise of the method). In a land-free uniformly deep sea, the latter premise would hold for a compact source with a small angular size as seen from the detector. In a real ocean, this premise is often facilitated by strong near-shore bathymetry, by narrowing the range of angles at which waves from a larger EQ area might arrive at the detector.

Figure 5 shows the sea surface in the 2010 Chile tsunami simulation at 4 h and 40 min after the tsunami origination. The tsunami simulation was initiated with an estimated source function for the 2010/02/27 event, produced with the SIFT system at the PMEL (http://nctr.pmel.noaa.gov/chile20100227/). Curve \(\Gamma\), shown in Fig. 5 by a red arc (after minor smoothing), conforms to the wavefront simulated with a realistic source function. As seen in the simulation, the radial structure of the wave changes at larger angles to the coastline, from a longer wavelength and a lower amplitude in the north-bound beam passing through 32412 to about twice shorter waves and a higher amplitude in the direction normal to the coastline. Consequently, the measurements at DART 32412 represent the actual wave time-history only in the north-going tsunami wave beam, and permit forecasting only the sites along the American coasts north of the DART station.

The response computations were initialized with an initial deformation built upon \(\Gamma\), as described in Sect. 2.1 above. The resulting time-histories computed at the sites being forecasted are shown in Fig. 6. The initial deformation leaves a pulse-like signal at 32412, 2 min long until the first zero crossing (Fig. 6, top left), half of which corresponds to the forward-traveling wave. The solutions at other stations represent the stations’ responses to DART 32412.

The predictions shown in Fig. 7 are obtained as convolutions of each site’s response (shown in a corresponding pane in Fig. 6) and the tsunami measurements at station 32412 (shown in Fig. 7, top left). Consistent agreement between the predictions and the observations suggests that for any tsunami originating in Chile, its future time history anywhere by the coast of North America can be directly derived from the tsunami’s record at the station 32412 (offshore Peru).

Chilean-born tsunamis typically excite edge waves on the adjacent continental shelf, which sustain wave radiation for many hours (Geist 2013; Catalan et al. 2015). A relevant feature of the response formalism is its ability to account for any part of the wave train passing the detector at any time during the event, including the later waves resulting from local resonances, edge waves, and other shelf phenomena. Owing to this feature, the resulting predictions are accurate for 10 h or longer.

## 4 Case Study: Predicting the 2011 Tohoku Tsunami Around the Pacific with Three DARTs Near Japan

### 4.1 Initial Shapes

### 4.2 Pulse Response Functions

Each PRF records transformations which the tsunami will experience between the detector and the site due to interactions with the complex bathymetry. As an example, we briefly examine PRFs at station 46407 off of the mid-Oregon coast and at station 51407 in Hawaii to each DART in the 3-DART set, shown in Fig. 10. The 46407’s PRFs tell us that the first waves are mostly determined by the signal at 21419, and therefore arrived by the northern (along the Aleutians) route. Pulse response to 21413 becomes noticeable only in the later part, which indicates waves re-directed to the site by scattering. About 1.5 h after the pulse arrival, the PRFs to both 21413 and especially 21418 exhibit an intense signal at a distinct frequency, which indicates a strong bathymetric feature (Mellish Seamount, as follows). Likewise, the observed Tohoku tsunami signal at 46407 also contains intense waves between 1.5 and 2 h after the tsunami arrival, which Tang et al. (2012) identified as a ray refracted from the Mellish Seamount. The 51407’s PRFs are remarkable in how little of high-frequency variance (typically dominating a PRF of a deep-sea site) is present in the pulse responses to 21418 and especially to 21419. As seen in the latter PRF, only a nearly harmonic wave with a 72-min period arrives to Hawaii from the 21419’s wave beam traveling along the Aleutians. It is possible that the underwater ridges connecting Hawaii and Aleutians serve as a wave guide at this particular wavelength (this hypothesis has not yet been evaluated).

### 4.3 Predictions

Given the pre-computed PRFs, the prediction at each site is obtained as a sum (4) of the responses to the wave motion registered at each DART in a set, each response being a convolution of a DART signal and a corresponding PRF. Figure 11, top row, displays the 2011 Tohoku tsunami measurements (de-tided) at stations 21413, 21418, and 21419 (the detectors), sampled at a 2-min interval, used to produce the predictions at 24 deep-water stations around the Pacific. These stations also recorded the Tohoku tsunami (Fig. 8), where it arrived between 1.6 and 20 h after it had passed the detectors. The resulting predictions computed as a combined response to the 2-detector set and to the 3-detector set, vs. the observed sea level variations, are shown in Figs. 11, 12 and 13. By visual comparison with the SIFT-made predictions (Tang et al. 2012) with an estimated source inferred from the measurements at two DARTs 21418 and 21401 (in the direction of 21419, out of service since 07/2014) and at three DARTs 21418, 21401, and 21413, the hindcasts computed with the response formalism display a similar accuracy.

