Extraction of tsunami source coefficients via inversion of DART\(^{\circledR}\) buoy data
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Abstract
The ability to accurately forecast potential hazards posed to coastal communities by tsunamis generated seismically in both the near and far field requires knowledge of so-called source coefficients, from which the strength of a tsunami can be deduced. Seismic information alone can be used to set the source coefficients, but the values so derived reflect the dynamics of movement at or below the seabed and hence might not accurately describe how this motion is manifested in the overlaying water column. We describe here a method for refining source coefficient estimates based on seismic information by making use of data from Deep-ocean Assessment and Reporting of Tsunamis (DART\(^{\circledR}\)) buoys (tsunameters). The method involves using these data to adjust precomputed models via an inversion algorithm so that residuals between the adjusted models and the DART\(^{\circledR}\) data are as small as possible in a least squares sense. The inversion algorithm is statistically based and hence has the ability to assess uncertainty in the estimated source coefficients. We describe this inversion algorithm in detail and apply it to the November 2006 Kuril Islands event as a case study.
Keywords
Tsunami forecasting Tsunami source estimation DART\(^{\circledR}\) data inversion 2006 Kuril Islands tsunami Tsunameter1 Introduction
Tsunamis have been recognized as a potential hazard to United States coastal communities since the mid-twentieth century when multiple destructive tsunamis caused damage to the states of Hawaii, Alaska, California, Oregon, and Washington. The National Oceanic and Atmospheric Administration (NOAA) responded to these disasters with the establishment of two Tsunami Warning Centers responsible for providing warnings to the United States and her territories (these centers also provide warnings internationally to participating nations in the Pacific, Atlantic, and Indian Oceans, as well as the Caribbean Sea Region). In addition, the agency assumed the leadership role in the area of tsunami observations and research and has been measuring tsunamis in the deep ocean for several decades. The scale of destruction and unprecedented loss of life following the December 2004 Sumatra tsunami prompted a strengthening of efforts to address the threats posed by tsunamis, and, on 20 December 2006, the United States Congress passed the “Tsunami Warning and Education Act” (Public Law 109–424, 109th Congress). Central to the goal of protecting United States coastlines is a “tsunami forecasting capability based on models and measurements, including tsunami inundation models and maps \(\ldots\).” To meet this congressionally mandated forecasting capability, the NOAA Center for Tsunami Research has developed the Short-term Inundation Forecast for Tsunamis (SIFT) application (Gica et al. 2008; Titov 2009). This application is designed to rapidly and efficiently forecast tsunami heights at specific coastal communities.
At each community, estimates of tsunami wave arrival time and amplitude are provided by combining real-time tsunami event data with numerical models. Several key components are integrated within SIFT: deep-ocean observations of tsunamis collected near the tsunami sources, a basin-wide precomputed propagation database of water level and flow velocities based on potential seismic unit sources, an inversion algorithm to estimate coefficients associated with unit sources based upon the deep-ocean observations, and high-resolution tsunami forecast models developed for specific at-risk coastal communities. As a tsunami wave propagates across the open ocean, Deep-ocean Assessment and Reporting of Tsunamis (DART\(^{\circledR}\)) buoys observe the passage of the wave and relay data related to its arrival time and amplitude in real or near real time for use with SIFT (these buoys are one form of a specialized instrument known as a tsunameter). The SIFT application uses the reported observations to refine an initial assessment of the magnitude of the tsunami that is based purely on seismic information. The refinement is done by comparing observations from the DART\(^{\circledR}\) buoys to models in the precomputed propagation database via an inversion algorithm.
In this article, we focus on the inversion algorithm, which combines data from DART\(^{\circledR}\) buoys with precomputed models to yield refined estimates of the source coefficients. We begin with an overview of the SIFT application (Sect. 2), after which we describe the data collected by the DART\(^{\circledR}\) buoys (Sect. 3). We then discuss the precomputed time series of model tsunamis for the DART\(^{\circledR}\) buoy locations in Sect. 4 in preparation for a detailed description of the inversion algorithm in Sect. 5. Here we also illustrate use of the algorithm on the November 15, 2006 Kuril Islands event (this provides a good example since this event was observed by eleven buoys in the North Pacific Ocean along the Aleutian Islands and a considerable portion of the North American coastline). The inversion algorithm is based upon a statistical model, and hence, we are able to assess the uncertainty in the resulting source coefficient estimates. We end the main body of the paper with a discussion of potential extensions and refinements to the SIFT application in Sect. 6 and with our conclusions in Sect. 7. (We gives some technical details in Appendix 1 on assessing the uncertainty in the estimated source coefficients. In Appendix 2, we use the coefficient estimates and their associated uncertainties to come up with a measure of the strength of the tsunami-generating event that is of interest to compare with the seismically determined moment magnitude.)
