The frequency-magnitude distribution of earthquakes is modeled by the Gutenberg–Richter (G-R) law: 1log⁡10(N)=a-bM where N is the cumulative number of earthquakes with a magnitude greater than or equal to M. The parameter b describes the relative proportion of small to large earthquakes, while the a-value represents the overall seismic productivity of a region.

To determine a stable b value, the catalog was filtered to include only the modern instrumental period (1965 onward), corresponding to the establishment of the World-Wide Standardized Seismograph Network (WWSSN). The Magnitude of Completeness (Mc) was initially assessed using the Maximum Curvature (MAXC) method , which defines the completeness threshold by identifying the maximum value of the first derivative of the frequency–magnitude curve. For this dataset, the raw MAXC peak occurs at Mw = 7.05. However, because raw MAXC estimates frequently fall within the catalog's detection roll-over region and can therefore underestimate true completeness, we adopted a conservative operational threshold of Mc = 7.4. We emphasize that this threshold represents a modeling choice to ensure absolute catalog completeness rather than an exact, immutable catalog property. To ensure robustness, we evaluated the stability of the estimated b value over nearby completeness thresholds (e.g., Mc = 7.3 to 7.5), confirming that the principal basin-scale recurrence patterns are not materially altered. Based on the resulting complete subset of 212 events, a b value of 1.05 ± 0.07 was calculated using the Aki–Utsu maximum likelihood estimator (Fig. d), where the standard error is given by σb = b/N .

A Magnitude of Completeness (Mc) of 7.4 was mathematically determined using the Maximum Curvature (MAXC) method, identifying the point of maximum first derivative in the frequency-magnitude curve. Based on the resulting subset of 212 events, a b value of 1.05 ± 0.07 was calculated using the Aki–Utsu maximum likelihood estimator (Fig. d) . This value is consistent with the global average (b = 1), indicating a robust fit .

For the sake of applicability, a polygonal curve Γ(Lat, Long) was first defined to pass through the historical epicenters, representing both known and potential earthquake zones. The curve Γ was constructed by interpolating through the historical epicenters and aligning with major subduction interfaces and fault boundaries identified from global tectonic maps.

The continuous curve Γ was constructed by tracing the major tsunamigenic fault systems and subduction interfaces globally, discretizing this path into 10 km intervals. To account for the spatial scatter of historical earthquakes, epicenters located within a [400] km search radius were orthogonally projected onto the nearest segment of this curve before applying the 1D Gaussian smoothing.

To account for spatial variability in historical seismic occurrence (Fig. b), a spatially varying a(Γ) value was derived using one-dimensional Kernel Density Estimation (KDE) along the along-trench coordinate Γ. This approach mitigates data sparsity by smoothing the influence of historical events over their surrounding segments. A Gaussian kernel f(Γ) was applied at the projected 1D location (Γi) of each event: 2f(Γ)=1σ2πe-Γ-Γi22σ2 While near-field inundation modeling requires resolving complex 3D fault geometries, basin-scale transoceanic exposure screening is predominantly governed by macro-scale, along-trench variations in seismic productivity . Therefore, this dimensional reduction to a 1D longitudinal projection is an established methodology for macro-scale hazard analysis . A smoothing bandwidth of σ = 500 km was selected to satisfy established empirical scaling laws for subduction interface earthquakes. Because great megathrust events (Mw ≥ 8.5) are restricted by down-dip width saturation at approximately 200 km, their ruptures are forced to propagate longitudinally . Established length-scaling relations indicate that the characteristic along-strike rupture length for Mw 8.8–9.0 earthquakes spans roughly 440–588 km . The 500 km KDE bandwidth therefore physically scales the probability density to match realistic rupture extents, eliminating the severe epicentral over-fitting that occurs with narrower bandwidths. A quantitative statistical evaluation (Fig. ) confirmed that spatial variance stabilizes at this 500 km threshold, while the macro-scale spatial clustering remains stable and independent of the selected bandwidth (r > 0.95). This curve was subsequently normalized to match the total observed annual rate of events above Mc from the modern catalog, so that local variations do not inflate regional seismic productivity, producing an absolute annual rate density for each source point.

The final weight assigned to each source point i is its total annual rate of significant tsunamigenic earthquakes (M ≥ 7.4), calculated by applying the spatially determined a(Γ) from the KDE to the Gutenberg–Richter relationship and integrating over the relevant magnitude bins. This yields a single representative annual rate, λ, which serves as the input likelihood for the tsunami ray-tracing models. The corresponding exposure map, showing the average return period T = 1/λ, is presented in Fig. .