More DARTs around the source detect more variability along the wavefront. In particular, as seen from the records at the three detectors (Fig. 11, top row), the tsunami has a larger wave height and a shorter wavelength in the central beam passing through 21418, while it has a smaller wave height and a longer wavelength in the side beams through 21413 and 21419. Thereby the predictions made with the 2-DART set do not resolve this central wave beam. Nevertheless, both the 2-DART and the 3-DART predictions have captured the major signal variance at each location. The 2-DART predictions fit the observations better at five Aleutians DARTs, which receive their wave, shaped by the Kurils and Aleutian trenches, directly from 21419 (see Fig. 8). Contribution from 21413 in the Aleutians is negligible, due to a very different beam direction. In the 3-DART configuration, however, Aleutian sites receive a signal from 21418 introduced by azimuthal interpolation. This signal is responsible for the false short peaks on top of correctly predicted waves.

It has been found that numerical forecasts systematically underestimate tsunami arrival time in the far field, which was explained by the neglect of seawater and seafloor compressibility and some other factors (Tsai et al. 2013; Watada et al. 2014). The propagation time errors embedded in the computed PRFs might potentially degrade the combined responses, should these errors be different among the site’s responses to different detectors. In the Tohoku case, the difference among the propagation time errors from a detector to the site in different tracks is not expected to exceed 1–3 min, which, given the much longer tsunami period, is not likely to significantly affect the results. Nevertheless, since the response formalism with multiple detectors is potentially sensitive to phase errors, the PRF computations in this section were performed using a depth correction method suggested by Wang (2015). Almost perfect timing between the predictions and the observations at all sites (no time shifts were applied to any time-history in Fig. 11, 12, 13) suggests that the propagation time errors have been largely eliminated. Without the depth correction, the predicted arrival times are up to 15 min early; while the predicted wave heights with and without depth correction differ very mildly for this event.

## 5 Case Study: Same Pulse Response Functions, Different Events

Pulse Response Functions connect a forecasted site and the deep-sea detectors, and depend on the location of the tsunami originating area. At the same time, PRFs do not depend on the tsunami source structure. No information about an EQ geometry except a location of its epicenter were used for deriving PRFs exercised in this work. We, therefore, expect that the same PRFs used for forecasting, for instance, the 2011 Tohoku tsunami, can be used for forecasting another tsunami originating in the same area—though how large of an area the same responses can serve varies from region to region and is a subject for a separate study. This proposition is illustrated here by applying the above PRFs used with the 2011 Tohoku tsunami (centroid at \(142.373^{\circ }{\rm{E}}\) and \(38.297^{\circ }{\rm{N}}\), at 29.0 km depth) for predicting two other, artificial tsunamis. The tsunamis were generated, accordingly, (1) by a hypothetical \(M_{\rm{w}}\) 8.0 trust fault EQ caused by a uniform 3.6 m slip of a 169 km \(\times\) 47 km plate (\(({\rm{strike},\, \rm{dip},\, \rm{rake}})=(187,15,90)\)) with a centroid at \(143.25^{\circ }{\rm{E}}\) and \(39.0^{\circ }{\rm{N}}\) at 20 km depth, 109 km north of the epicenter of the 2011 Tohoku EQ; and (2) by a hypothetical \(M_w\) 8.3 normal fault EQ caused by a uniform 5.6 m slip of a 247 km \(\times\) 58 km plate (\(({\rm{strike}, \rm{dip}, \rm{rake}})=(206,45,-90)\)) with a centroid at \(144.5^{\circ }{\rm{E}}\) and \(37.5^{\circ }{\rm{N}}\) at 26 km depth, 207 km south of the epicenter of the 2011 Tohoku EQ. For both hypothetical events, a shear modulus of 44.1 GPa is assumed.

The artificial tsunamis were propagated from the source to the DART stations using the RIFT model (Wang et al. 2012) developed and operated at the PTWC. The simulations were carried out at a 4 arc-min resolution. The bathymetric grid was derived from the GEBCO 30-arc-sec data, and the ocean-depth correction method of Wang (2015) was applied to reduce the tsunami travel time errors. Compared to the 2011 Tohoku tsunami, the artificial tsunamis are composed of much shorter waves. For instance, the duration of the full leading wave of the Tohoku tsunami at 21413/21419 is about 30 min vs. 20 min (Fig. 14) for the first artificial tsunami and 12 min for the second artificial tsunami, followed by even shorter waves (Fig. 15).

^{2}successfully forecasted with the response formalism:

azimuthal resolution, roughly determined by a number of detectors around the source area;

temporal sampling interval at the detectors (\(\Delta t_0=2\) min, throughout this work), prescribing the width of the initial shape for computing the PRFs and the shortest possible period (\(2\Delta t_0\)) in the detected signal;

numerical properties of the model(s) in the short (low-resolved) wave range.

