APPLICATION TO UNDULATING CRUST
With the above solution, it is interesting to consider about an application of the model introduced in this work in order to realize a secular upheaval of datum level in relation to a local case of the past tsunamigenic earthquakes as the actual seismic events.
One of the specific examples is the case at a couple of the earthquakes in 1944 and 1946. This case might be helpful to give a dynamical understanding of the crustal upheaval around the seismic events.
In order to show a typical geodetic process around the seismic events, an illustration of the geodetic surveys before and after the seismic events in a specific area are shown in Figure 5. The curve X shows a preseismic pattern of the crustal surface, and each of the curves Y and Y_{0} shows coseismic pattern around an island (Fig.5). The curve X is the crustal upheaval pattern in 1929 relative to the geodetic reference of 1887, and the curve Y is for that in 1950 relative to the reference of 1929, though the curve Y_{0} is that in 1950 relative to that in 1887. The illustration as shown in Figure 5 was first appeared in the publication which was a notice to the hazardous earthquakes written by Kenzo Sassa (1951).
Figure 5. Geodetic process around a couple of the earthquakes
(The survey line is along the coastline of an island. Relative crustal upheavals X, Y, and Y_{0} are shown.)
Now, consider one of the coseismic patterns in Figure 5 for convenience. When the preseismic pattern is expressed by a function Z= F(S, T) with a distance S from the reference station A, and the coseismic pattern is for a function Z= F(S,T'), then, the difference of the two functions may be the crustal upheaval pattern around the couple of the seismic events. That is,
DZ(S;T,T')=F(S,T')F(S,T) (11)
Where T'>T. Referring to the functions F(S,T') and F(S,T), it can be considered a corresponding pattern along a meridional line for convenience. When a meridional line passing the station A on the geodetic survey line (Fig.5), is considered, then, a pattern of crustal upheaval along the meridional line could be drawn boldly.
As for the bold crustal upheaval pattern or the topographical pattern along the meridional line passing the station A and D, the pattern along the line as the X axis can be shown as shown at the top in Figure 6. In this case, the distance S in (11) should be read as X, and, DZ(X;T,T')=F(X,T')F(X,T).
Assuming an identified forcing at the seismic events to give some difference of the crustal upheaval along the X axis, it can be seen an undulating crust surface (mid Fig.6).
Figure 6. A consistent model of undulating crust around a couple of the
earthquakes
Then, the pattern of DZ as a function of X may be shown as that at the bottom in Figure 6. When this pattern is taken to corresponding to the pattern in the range between the positions X for D and A, then, it can be taken that a consistent trend to the pattern as seen in the range between the stations D and A is found with a background of a dynamical theory. Here, it is necessary to see that the consistent pattern can be obtained simply by an elastic modeling even though a couple of the seismic events might exhaust some energy at each event of the seismic fault formation.
This may be one of the important keys to have a final solution of the hazardous tsunamigenic earthquake.
NOTICE TO SATELLITE ALTIMETRY
There might be an idea to utilize an available satellite altimetry for detecting crustal upheaval in practice with a careful processing of the data for obtaining the crust surface pattern as a topographical pattern, even though the satellite altimetry at present has only a short history in the fields of geophysical sciences, especially, in the interdecadal problems of geophysical processes. Recent advance of the satellite altimetry has been reported and its data set is supplied for application to the geographical mapping of the land surface and on the sea surface in a certain precision referring to the geoid, which is taken as the reference at present. Nevertheless, the author has unfortunately not yet any information about whether the accuracy of the satellite altimetry is satisfiable at applying it to the evaluation of the sea level in the coastal zone. Hence, it is hard to consider that the interannual variations of the sea levels at the tide stations can be equivalent to the satellite altimetry variations at the corresponding tide station. Now, we have to require that the accuracy of the available satellite altimetry would be equivalent to that of the tidal stations on the coast for the specific geophysical research purposes.
In the case as shown in Figure 6, the existing and available data set of the geodetic survey is used. This geodetic data set was obtained for about one hundred years, though the author used the data set for the time period from 1888 to 1950 in this work. This geodetic data set is obtained in a certain high precision by using the advanced geodimeters with the welltrained and skilled survey groups consisted by the geodetic specialists.
The satellite altimetry is yet under an improving stage in the scope of geodetic sciences, especially, for the problems concerning on the crustal upheaval in the coastal zone as the author has studied.
Therefore, it should be aware in fact that the accuracy of the available satellite altimetry at present in the coastal zone cannot be satisfiable for the purpose of studies on the problems of undulating crust in the coastal zone.
Adding to the above, it should be noticed that it is necessary to take a long time for compilation of the data of the global satellite altimetry for the author's purpose in this work. It should be required the accurate and precise data set of the advanced satellite altimetry for more than several ten years.
A more advanced technique in future is also expected for active practice of the result of the author's works and of the related contributions in the field of geodesy and geophysical sciences.
CONCLUSIONS
A theoretical model is introduced for realizing a crustal upheaval in the coastal zone by the active effect of possible water loading to the crust. The increase of the ocean water loading can be expected as a resultant effect of the global warming, i.e., as the effects of the climatological warming and glacial melting on the earth's surface. The model is a simple linealized elastic model of a thin plate assumed to be equivalent to the existing tectonic plate or the crust covering the earth. This model plate is for demonstrating and evaluation an expected crustal upheaval just around the coastal zone. A set of the differential equations for an equivalent elastic thin plate is solved with some equivalent elastic constants specifying the properties of the model plate. The solution under some assumed conditions suggests that the ocean water loading is effective to give a set back of the coastline. The solution is essentially for a final equilibrium crustal pattern. Then, the author has given a brief note on time factor for convenience. This solution may be well applied for obtaining a key to some other geophysical problems, for example, for a dynamical understanding of an existing profile of geographic section and of a dynamical mechanism about a process of a coseismic crustal upheaval pattern. An additional notice to satellite altimetry is given for detecting the crustal upheaval in the coastal zone in practice.
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