The resulting exposure map reveals significant spatial heterogeneity in earthquake recurrence. The Kuril–Kamchatka and Japan Trenches in the northwest Pacific, alongside the Peru–Chile Trench in the southeast, emerge as the most seismically active source zones, exhibiting return periods between 100 and 1000 years (red to orange segments). This creates a dominant “exposure dipole” across the Pacific Basin. Additionally, the Aleutian Trench and the Sumatra–Andaman subduction zone show high recurrence rates, highlighting them as critical sources for transoceanic tsunami generation. In contrast, mid-ocean ridges and transform boundaries generally exhibit longer return periods (blue segments), reflecting their lower potential for generating large tsunamigenic displacements.

In this study we adopt a recurrence-based weighting, though we also examined whether magnitude-energy formulation would alter the resulting coastal exposure patterns. Using the same magnitude bins between M = 6.5 and 9.5, we computed an annualized energy metric Λj for each segment j by multiplying each bin's rate by E(M)∝101.5M and summing across bins. However, since the PTHA implementation employs a single regional b value and the same magnitude range for all source segments, the Gutenberg–Richter distribution has an identical shape everywhere. Under these assumptions Λj reduces to a constant multiple of the recurrence rate (Λj = Cλj), and thus energy-weighted ray-density field would be indistinguishable from the recurrence-weighted field up to a uniform rescaling. For this reason we present the recurrence-based maps, explicitly.

The model employs a kinematic ray-tracing approach. Rather than solving the dynamic transport equation for wave amplitude, which is prone to caustic singularities over complex deep-ocean bathymetry, we strictly solve the kinematic equations to track the trajectories of discrete wave energy packets. The state of a wave ray is defined by its position vector, x = (x,y), and wave number vector, k = (kx,ky). Their evolution in time t is governed by: 3dxdt=cg+U4dkdt=-∇hω-k⋅∇hU Here, U = (U,V) is the ambient current velocity vector, cg is the intrinsic group velocity vector, and ω is the intrinsic angular frequency of the wave. For far-field tsunami applications, ambient currents are negligible compared to the wave speed, so we set U = 0 in Eqs. ()–(). The angular frequency ω is related to the wave number magnitude k = |k|=kx2+ky2 and local water depth d through the linear dispersion relation: 5ω2=gktanh⁡(kd)

Wave refraction is governed by the term -∇hω in Eq. () (where U = 0). For the characteristic period spectrum evaluated in this study, the waves strictly satisfy the shallow-water limit even in the deepest oceanic trenches. Because the phase speed remains overwhelmingly depth-dependent, the bathymetric refraction gradients that dictate the ray trajectories are fundamentally unaffected by frequency dispersion. Since ω depends on both k and d, its horizontal gradient is primarily driven by bathymetric variations: 6∇hω=∂ω∂d∇hd and because no diffraction term is included here, rays cannot reproduce energy leakage around obstacles. This shows that changes in the wave number vector, and thus the propagation direction, are directly proportional to the local sea-floor gradient, ∇hd, which is the main mechanism for tsunami refraction across ocean basins.

The system of ODEs (Eqs.  and ) is solved numerically to trace each ray's trajectory using a fourth-order Runge–Kutta scheme. Rays propagate over a global, unstructured triangular mesh comprising 1 726 063 nodes and 3 452 122 elements, with spatial resolution varying from 10.77 to 28.35 km (Fig. ). The variable-resolution unstructured mesh is generated based on the shallow-water wave, ensuring that tsunami wavelengths are adequately resolved. During the mesh generation process, bathymetric gradients and all relevant geometric features that affect the wave refraction under study are explicitly taken into account. This feature-tracking preserves the steep bathymetric gradients necessary for accurate refraction, forcing finer nodal placement (down to 10.77 km) along critical structures like seamount peaks and trench fracture lines, while allowing coarser resolutions (up to 28.35 km) in relatively flat abyssal plains, without requiring a uniform, ultra-high resolution domain.

Bathymetric data are sourced from 2025 General Bathymetric Chart of the Oceans dataset with 15 arcsec resolution. Using a time step of 180 s and Courant–Friedrichs–Lewy CFL < 1, the depth-dependent resolution is set as the product of the time step and local group velocity, ensuring that no ray travels more than one element per time step. At each time step, environmental parameters are evaluated dynamically, as rays can lie anywhere within a mesh element. Water depth at a ray's latitude–longitude location is computed using barycentric interpolation from the nodal values of the containing element. The bathymetric gradient, ∇hd, which drives wave refraction, is precomputed for all elements to improve computational efficiency.