## 6 Forecasting at Tide Gages

The application of the response formalism is limited by the condition of linearity of the propagation tract from the detector to the site (hereafter, a wave is termed ‘linear’ if its amplitude at a point of interest is linearly related to the wave amplitude in the deep sea, and termed ‘non-linear’ otherwise). Thus the response method might not directly apply to forecasting large waves in shallow bays and harbors or to predicting the details of land inundation, but can predict relatively small waves, as in the above examples. Moreover, the response method can be modified to allow forecasting some non-linear environments, in particular, moderately big waves in some bays and harbors, as described in Sects. 6.1 and 6.2 below.

### 6.1 Equations for a Non-linear Resonator

*s*(the forecast):

*G*, analogous to PRF, depends on the system’s parameters. A non-linear resonator is often described by a similar equation,

^{3}where the system’s parameters depend on the system’s state

*G*in the solution process, but using a pre-computed

*G*set instead. We note that (10) is equivalent to a system of two equations with respect to two functions

*s*and \(\xi\):

*K*functions such that \(||\xi _1||<||\xi _2||< \cdots <||\xi _K||\), and the set \(||\xi _k||\) samples the range of targeted signal levels in

*s*. Let \(s_k\) be a forecast computed with (11) for \(\xi =\xi _k\). If there exists a unique index

*j*for which \(||s_j|| \approx ||\xi _j||\), then \(s=s_j\) is the sought solution. Next, this approach is elaborated for forecasting a harbor with a set of PRFs.

### 6.2 Harbor Forecasting with Scaled PRFs

*a*at the deep-ocean detector(s) as

*a*. Apparently, for \(a_1<a_2< \cdots <a_K\),

*a*for computing scaled PRFs is selected to provide a desired set of levels \(||P_a||\) which sample the range of targeted signal levels in the harbor.

*a*exciting small waves in a harbor, the harbor’s scaled PRFs \(p_a(t)\) approach its linear limit

*p*(

*t*) independent of

*a*. The bulk of the energy loss for a small-amplitude wave is due to radiation out of the harbor rather than due to quadratic friction, so the harbor behaves as a linear system. For a larger wave amplitude, a scaled PRF \(p_a(t)\) describes a resonator with larger frictional losses. A scaled PRF is, therefore, lower than the linear-limit PRF. The greater the

*a*value, the lower the scaled PRF. Therefore predictions \(s_k\) produced with scaled PRFs satisfy

*s*in the range \(0 \le ||s|| \le ||P_{a_K}||\), as depicted in Fig. 17. Prediction \(s_j\) closest to the intersection is therefore the sought forecast.

When forecasting with more than one DART, scaled PRF computations should use the common set of levels \(||P_a||\), whereas an *a* set can vary among the DARTs. Contributions from individual detectors in the sum (4) should use scaled PRFs corresponding to the same ||*P*|| level. Therefore, scaled PRFs can provide as many predictions as there are selected ||*P*|| levels, regardless of the number of DARTs.

Prediction levels greater than \(||P_{a_K}||\) indicate very high waves outside the targeted range. The response formalism won’t be able to evaluate the height of such waves, but will clearly predict their occurrence.

### 6.3 Case Study: Forecasting Crescent City, CA

Unlike the 2010 event, the harbor’s response in the 2011 Tohoku tsunami was strongly non-linear. Figure 19 shows 2-DART hindcasts of the 2011 Tohoku event at the harbor’s tide gage. The top two panels display responses \(P_a(t)\) in m and dimensionless PRFs \(p_a(t)\) of the gage to the DARTs 21413 and 21419. The linear-limit PRFs are drawn with black lines. Gray lines show responses \(P_a(t)\) to a 6-m high initial shape at 21413 and a 2-m high shape at 21419, respectively. The amplitudes of \(a=6\) m at 21413 and \(a=2\) m at 21419 were selected to produce waves in the harbor up to 2 m in amplitude, in response to each initial shape. Red lines show the corresponding scaled PRFs, which are lower than the linear-limit PRFs over the later part of the signal, but coincide with them over the first period (non-linearity takes time to develop). The scaled PRFs yield a prediction (red in the bottom panel) varying up to 2 m in amplitude as the responses \(P_a(t)\) to either DART do. Hence we have matched signal levels in a response to a scaled pulse at each DART and in a 2-DART prediction produced with the scaled PRFs. The prediction agrees well with the observations (black dots, bottom panel), in spite of the non-linearity of the system. As expected, a prediction made with the linear-limit PRFs (gray in the bottom panel) significantly over-estimated the harbor’s actual response for the more energetic part of the wave train.