2 Short-term inundation forecast for tsunamis (SIFT)
The SIFT application exploits the fact that the ocean acts as a low-pass filter, allowing long-period phenomenon such as tsunamis to be detected by measurement of pressure at a fixed point on the seafloor (Meinig et al. 2005). The strategy behind SIFT is to assess the potential effect of a tsunami by combining pressure measurements collected in real time with models, thus refining an initial assessment based purely on seismic data available soon after an earthquake. SIFT is intended to be an operational system that provides its assessments in a timely manner. Given that computations concerning wave generation, propagation, and inundation must be done under time constraints, SIFT makes use of a precomputed propagation database containing water elevations and flow velocities that are generated by standardized earthquakes located within “unit sources,” which are strategically placed near ocean basin subduction zones (Gica et al. 2008). Within SIFT, model time series are extracted from the numerical solution to the propagation of tsunami waves throughout the ocean basin as generated at the unit sources. Dynamics of these tsunami waves in the open ocean allow them to be linearly combined to mimic observed data. The SIFT application is designed primarily to predict trans-oceanic tsunamis rather than smaller regional events because of operational considerations and has certain other limitations in its current implementation (e.g., the unit sources all have the same fixed dimensions).
An inversion algorithm is used to estimate source coefficients that adjust the amplitudes of the precomputed models from each unit source using deep-ocean measurements collected by DART\(^{\circledR}\) buoys. These coefficients, once estimated by the inversion algorithm, provide the boundary conditions under which nonlinear inundation models are run to provide forecasts of incoming tsunami waves at threatened coastal communities. These models are run independent of one another in real time while a tsunami is propagating across the open ocean. The models provide an estimate of wave arrival time, wave height, and inundation following tsunami generation. Each inundation model has been designed and tested against historical events to perform under very stringent time constraints, given that time is generally the single limiting factor in saving lives and property. A total of seventy-five community inundation models are scheduled for completion at the end of federal fiscal year 2012.
3 Bottom pressure measurements from DART\(^{\circledR}\) buoys
A DART\(^{\circledR}\) buoy actually consists of two separate units, namely, a surface buoy and a bottom unit with a pressure recorder (NOAA Data Management Committee 2008). These units communicate with each other via acoustic telemetry, and the surface buoy in turn communicates with tsunami warning centers via the Iridium Satellite System. The bottom pressure recorder internally measures water pressure, which, for operational purposes, is converted within the bottom unit to equivalent sea surface elevation using the factor 670.0 mm/psi, i.e., 0.09718 mm/Pa (Mofjeld 2009). Some research studies involving bottom pressure might require a more exact conversion factor, an example being the tidal study by Mofjeld et al. (1995). The local water-column averaged product of the in-situ water density and acceleration of gravity is then used to compute the factor.
The water pressure is integrated over nonoverlapping 15-s time windows, so there are \(60\,\times\,4 = 240\) measurements every hour. We associate each window with an integer-valued time index n. For simplicity, we adopt the convention that \(n\,=\,0\) corresponds to the 15-s time window during which a particular tsunami-generating earthquake of interest commenced. The actual time associated with the nth time window is \(t_n = a + n \Updelta\) (in hours), where a is a fixed offset and \(\Updelta\,=\,1/240\,\doteq\,0.004167\) h. In what follows, it is convenient to set \(a\,=\,0\) so that t_{n} is the elapsed time from the 15-s window containing the earthquake event. We denote the internal measurements by x_{n}, where \(n\,<\,0\) (or \(n\,>\,0\)) is the index for a measurement recorded before (or after) the earthquake.
The internally recorded x_{n} measurements only become fully available when the bottom unit is lifted to the surface for servicing (about once every two years). Normally, the buoy operates in a monitoring mode in which the bottom unit packages together one measurement every 15 minutes (a 60 fold reduction in data) over a 6-h block for transmission up to the surface buoy once every 6 h. We refer to measurements from this monitoring mode as coming from the “15-min stream.” Let \(n_l,\,l= -1, -2, \ldots\), represent the indices associated with the portion of the 15-min stream that occurs just prior to the \(n\,=\,0\) measurement (the measurement x_{0} itself might or might not be available). Typically, we have \(n_{l} - n_{l-1} = 60\), but this need not be true for all l due to data dropouts. Also note that n_{−1} itself need not be a multiple of 60 since the earthquake can occur anywhere within the 15-min reporting cycle.