Rays are initialized along potential tsunami-generating fault lines, shown in Fig. , at 25 km intervals. To ensure comprehensive coverage of potential refractive pathways, rays are initialized across a discrete spectrum of characteristic tsunami periods spanning the typical empirical range of 15 min to 3 h at each source location. By aggregating this discrete period spectrum globally, the kinematic model effectively bounds the frequency-dependent variations in transoceanic energy focusing to construct the final exposure footprint. While real near-field radiation directivity depends on detailed 3D rupture kinematics – such as variable slip distribution, rake, and dip, the dominant macro-scale energy flux of large subduction megathrust earthquakes radiates predominantly orthogonal to the fault strike. Therefore, the initial propagation direction for all rays is set perpendicular to the local fault segment. This simplification natively encodes the primary radiation lobes required for transoceanic far-field propagation without necessitating complex 3D volumetric initializations. Furthermore, rays are oriented exclusively oceanward; shoreward-directed waves are excluded because their propagation is governed by local near-source complexities and non-linear shoaling, which fall outside the scope of basin-scale transoceanic exposure screening.

A ray is terminated if it reaches land (where water depth becomes zero or negative) or enters a user-defined shallow coastal zone (500 m depth). This cutoff is imposed to exclude the complex nearshore region where nonlinear shoaling dominates and where strong refraction often leads to ray crossing (caustics), rendering the standard geometrical optics approximation insufficient. Furthermore, as waves enter these shallower depths, their decreasing group velocities and shorter wavelengths would require a proportionally higher-resolution mesh. Implementing such fine resolution along global coastlines would impose prohibitive computational costs, directly defeating the primary utility of this framework as a rapid transoceanic exposure screening tool. In total, 9000 rays were released, and the simulation was run for 24 h, sufficient for all rays to reach a termination criterion. Upon termination, the spatial density of ray impacts along the coastline is calculated. By utilizing a ray-density proxy rather than a classical ray-tube energy flux calculation, the model natively bypasses the mathematical singularities associated with deep-water caustics (ray crossings). In this kinematic framework, caustics simply and stably register as localized zones of high ray concentration. This normalized ray termination count (ranging from 0 to 1) serves as a proxy for the relative probability of exposure (focusing potential), rather than a measure of dynamic wave amplitude or absolute energy flux, effectively mapping regions of energy convergence (focusing) and divergence (shadowing). This framework allows efficient and accurate simulation of tsunami propagation across complex global bathymetry, illustrating how basin geometry controls wave energy distribution. The simulations were performed on a 32-core Mac Studio with 512 GB of unified memory. The code is parallelized, and since each ray is treated independently, the 24-hour simulation completed in approximately 12 h. This ray-tracing approach captures tsunami exposure primarily for transoceanic propagation from the source across the basin to distant locations. Hazards in the immediate vicinity of the earthquake are not considered in this study, as our focus is on the effects of ocean-basin geometry on tsunami travel; near-source tsunamis may indeed be more catastrophic, but they fall outside the scope of this work.

To systematically examine how source geometry, recurrence intervals, and bathymetric refraction shape global tsunami exposure, we designed four targeted numerical experiments. Each scenario isolates a fundamental driver of tsunami propagation: (i) fault geometry, which controls the initial azimuth and directional bias of energy release; (ii) variable bathymetry, which refracts and redirects wave trajectories across the basin; and (iii) probabilistic weighting, which modulates exposure intensity based on earthquake return periods. Figure  summarizes the outcomes, where basin-scale ray density is shown using a white-to-blue scale, and coastal focusing is highlighted by a green–yellow–red scale. All results are normalized by the global maximum to enable direct comparison across experiments.

Figure a–b adopt a constant-depth approximation, forcing rays to follow great circle paths (GC) and, by design, removing any effect of bathymetric refraction. This configuration exposes the sole influence of fault geometry and position, effectively reducing the ocean basin to a two-dimensional line-of-sight system. Under these conditions, a highly energetic corridor extends across the Pacific Basin, originating from the Kuril–Kamchatka and Japan trenches. This direct projection leads to a broad signature of exposure, producing normalized coastal exposure indices between 0.8 and 1.0 along the northeast coast of Australia, Papua New Guinea, and the Solomon Islands, as well as along South America. Panel (a), using uniform ray release, overemphasizes the role of smaller fault systems; panel (b) incorporates the probabilistic weights (λ) from Sect. , refining the magnitude but retaining the linear propagation patterns inherent to the constant-depth assumption, keeping exposure indices artificially high (∼ 0.8–1.0) along these direct line-of-sight coastlines.

