Absract.  Until  recently, the Bering  coast  of Kamchatka 
was not considered as an area  with  a high   level   of  tsunami  
risk despite the occurrence   of   7.6   magnitude   tsunamigenic
earthquake  near  Ozernoy Cape  in  November of 1969.  However, a 
series  of  6.9-7.1  earthquakes occurred  in  1989-1991  in  the 
northern part of the  Koryak  autonomous region, have  risen  the
public  concern  about  the   impeding   large earthquake in this 
area.  Since a large part  of the  potential  seismic-prone  area  
is located under the sea bottom, a future earthquake can generate   
a hazardous tsunami with  the destructive  effect  for   numerous    
fishing  villages  located  on  the  sand spits,  near  the river 
estuaries and other low-lying areas. That is  why  the evaluation 
of  tsunami  risk  for  the  Bering  coast  of Kamchatka  was  an 
essential  part  of  the  project on re-estimation of the seismic 
hazard   for  the  Koryak  autonomous    region.  Application  of  
the   conventional  method   of  tsunamizoning,  based   on   the  
straightforward stochastic evaluation of historical  data, is not  
possible  for this area due to  nearly the absence of  historical 
tsunami data.  In this study we use the deterministic approach to  
this problem,  based  on  delineation  of  potential tsunamigenic   
zones,   assigning   the  source  parameters for  the   so-called    
'design earthquake', application of numerical models  of  tsunami  
generation  and  propagation  in order  to  obtain  the  computed 
mareograms  at  the  important  coastal  points and their further  
analysis to estimate the possible run-up heights along  the coast.

1. Introduction

A considerable part of the earthquake-prone zone in the north-eastern Kamchatka is located under the sea bottom, that produces a potential tsunami danger for the eastern coast of the Koryak Autonomous Region (ÊÀR). During many years the reality of this danger was neglected. It is enough to say, that in all studies of tsunami risk for the Kuril- Kamchatka coast (see, for example, Atlas..., 1978; Go et al., 1984, 1986; Pelinovsky, Plink, 1980) the part of Kamchatka coast to the north from Komandorskiy Islands was not considered at all. The occurrence of a strong (magnitude 7.7) tsunamigenic earthquake on November 23, 1969, which generated tsunami waves with heights up to 10-15 meters near the Ozernoy Cape, has appeared to be completely unexpected. However, since it has occurred near the practically unsettled part of Kamchatka coast and has not caused a significant material loss (and furthermore, human victims), this event did not result in the revision of the tzunamizoning maps and the re-estimation of tsunami risk for north-eastern Kamchatka.

The recent findings of geological traces of paleotsunamis on the south coast of the Karaginskiy Island and in several places on the western coast of the Karaginskiy Bay gave direct evidence that this area experienced tsunami attacks in the past. Many fishing settlments on the Bering coast of ÊÀR are located on pebble-sand spits near the coastal lagoons and the mouth of rivers with elevation not higher than 2-3 meters above the sea level. Some of them do not have safe ways and places for evacuation, that makes them especially vulnerable even in case of a moderate tsunami. That is why the study of potential tsunami risk for the Bering coast of Kamchatka is the important part of the on-going research project on revision of seismic zonation maps for this area which started in 1993 at the Institute of Volcanology in Kamchatka after a series of felt earthquakes with magnitudes 6.5-7.1 occurred in 1989-1991.

1. Possible approaches to the problem of evaluation of tsunami risk

The available methods of the tsunami hazard estimation fall into two main categories: the straightforward historical stochastic evaluation and the so-called fully deterministic approach.

The stochastic method is based on available mareograph records and tsunami run-up measurements and does not involve any numerical models or seismotectonic consideration. Most effectively it can be applied to tsunami catalogs spanning period longer than the local seismic cycle. Its application to the given tsunamigenic area involves the following basic steps:

1. compilation of historical tsunami observations for a particular site;
2. assuming the type of statistics;
3. obtaining the historical exceedance rate graphic;
4. obtaining annual probability of exceedance.

The primary disadvantage of this method is its unreliability at annual probabilities lower than the inverse period of the catalog (provided that within this period the catalog can be considered as complete, that is not a case for many tsunamigenic regions). One of advantages is that this method naturally integrates the estimates of risk for local and distant tsunamis.