## 7 Case Study: Forecast Timing

During a tsunami event, forecast would be attempted as soon as, say, first tsunami peak has passed the detector. Figure 20 shows a prediction of the 2010 Chilean tsunami in Crescent City made with the first 20-min of the tsunami’s recording at DART 32412, which contain the first wave peak. For comparison, the figure also shows a prediction generated with the longer time-history at 32412. The two predictions coincide for the first 20 min—the duration of the observations at 32412 shared by the two predictions. Past the duration of the observations at the DART, without contribution from the later-arriving negative wave, the early, 20-min-based prediction slightly over-estimated the observed signal in the next hour, though still appears fairly accurate. So the response-based forecast can be made early in the event, and then refined as more data become available.

Forecast timing is different between the PRF method and the inversion-based methods, since the two approaches feature different use of data. The PRF approach takes a fraction of a second to make a forecast once DART observations are obtained, but the prediction is “final” only for the duration of the available observations, especially early in the event. The prediction describes the site’s response to the observed part of the wave train. Past the respective duration of the observations, the prediction lacks contribution from the waves which had not yet arrived at the detector(s). These later waves, if not small, can be included in the prediction later in the event.

For the inversion-based forecast, the observations need to accumulate until there is enough signal for the inversion (say, 30 min—up to one full wave), followed by high-resolution simulations focusing on the locations of interest. Once this is done, the resulting prediction potentially describes the entire duration of the event. However, since in the inversion process, the models are selected to fit only the first observed wave peak (Tang et al. 2012; Percival et al 2011), the forecast accuracy might reduce past the duration of the site’s response to the first wave(s) at the DART(s) being fitted. The inversion-based forecasts at all locations in Figs. 16, 17, 18 and 19 produced with the SIFT system can be seen on NCTR’s Events pages (http://nctr.pmel.noaa.gov/honshu20110311/, http://nctr.pmel.noaa.gov/chile20100227/).

## 8 Concluding Remarks

Predicting tsunami impacts at remote coasts largely relies on tsunami height measurements in an open ocean. In this work, we demonstrated that tsunami measurements can be used to formulate a boundary value problem and numerically “propagate” the wave directly from the detector(s), differently from the common practice of using the measurements to infer the tsunami source function. We suggested a formalism of specific response functions, which provide an instant solution to the above boundary problem, as long as non-linear effects are negligible. The tsunami predictions in the far field are generated as a response or a combination of responses to one or more tsunameters, with each response obtained as a convolution of real-time tsunameter measurements and a pre-computed pulse response function.

In the response formalism, wave field is assembled from wave beams assigned to individual detectors; and therefore the method’s ability to resolve the wave azimuthal structure, or the directional diagram, depends on the detector’s azimuthal coverage. Consequently, in the forecast examples presented, the method was applied to forecasting only the North America in the 2010 Chile tsunami, but almost the entire Pacific in the 2011-Tohoku event. With the employment of the ocean-depth correction method of Wang (2015), the tsunami arrival time errors common to tsunami models were largely eliminated in the computation of the response functions.

For forecasting non-linear coastal environments, the response formalism can be used to generate offshore boundary inputs into fine-resolution grids focused on desired coastal sites. We also suggested an adaptation of the response formalism for providing direct forecasts in some non-linear environments such as a shallow harbor, although the limits of validity and the robustness of this adaptation are areas for further study. Hindcasts of the two largest trans-Pacific tsunamis of the twenty first century along the US West Coast and beyond have demonstrated high accuracy (not limited to the early arriving waves) of the PRF methodology.

- 1.
The PRF method uses measurements for computing tsunami’s forward propagation, which in mathematical terms represents a well-posed problem.

- 2.
The method scales to any number of detectors, with the forecast accuracy and coverage gradually progressing as input from more detectors is incorporated.

- 3.
The method transmits the exact measurements for the entire duration of the event.

- 4.
The method has shown a promise for forecasting the later-arriving waves.

- 5.
The method uses no assumptions about the EQ dynamics (such as an instantaneous co-seismic sea floor deformation), nor about the dynamics of the tsunami generation.

- 6.
Given an adequate detectors’ coverage and pre-computed PRFs, the method can provide instant predictions in a tsunami event.

Thereby, depending on the detector coverage between the EQ location and the forecasted coast, the response formalism might provide a useful complement to existing forecasting technologies. While further work is needed to investigate in more detail different aspects of the response method, the existing tsunameter network already permits operating the response forecasting method with promising results, for some EQ locations and forecasted sites.

## Footnotes

## Notes

### Acknowledgements

We acknowledge NOAA/NDBC for providing the DART records, NOAA/NOS for providing the tide gauge records, Ocean Networks Canada for providing the NEPTUNE records, and NOAA/NCEI for providing the bathymetry used in the numerical simulations. Dmitry Nicolsky acknowledges support for his work from the state of Alaska.

## Supplementary material

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