The time series in Fig. 1 have a prominent tidal component that must be removed prior to use of the inversion algorithm described in Sect. 5. Detiding must be done nearly in real time and is not a simple matter. We have explored approaches based on harmonic models, Kalman filtering/smoothing, empirical orthogonal functions and low-order polynomial fits (Percival et al. 2011). In what follows, we assume that data x_{n} from the 15-s or 15-min streams or data \(\bar{x}_n\) from the 1-min stream have been suitably detided. We denote the detided data by d_{n} and \(\bar{d}_n\). (We detided the data using a Kalman filter/smoother in all the examples presented below.)
4 Models for DART\(^{\circledR}\) buoy data
Each adjustable model was constructed with a 15-s time step, but, to save space in the database, was subsampled down to a discrete grid of times with a 1-min spacing. In general, the times used in a precomputed model might or might not correspond to the times at which the DART\(^{\circledR}\) buoy data were actually collected relative to the start of the earthquake. To facilitate matching the observed data with an adjustable model, we use cubic splines to interpolate the model. Let g(t) represent the spline-interpolated model at an arbitrary time t for a particular unit source and DART\(^{\circledR}\) buoy. The adjustable model value corresponding to a measurement \(x_{n_l}\) from that buoy over a 15-s time window associated with the elapsed time \(t_{n_l}\) is just \(g(t_{n_l})\). A 1-min average \(\bar{x}_{n_l}\) consists of an average of \(x_{n_l-3},\,x_{n_l-2},\,x_{n_l-1}\) and \( x_{n_l}\), so its associated adjustable model is an average of \(g(t_{n_l-3}),\,g(t_{n_l-2}),\,g(t_{n_l-1})\) and \(g(t_{n_l})\).
Figure 2 shows an example of a spline-interpolated adjustable model g(t) (black curve), which is based upon values precomputed at 1-min intervals and stored in the database (black dots). This model is for DART\(^{\circledR}\) buoy 21414 for an earthquake originating from unit source a12, which is in the Kamchatka–Kuril–Japan source region (see Fig. 3 below). During the November 15, 2006 Kuril Islands event, this buoy transmitted a 1-min stream \(\bar{x}_{n_l}\) at times \(t_{n_l}\). These times did not coincide exactly with those of the precomputed model. The circles in the plot show the spline-interpolated values \(g(t_{n_l})\) versus \(t_{n_l}\), each of which would be the adjustable prediction for a single (unavailable) 15-s average \(x_{n_l}\) (the corresponding prediction for the available \(\bar{x}_{n_l}\) would be the average of \(g(t_{n_l})\) and three values associated with times occurring 15, 30, and 45 s earlier). As the example shows, the cubic spline interpolation provides accurate estimates of the model values at the times of the DART\(^{\circledR}\) observations.
5 Inversion algorithm for extracting source coefficients
The purpose of the inversion algorithm is to use data collected by DART\(^{\circledR}\) buoys (after appropriate detiding) to produce an estimate of the initial fluid conditions or source of an earthquake-generated tsunami. As noted in the previous section, the inversion algorithm depends upon a database of precomputed models to produce synthetic boundary conditions of water elevation to initiate the forecast model computation. These models are scaled to unit sources that are based upon a moment magnitude \(M_W\,=\,7.5\) inverse-thrust fault earthquake with the assumed parameters, each corresponding to a segment of a subduction zone. There is a model in the database for every pairing of a particular unit source with a particular buoy. This model predicts what would be observed at the buoy if an earthquake was to originate from a selected unit source. The inversion algorithm adjusts the amplitude of the precomputed model to match the time series observed at the DART\(^{\circledR}\) buoys. The adjustment takes the form of a multiplicative factor, which we denote by α and refer to as the source coefficient. This coefficient in effect defines the tsunami source or fluid origin of a tsunami as the initial sea-surface deformation resulting from the transfer of energy released by geophysical processes to the fluid body. A value for α that is greater (less) than unity means that the modeled initial sea-surface deformation is greater (less) than what would arise from the standard \(M_W\,=\,7.5\) earthquake scenario.