The fully deterministic approach involves the application of numerical models of tsunami generation, propagation and run- up for the calculation of expected wave heights along the coast for one or more hypothetical sources representing potential tsunamigenic earthquakes. Assuming some set of parameters of these sources ('designed earthquake') and having the digital bathymetry of the area with necessary degree of detalization, it is possible to calculate the expected run- up heights at any particular coast. One of advantages of this method is its applicability to any coastal point regardless the availability and completeness of historical data on tsunami occurrence. However, within this approach it is quite difficult to estimate the probabilities of excedance of the obtained run-up heights.

Application of the stochastic approach to the estimation of tsunami risk of the Bering coast of Kamchatka is not possible due to nearly the full absence of historical tsunami data for this area. In fact, the regional tsunami catalog contains data only for two events - the 1960 Chilean tsunami and the 1969 Ozernoy tsunami. That is why in the this study we have to use one of modification of the deterministic method based on the so-called 'scenario approach'. This approach is widely used for tsunamizoning of areas with the lack of historical data and insufficient knowledge of the seismicity pattern of the territory (Hebenstreit, Whitaker, 1981: Whitemore, 1993; Bernard et al, 1994). This approach involves the following basic steps:

1. delineation of potential tsunamigenic zones;
2. assigning the source parameters for the so-called 'designed earthquake';
3. application of numerical models of tsunami generation and propagation in order to obtain the computed mareograms at the important coastal points;
4. detailed investigation of selected areas at the coast on the basis of application of the run-up models.

The present work describes the results obtained at the first three stages of application of the above approach to the evaluation of tsunami risk for the Bering coast of Kamchatka.

2. Model of tsunami generation

For the study of tsunami generation, the most adequate mathematical model is the solution of a closed system of equations of the dynamic theory of elasticity, describing the oscillations of layered elastic half-space (the model of the Earth crust and the upper mantle) coupled with an overlaying compressed liquid layer (the model of the ocean). This approach to the tsunami generation was first proposed by Podyapolskiy (1968) and than used by Gusiakov (1974): Yamashita, Sato (1974); Ward (1980). Comer (1984) has shown, that within the framework of long-wave approximation, the solution of the fully coupled problem of tsunami generation is equivalent to the consecutive solution of two separate problems: (1) determination of static bottom deformation due to a buried seismic source and (2) calculation of tsunami propagation within the framework of the long wave theory in an ocean with the variable depth using the solution obtained at the first stage as the initial condition for the tsunami generation. This approach is widely applied in the numerical modeling of real historical tsunamis in the different parts of the Pacific and elsewhere and in the cases, when parameters of seismic source are known, allows to obtain the reasonable agreement of computed and observed mareograms.

As the first stage of the tsunami generation problem, we solve the problem of calculation of static displacement on the surface of an elastic halfspace induced by the action of distributed internal source of dislocation type. This problem is reduced to the solution of equilibrium equation

(l+m) grad div U + mdU + F = 0 (1)

with the following boundary condition at the free surface of a halfspace (at z=0)

G_z=t_xz=t_yz=0 (2)

Here U(x,y,z) - displacement vector, m and l - Lame's parameters of the elastic media, F - internal force of an unite value. The solution of problem (1) - (2) is built first for the single vertical and horizontal point forces, then it is generalized for dipole-type point sources modeling the dislocation over infinite element of internal rupture. By numerical integration, the solution is generalized for an finite rectangular source modeling an inclined shear buried fault. This source is described by 6 parameters such as length of a fault L, its width W, depth of the upper edge of a fault h₀, dip-slip angle, strike-slip angle g and amount of displacement over a fault D₀. As a measure of intensity of such a source, its seismic moment, defined as

M₀ = m * L * W * D₀ (3)

is used. The static deformation of the surface of an isotropic elastic halfspace induced by this source is calculated by the program described in (Gusiakov, 1978) and than used as the initial conditions for the program of calculation of tsunami propagation in the ocean with a real bathymetry.

According the regional seismotectonics, the most typical source mechanism for the earthquakes in this region is the reverse dip-slip fault with dip angle s=70°, strike slip y = 90° and strike direction (azimuth) 0 = 225°. Its other parameters is assumed to be as follows: length of the fault L = 100 km, width of the fault W = 40 km and displacement Do = 2 m. The seismic moment of this source calculated by (3) is equal to 4 * 10¹⁰ N*m . According to the correlation formula (Aki, 1972)

M_s = (lg M₀ - 9.0)/1.5 (4)

this value roughly corresponds an earthquake with the surface wave magnitude of 7.6.