In Fig. c–d, where realistic bathymetry is introduced, refraction and scattering fundamentally reorganize the tsunami wave-field. For example, the exposure corridor toward Northeast Australia is reduced by approximately 80 % (dropping from an exposure index of 0.8–1.0 down to 0.1–0.2) as rays encounter the complex topography of the Southwest Pacific – namely the Melanesian Arc and the Solomon Rise. In contrast, topographic structures in the North Pacific, such as the Emperor Seamount Chain and the Mendocino Fracture Zone, act as waveguides, channeling energy toward North America, increasing normalized coastal focusing in the Pacific Northwest by over 50 % (elevating the index from diffuse values of 0.4–0.6 up to highly localized peaks of 0.9–1.0) relative to the uniform depth case.

Panel (d) represents the most complete exposure characterization, combining realistic bathymetry with the probabilistic source weighting (λ). By modulating the intensity of each ray based on its source's annual occurrence rate, this configuration filters out low-probability contributions seen in panel (c). Notably, it reveals a reciprocal focusing pathway between South America and East Asia: the Kuril–Kamchatka system preferentially feeds energy toward Chile (index > 0.8), and the Peru–Chile trench returns a concentrated ray field toward Japan and the Philippines. Additionally, the model identifies a pronounced shadow zone in the Southeast Pacific – west of the East Pacific Rise – where ambient basin ray densities of 0.3–0.4 (seen in panel b) are reduced to near zero (< 0.05). Finally, the probabilistic approach intensifies the signal over the Bay of Bengal and the Indonesian archipelago, elevating the normalized exposure index from background levels (∼ 0.3–0.5 in panel c) to peak values exceeding 0.8, where high recurrence rates of tsunamigenic earthquakes lead to elevated exposure levels.

For the simulation of tsunami wave propagation, we use the Boussinesq-type wave model, FUNWAVE, in spherical coordinates . The model has been validated against a large set of analytical, laboratory, and field benchmarks as part of the development of tsunami hazard maps for the US East Coast . An optimized MPI parallel implementation of FUNWAVE was developed, which has highly scalable algorithms, with a typical parallel scaling efficiency of more than 90 % of the number of cores in a typical medium-sized computer cluster. This makes it possible to run the model over large ocean basin-scale grids with a sufficiently fine resolution. The grid used in FUNWAVE simulations was developed from , which was downscaled from a resolution of 15 arcsec to 4 arcmin and cropped to focus on the Pacific Ocean, resulting in a final rectilinear grid comprising over 5 million points. Simulations were executed on an AMD EPYC 7H12 processor with a 2.6 GHz 64-core (128-thread) architecture, requiring approximately 50 min of real time to simulate 32 h of tsunami propagation.

To mitigate wetting and drying numerical artifacts along the coastline while preserving the steep bathymetric gradients of the Kuril–Kamchatka subduction zone essential for accurate wave generation, a spatially selective smoothing approach was adopted. A ramp function (Fig. ) was constructed around the 10 m depth contour to linearly transition from the original bathymetry in the deep ocean to a Gaussian-smoothed grid in shallow coastal waters. This approach effectively ensures numerical stability in the nearshore region without compromising the bathymetric fidelity at the earthquake epicenter. Additionally, sensitivity simulations conducted on lower resolution 8 and 16 arcmin grids yielded negligible differences in results, confirming that the chosen 4 arcmin resolution is adequate for resolving the initial seafloor displacement and modeling transoceanic tsunami propagation.

For both cases, NOAA and USGS, FUNWAVE simulations were performed using the bed deformation as the initial free surface water height. Figure  depicts the time history of generation and propagation of tsunami waves in the computational domain for the NOAA case with zero initial velocities. Simulations initialized with velocities from the elastic half-space model overestimate wave heights at DART stations for both the NOAA and USGS cases. In contrast, simulations with zero initial velocities provide more accurate predictions of wave heights at the DART locations (disregarding the phase lag), exhibiting minimal over-prediction or under-prediction of wave heights. This result indicates that within the framework of Boussinesq-type models like FUNWAVE, the initial bed-deformation is sufficient for effectively modeling tsunami wave propagation accurately.