The inversion algorithm estimates α by matching the precomputed model and the detided data from the buoy via a least squares procedure. As discussed below, the algorithm takes into account the possibilities that the earthquake might be attributable to more than just a single unit source (so that the adjustments take the form of a vector \(\varvec{\alpha}\) of multiplicative factors) and that more than one buoy might have collected data relevant to a particular event. In Sect. 5.1, we present the inversion algorithm under the simplifying assumptions that we know (1) the appropriate unit sources associated with the earthquake and (2) the portions of the detided buoy data that are relevant for assessing the tsunami event (we pay particular attention to assessing the effect of sampling variability on our estimates of \(\varvec{\alpha}\)). Proper selection of the unit sources and of the relevant data is vital for getting good results from the inversion algorithm. We discuss source selection in Sect. 5.2 and data selection in Sect. 5.3. (For earlier related work on inversion algorithms, see Johnson et al. 1996; Wei et al. 2003.)
5.1 Estimation and confidence limits for \(\varvec{\alpha}\)
The sum of the estimated α_{k}’s is also of interest since this measure of overall source strength provides initial conditions for models that predict coastal inundation. The bottom plot of Fig. 5 shows this sum for each of the 55 pairs, along with associated 95% CIs (standard statistical theory says that the standard error for this sum is given by the square root of the sum of all the elements in \(\Upsigma\)). Taking sampling variability into account, this sum is remarkably consistent among pairs with at least one buoy in the main beam. There are three pairs that do not fit this description, namely, (5,6), (5,7), and (6,7). These are associated with the three \(\sum \hat{\alpha}_k\)’s that are markedly smaller than the other 52. Thus, for this particular event, 52 of the 55 pairs would have produced similar initial conditions for the inundation models, but more cases need to be studied to ascertain whether we can expect this degree of consistency to occur routinely. Note that, even though the overall sum is relatively stable, the partitioning of the sum among its three \(\hat{\alpha}_k\)’s differs a fair amount across the 52 pairs; however, in the far field, well away from the source, the ambiguity in partitioning of source strength among the three unit sources should not markedly influence any inundation forecasts since the sources are so close together relative to the distance separating them from the region to be forecast.
5.2 Selection of sources
Initial determination of number K of unit sources contributing to an earthquake event based solely on the moment magnitude M_{W} (Papazachos et al. 2004)
Criteria | K |
---|---|
\(M_W\,\leq\,7.9\) | 1 |
\(7.9\,<\,M_W\,\leq\,8.05\) | 2 |
\(8.05\,<\,M_W\,\leq\,8.3\) | 4 |
\(8.3\,<\,M_W\,\leq\,8.6\) | 6 |
\(8.6\,<\,M_W\,\leq\,8.8\) | 8 |
\(8.8\,<\,M_W\,\leq\,8.95\) | 15 |
\(8.95\,<\,M_W\,\leq\,9.09\) | 18 |
\(9.09\,<\,M_W\,\leq\,9.2\) | 21 |
\(9.2\,<\,M_W\,\leq\,9.3\) | 24 |
Once sufficient DART\(^{\circledR}\) data become available, we can use the inversion algorithm with the initial selection of unit sources to obtain estimates \(\hat{\varvec{\alpha}}\) of the source coefficients—these estimates are refinements of the initial determination based on seismic information alone. Because of the nonnegativity constraints, it is possible that some, say K′, of the source coefficients will be set to zero, so that only K − K′ unit sources are retained in the model. An examination of the CIs for the remaining coefficients might recommend dropping additional unit sources whose corresponding \(\hat{\alpha}_k\)’s are not significantly different from zero. The solid curve in Fig. 6(b) is the model that results from using the subset of data from buoy 21414 (indicated by the gray circles in the bottom panel of Fig. 3) to obtain the least squares estimates \(\hat{\alpha}_k\). Here, \(K'\,=\,3\) of the coefficients were set to zero, thus eliminating unit sources z13, a14, and z14 from the model, while retaining only a13; however, the 95% CI for the α_{k} corresponding to a13 indicates that its estimated coefficient is not significantly different from zero. The match between the models and the observed data is poor in both Fig. 6a, b—in fact the match in (a) based on seismic information alone is arguably more appealing visually than the one in (b) that makes use of the buoy data! By comparison, Fig. 6c shows that we can get a much better match with a different set of unit sources, namely, a12, a13, and a14 (as used in Figs. 3 and 4). This set was selected by trial and error from among all the units sources close to the epicenter. When source coefficients are subsequently used to provide initial conditions for models that forecast inundation, the contrast between Fig. 6a, b points out the possibility of actually degrading a seismically based forecast due to an inappropriate selection of unit sources in the inversion algorithm; on the other hand, the contrast between Fig. 6b, c suggests that an appropriate choice of unit sources can offer an improvement over the seismically based forecast. The interface for the SIFT application is designed to make it easy for an operator to add or remove unit sources, hence facilitating experimentation with various models.