3. Numerical model of tsunami propagation

As a model of tsunami propagation in an ocean with variable depth, we use the nonlinear shallow-water system which is solved numerically by the finite-difference method:

u_t + uu_x + vu_y + gn_x = gD_x

v_t + uv_x + vv_y + gn_y = gD_y (5)

H_t +(uH)_x + (vH)_y = 0

Here u(x,y,t) and v(x,y,t) - horizontal components of velocity of fluid particles in x and y directions, correspondingly, h(x,y,t) - the vertical displacement of the free surface of liquid, t - time, g - acceleration of gravity, H(x,y,t)= h(x,y,t) + D(x,y,t) - ocean depth. The movement of the bottom is described by function D(x,y,t) = U_z(x,y)f(t), where f(t)- time-dependent function describing the bottom movement in the source area.. Function D(x,y,t) is given in some area S_o-S, which is called a tsunami source area; S - area of the solution of system (3), having boundary G, consisting of two parts: G₁, representing the coastline, and G₂ , representing a free boundary in the open ocean. At the boundary G₁ we put the conditions of full reflection from the vertical wall:

d(h)/dn = 0 and Un = 0 (6)

(n is the single normal vector). At the boundary G2 we set the 'free passage' conditions, simulating the free exit of wave disturbance from the region of computation. The system (3) is solved numerically using so-called splitting method described in (Titov, 1988) and variable computational grids. The time-dependence function f(t) is taken as:

0, at t < 0

f(t)=t/t, at 0 < t < t (7)

1, at t > t

where t is the characteristic time of the source function.

The available observations show, that the main part of displacement in an earthquake source is accomplished in rather short time (less than 1 min), which is rather small as compared to the typical period of tsunami (10 - 15 min). Numerical experiments show, that variations of this parameter within 10 - 100 sec practically do not change the amplitude and the form of tsunami waves arising within this model, so that the above parameter was accepted equal 3 s (one time step of numerical scheme) used for the solution of system (3). The displacement of bottom is calculated within the rectangular area with dimensions 100 by 150 km in nodes of the same grid, which is used for calculation of tsunami propagation. Estimates show, that this area contains more than 90% of the total volume of bottom displacement, so that we can neglect the input in the tsunami generation from the remaining area S-S_o. The calculation of tsunami propagation is carried out on numerical grid with dimension 240 on 300 for tsunamigenic zone 1 and 160 on 360 for zone 2. The gridded 1- min bathymetry of this area was obtained by digitizing of specially prepared bathymetric chart of scale 1:1000000.

4. Position of model sources

As a potential tsunamigenic zone, we adopt the eastern submarine part of the earthquake-prone zone obtained as the result of re-estimation of seismic risk for the territory of the Koryak Autonomous Region. We divide it into two parts: the south-western part (zone1) and the north-eastern part (zone 2) outlined by dotted line in Fig.1.

Fig.1 Map of seismicity of north-western Kamchatka for the period from 1912 to 1992. The dotted lines show the position of potentially tsunami-prone zones.

Fig.2. Positions of model earthquake sources for tsunami computation.

Fig.3. Static bottom displacement for source N 1 (see Fig.2). Digits near isolines mean the vertical bottom displacement in cm. The solid dots indicate to the coastal points where computed mareograms were obtained. The inserted figure shows the position of these points on Il'pyrskaya Spit.

Fig.4. Perspective view (from the south-east) of the initial displacement of water surface for model source N1 (above) and the wave field at t=30 min after an earthquake (below).

Fig.5. Computed waveforms for model source N1 (position shown in Fig.2). The horizontal axis shows the time after an earthquake (in hours). The scale of the vertical axis (water level displacement) is shown on the left. Numbers near the computed waveforms correspond to the numbers of coastal points shown in Fig.3.

The positions of both zones roughly coincide with the position of the continental slope in this part of the Bering Sea. Because there is not enough data for deeper detalization of seismotectonic structures beneath the Karaginskiy Bay, we consider that the probability of tsunami occurrence and the type of a source mechanism are the same in each of the selected tsunamigenic zones. Altogether, we have considered six variants of the position of a tsunami source numbered from 1 to 8 and shown in Fig.2 as the projection of a fault plane to the surface.

This set of sources covers practically all area producing the potential tsunami threat to the coastal villages located at the Bering coast of the KAR (their locations are also shown in Fig.2).

5. Results of calculations

The frameworks of this paper do not allow us to give a detailed description of results of the whole set of numerical experiments made (for details, see Gusiakov, Marchuk, 1996). Here we present the results of tsunami calculation for model source 1 (Fig.2) and then briefly describe changes in the wave pattern for other positions of a source.