With the hydrodynamic model successfully validated against DART observations, we now evaluate the spatial distribution of tsunami energy across the Pacific basin. This step bridges the probabilistic and geometric frameworks outlined in Sects.  and  with the deterministic, high-fidelity simulation of the 2025 Kamchatka Peninsula event.

To evaluate the predictive capability of the ray-tracing framework, the model was initialized using the fault parameters of the Kamchatka event (NOAA scenario) listed in Table . A total of 270 rays were emitted perpendicular to the 540 km fault line (strike 217°) with a 3200 s (∼ 53 min) characteristic period, enabling a realistic representation of the initial radiation pattern, and were then tracked across the Pacific basin to identify convergence zones.

The resulting normalized ray termination counts are shown in Fig. a. Note that ray-based shadow zones appear sharper because diffractive spreading is absent in geometric optics. The simulation reveals clear corridors of energy focusing. In the near-field, high ray densities emerge along the Aleutian Islands and Alaska, primarily governed by the fault orientation and source proximity. In the far-field, bathymetric refraction organizes tsunami propagation into a dominant southeastward beam that first targets the Hawaiian Islands and then continues into the Southeast Pacific, ultimately forming a coherent refractive corridor that extends southward toward the coast of Chile. A second beam projects eastward toward the North American coast, notably targeting Washington, Oregon, and Northern California.

These geometric patterns are strongly supported by the full hydrodynamic simulation. Figure b shows the maximum wave amplitude envelope computed by FUNWAVE. The spatial distribution closely aligns with the convergence zones identified in panel (a), while also capturing additional physical processes absent in the ray-tracing model. First, the ray-based simulation produces sharp shadow zones behind island chains (e.g., southeast of Hawaii), an expected consequence of its geometrical optics approximation. In contrast, FUNWAVE shows softer gradients, where diffraction allows partial energy penetration into these shadow regions. Second, the hydrodynamic solution reveals faint concentric ripples radiating from the continental margins-reflected energy that is naturally captured in FUNWAVE but purposely omitted in the ray-tracing model, where rays terminate upon coastal impact.

While a strong qualitative agreement is visually evident in the spatial alignment of energy corridors shown in Fig. , we further evaluated the model's performance statistically. To evaluate far-field predictive performance against the fully coupled FUNWAVE hydrodynamic simulations, a Spearman rank correlation was calculated for spatially co-located coastal amplitude distributions. To isolate far-field refractive propagation from localized static coseismic deformation, 161 coastal observation points were retained after explicitly excluding all points within a 1000 km radius of the epicenter, yielding a baseline correlation of ρ = 0.65.

Because standard significance testing ignores the spatial autocorrelation of coastal tsunami amplitudes, a spatial block-bootstrap resampling scheme was implemented to estimate unbiased confidence bounds. The remaining coastline was partitioned into contiguous 500 km segments along the coastal trajectory, which were randomly sampled with replacement over 10 000 iterations to preserve local spatial dependence. This methodology yielded an unbiased 95 % confidence interval of [0.35,0.83]. Because the confidence bounds remain strictly positive, the correlation is statistically significant (p < 0.01). Sensitivity analyses across 300 to 800 km block sizes confirmed that while confidence interval widths scale inversely with effective degrees of freedom, the lower bounds remain confidently above zero, confirming that the correlation is robust and independent of the selected bin size. Ultimately, despite underpredicting amplitudes in diffraction-dependent coastal shadow zones, the spatially adjusted analysis confirms a highly significant, physical monotonic relationship between the rapid screening predictions and high-resolution hydrodynamic benchmarks in the far-field.

Finally, these modeled hotspots are corroborated by in-situ observations. Figure  presents the maximum water levels recorded at NOAA tide gauge stations. The observational record exhibits strong agreement with both the ray-tracing and hydrodynamic models. The highest amplitudes (red and orange markers) occur precisely where both frameworks predicted energy focusing – along the Hawaiian Islands and the U.S. West Coast. Importantly, the secondary southern beam identified in both the ray-tracing solution (Fig. a) and the hydrodynamic envelope (Fig. b) is confirmed by an elevated water level at a gauge in the Southeast Pacific (Fig. ), demonstrating the model's ability to resolve narrow, refractive transoceanic pathways over great distances.

Collectively, these results show that when informed by accurate source mechanics, the ray-tracing framework serves as a robust and computationally efficient proxy for identifying far-field coastal exposure and quantifying basin-scale tsunami exposure.