5.3 Selection of data from buoys
Once we have an appropriate selection of unit sources, the inversion algorithm estimates \(\varvec{\alpha}\) based upon whatever selection of DART\(^{\circledR}\) buoy data we hand to it. It might seem obvious we would want to use as much data as possible since statistical theory would seem to suggest that, as the amount of data increases, the variability in \(\hat{\varvec{\alpha}}\) should tend to decrease, leading to a better estimate of \(\varvec{\alpha}\). There are, however, at least two reasons for entertaining smaller amounts of data. First, warnings to coastal communities must be provided in a timely manner—there is no luxury during a tsunami event of waiting for all possible relevant data to arrive from a DART\(^{\circledR}\) buoy. Second, empirical evidence suggests that models and data are not equally well matched across time. The quality of the match is time-dependent, suggesting that we focus on particular segments of the data for purposes of fitting the model. Here, we illustrate these points by considering the effect of data selection on the estimation of \(\varvec{\alpha}\) for the 15 November 2006 Kuril Islands tsunami.
This Kuril Islands tsunami event is one in which the very first wave is the largest when observed at buoy 21414. This is because there is an unobstructed path for the tsunami to propagate from the source of the event to this buoy (see Fig. 3). For this case, we are thus better off just using the data up to the first complete wave to estimate \(\varvec{\alpha}\) since the data and the model disagree substantially beyond that point; i.e., the explanatory power of the model decreases beyond the first complete wave observed at the buoy. There are other situations in which later waves can be larger, in which case it would be desirable to use a longer stretch of the data for estimating the coefficients \(\varvec{\alpha}\). The interface for the SIFT application makes it easy for an operator to select the data to be used in the inversion algorithm.
6 Discussion
While the current version of the SIFT application is fully functional, here we discuss some possible extensions to the software that might impact upcoming versions.
The SIFT application is capable of estimating tsunami source coefficients in near real time, but there is a need to provide operators with help in its use during a tsunami event. As discussed above, two critical elements in successful use of the SIFT application are choosing a set of appropriate unit sources and selecting appropriate subsets of DART\(^{\circledR}\) buoy data. Currently, these choices depend upon experienced operators, but, for operators with limited experience (and as potential guidance for experienced operators under time constraints during a tsunami event), it is desirable to look for ways to automate the selection procedures. The problem of selecting unit sources is closely related to the topic of variable selection in linear regression analysis, for which there is a considerable literature we can draw upon for ideas. Complicating factors are the dynamic nature in which the data arrive, the potential desire to have spatially coherent unit sources, the correlated nature of the errors, and the possible interplay between selecting unit sources and subsetting the DART\(^{\circledR}\) data. The problem of selecting appropriate subsets of DART\(^{\circledR}\) buoy data is related to the topic of isolating transients, for which wavelets and other techniques for extracting a signal from a time series with nonstationary behavior can be looked to for guidance. How best to automatically select unit sources and to subset the DART\(^{\circledR}\) data are subjects of ongoing research.
A complicating factor we have not discussed is contamination of the DART\(^{\circledR}\) buoy data from seismic noise. While the November 2006 Kuril Islands tsunami event and many others are relatively free from such noise, there are cases where seismic noise is co-located in time with the tsunami event itself in the DART\(^{\circledR}\) data (see Uslu et al. 2011 for one example). How best to eliminate this noise is also the subject of ongoing research.
Another complication is that a tsunami can arise from an earthquake whose epicenter falls outside the set of all predefined unit sources. This happened with the 29 September 2009 Samoa event, which—after the event—prompted the addition of new unit sources to the database. In such a case, the current strategy within SIFT is to pick unit sources whose distance from the epicenter is as small as possible. For the Samoa event, it was possible to get good fits to data from individual buoys using this strategy, but not to data from a combination of buoys. This occurrence points out the need for a fail-safe option within the SIFT application for an operator to be able to set up new unit sources on the fly. Currently, implementation of this option faces substantial technical challenges due to the amount of time needed to compute the models.