The bottom static displacement for variant 1 is shown in Fig.3. The left (land-side) part of the source area uplifts with the maximum amplitude of about 1 m, the right (ocean- side) part subsides to the value of about 25 cm. The total volume of the bottom deformation turns out to be equal to 4.3 km³, and the initial (static) tsunami energy is equal to 9* 10¹² J. The perspective view of the wave surface for source 1 is shown in Fig.4a, Fig.4b for two moments of time. The computed wave forms at 18 coastal points are shown in Fig.5.

For variant 1, the waves of maximum heights (of about 2.2 m) come to point 12 (near the Il'pinskiy Cape), which is located in the direction of maximum energy radiation from this type of source. For coastal points from 13 to 16, located inside the Korf Bay, the amplitude of waves continuously decreases due to the refraction on the shallow water surrounding the Cape Goven. At points 10 and 11 on the Il'pyrskaya Spit (where the fishing village Il'pyrskiy, having population of about 400 people, is located) the calculated wave heights vary from 1.2 to 1.4 m. Taking into account the increment of the wave height during run-up process and possible combination of tsunami and high tide (with amplitude up to 2.2m) , the resulting wave height in

this part of the coast can reach the mark of 4.0-4.5 m. In this case, the waves can overflow the narrow (less than 300 m) and low (2-4 m) sand spit and produce the heavy damage in the Il'pyrskiy village. Besides all, due to its geographical position, this village does not have any safety ways and places for evacuation of its inhabitants.

Despite the tsunami source in variant 1 is located pretty close to the coastline, due to shallow depths in the northern part of the Karaginskiy Bay, the propagation time for the first wave is about 35 min to the nearest coastal village (point 13) and of about 60 min for Tilichiki, the largest coastal village in this area. This allows some limited time (20-25 min) for countermeasures, provided that the tsunami warning is timely issued by the regional Tsunami Center in Petropavlovsk-Kamchatskiy.

Now we will briefly consider the changes in wave patterns caused by changes in the position of the seismic source. It is well known, that for other equal conditions, the depth of water in the source region is one of the main factors controlling the tsunami generation. In variant 2, the increased water depths result in the increase of the maximum wave heights at the coast by about 20 %. In another case (variant 3), where average water depth is much shallower, the resulted amplitudes decrease by 25-30 % as compared to variant 1. The location of the source entirely inside the Korf Bay (variant 4) results in the considerable decrease of wave heights along the whole coastline (mainly, due the shallow depth in this bay). In this case, seiche oscillations of the bay may occur, resulting in water oscillations with amplitudes up to 1 m and lasting up to 5-6 hours after the event.

In variant 5, the source locates on the continental slope of Karaginskiy Island, so that the significant part of residual displacement falls directly on the island. This fact as well as the shielding effect of the island itself results in significant (in 4-5 times) reduction of wave heights at almost all the coastal points considered (except for point 18). This fact allows us to conclude that the major earthquake in the area of Karaginskiy Island does not impose a serious tsunami threat for the northern part of the Being Coast of the KAR.

In variant 6, where the source is located between the Ozernoy Cape and the southern end of Karaginskiy Island), the amplitudes of waves in the northern part of the Bay decrease considerably with the maximum height of about 30-50 cm. The propagation time of the first wave to the northern part of the Bay grows up to 2 hours. In this variant, only the settlements Khailyulya and Ivashka turn out to be in the potentially dangerous zone (wave heights are 1.5 and 1.8 ì, respectively).

Of the two considered positions of a seismic source within tsunamigenic zone 2 ( Fig.2), variant 8 is most dangerous for both main villages, Pakhacha and Apuka, located at the northern coast of the Olyutarskiy Bay. The maximum run- up here can be as high as 4 m. In this case, a considerable part of the wave energy can also penetrate inside the northern part of Karaginskiy Bay, resulting in 1.5 m waves near Il'pyrskaya Spit (at points 10 and 11).

6. Conclusion

The main result of our study is the verification of the statement, that a major submarine earthquake with magnitude of the order of 7.6 with the epicenter on the continental slope of the Komandorskiy Basin can produce a real tsunami threat for numerous fishing villages located in the northern part of Karaginskiy Bay. The expected tsunami wave at the nearest parts of the coast may be as high as 4 meters. Various secondary effects (coastal and underwater landslides), the probability of which is rather high in this region, characterized by a low level of background seismicity and by a large recurrence period of major earthquakes, can result in the significant (2-3 times) increase of heights of waves in comparison with those obtained in our calculations.

The authors wish to thank A.V.Osipova for the help in data collection and figure preparation. This study was supported by RFBR grants 96-05-65938, 95-07-19335 and 97-05- 96628.

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