The primary use for the source coefficients that the inversion algorithm produces is to provide initial conditions for models that forecast inundation in particular coastal regions. Currently, the forecasts of wave heights and run-up in areas likely to be impacted by a tsunami do not take into account the uncertainty in the source coefficient estimates. Research is needed to determine how this uncertainty impacts these forecasts and how best to present this uncertainty to managers in charge of issuing warnings to coastal communities.
A secondary use for the source coefficients is to check that the size of the tsunami event is in keeping with the seismically determined moment magnitude M_{W}. The fact that we can assess the sampling variability in the estimated source coefficients allows us to say whether the strength of the generating event as determined by the inversion algorithm is significantly different statistically from M_{W}. Details are provided in Appendix 2, where we show that, for the Kuril Islands event, the two ways of assessing the strength give comparable results when sampling variability is taken into account.
The automated system in the current version of the SIFT application is designed to work for events with moment magnitudes M_{W} at or below 9.3 (implying an initial selection of up to 24 unit source as per Table 1). An operator at a tsunami warning center can manually match a set of sources to the DART\(^{\circledR}\) data that extend over large distances along a fault zone if the automated system is deemed to have picked an inappropriate distribution of sources. A possible avenue for future development would be to enhance the automated system for handling larger events, but there are a number of technical issues to overcome (e.g., the timing of the contribution from the sources might need to be adjusted when sources are spread out widely because the assumption of simultaneity might be violated).
Finally, we note that all computations and graphics in this article were done in the R language (Ihaka and Gentleman 1996; R Development Core Team 2010). Portions of the computational R code were translated into Java, which, with augmentations, is the basis for the SIFT application.
7 Summary and conclusions
The SIFT application is a tool developed at the NOAA Center for Tsunami Research to estimate source coefficients during an on-going tsunami event. These source coefficients are needed in a timely manner as input to inundation models that can forecast tsunamis at coastal communities. While the source coefficients can be set initially based solely on seismic information, experience has shown that these initial settings can be improved upon substantially by estimating the coefficients based upon DART\(^{\circledR}\) buoy data collected during the on-going event. The SIFT application is designed to compute these refined estimates soon after the DART\(^{\circledR}\) data become available by making use of a database of precomputed models. These geophysically based models predict what would be observed at each buoy given a standardized earthquake emanating from a set of unit sources. The refined estimates of the source coefficients are computed within SIFT via an inversion algorithm, which relates the data to the geophysically based models via a linear regression model. With suitable nonnegativity or nonpositivity constraints, this statistical model allows for physically interpretable source coefficient estimates, along with an assessment of their sampling variability. The model is formulated in a manner flexible enough to allow for arbitrary combinations of the different types of data reported by the DART\(^{\circledR}\) buoys (either pressure measurements integrated over 15-s time windows or averages of four such measurements, i.e., 1-min averages).
We demonstrated the efficacy of the inversion algorithm by applying it to data from the November 15, 2006 Kuril Islands event. This example shows that, assuming an appropriate choice of unit sources, estimates of the source coefficients based upon data from a single buoy produce a much better match to the observed DART\(^{\circledR}\) buoy data than what is provided by coefficients set using just seismic information (see Fig. 6). These refined estimates in principle would have been available no more than 2.5 h after the occurrence of the earthquake generating the tsunami. Use of data from an additional one to three buoys (available within 3–4 h after the earthquake) produces estimates of the source coefficients with sampling variabilities that are not substantially improved upon by using data from distantly located buoys (see Fig. 4). Operationally, this finding suggests that there is no need to wait for the tsunami to pass by more than two or three buoys in the hope of getting better estimates of the source coefficients (see Fig. 5 also). Models for the data fit using four buoys were able to predict quite well the pattern—but not the exact timing—of the tsunami as it passed by distantly located buoys, demonstrating the ability to model tsunami events on ocean-wide scales based on just three freely adjustable source coefficients (see Fig. 3).
While work is in progress to add more functionality to the SIFT application, it has already proven to be a valuable tool for assessing the potential hazards of tsunamis to coastal communities, in part due to the inversion algorithm that is the focus of this article. Pending successful completion of the ongoing research described in Sect. 6, future versions of the SIFT application will have features that should make it easier for operators to specify the input required for successful use of the inversion algorithm.
Notes
Acknowledgments
This work was funded by the Joint Institute for the Study of the Atmosphere and Ocean (JISAO) under NOAA Cooperative Agreement No. NA17RJ1232 and is JISAO Contribution No. 1783. This work is also Contribution No. 3557 from NOAA/Pacific Marine Environmental Laboratory. The authors would like to thank Bradley Bell for computer code and discussions on constrained least squares.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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