Another formulation and method of solving weak constraint problem in variational objective analysis is shown. Weak constraint problem is solved by introducing new variables and imposing strong constraints which are the relations between old variables and new variables. The Euler-Lagrange equation obtained by the proposed formulation of weak constraint differ a little in an additional perturbation term from those obtained by strong constraint, while those obtained by the conventional formulation of weak constraint differ much from those obtained by strong constraint. Then this method of solving weak constraint problem serves a practical purpose. Weak constraints are viewed as transformation of old variables into new variables, the weights of which are given in the weighted least square error expression. The formulation of weak constraint shows the relationship between weak constaint and strong constraint more clearly than the conventional formulation of weak constraint. Especially when the weight of weak constraint becomes larger, in other words, weak constraint becomes stronger, the alternative solving method shows more naturally that the solution of weak constraint tends to the solution of strong constraint than the conventional one does. It also gives a unified expression of weak and strong constraints. The unified expression of weak and strong constraints gives us flexibility in solving some problems, and we can change weak constraint into strong constraint and vice versa in the expression.
A generalized method of multivariate optimum interpolation method is shown. Generalization is done in two points. The first is that homogeneous linear constraints are incorporated in formulation explicitly. The second is that the generalized method makes use of both climatological information (climatological mean data and their covariance) and forecast data in an integrated way, while multivariate optimum interpolation makes use of either climatological information (climatological mean data and their covariance) or forecast data as its first guess. The role of covariance matrix is made clear in a different way from that of Phillips (1982) . It is shown that covariance matrix consistent with linear constrains works as a projection operator or as a filter, just as explicit linear constrains in variational objective analysis do. The method is considered to be an improved method of variational objective analysis in two points. The first is that the method produces statistically optimum analyzed data while statistical meaning is vague in variational objective analysis. The second is that the method gets analyzed data on grid points directly from randomly distributed observed data, while variational objective analysis needs pre-interpolation of observed data onto grid points. Some examples of application of the method is shown. The statistical meaning of weak constraint in variational objective analysis is illustrated in an example. A divergent covariance model is also discussed. The method includes both multivariate optimum interpolation method and variational objective analysis with linear constraints as its special cases. Via the method, the relationship between multivariate optimum interpolation method and variational objective analysis can be understood basically.
The pH of meteoric water was determined for 995 and 736 samples respectively in Tokyo (1973-1980) and Tsukuba (1980-Apr. 1984). The mean value of pH during the observation period was 4.5 both in Tokyo and Tsukuba. It was found that the distribution of the hydrogen ion concentration in meteoric precipitation is of the lognormal type. The percentages of occurrence of acid rain (pH<4) were 26% in Tokyo and 10% in Tsukuba, while those of the precipitation amount were 7.0% in Tokyo and 4.3% in Tsukuba. The apparent constancy for many years of pH of meteoric water in Tokyo may be due to the buffering action of calcium carbonate suspension against nitric acid.
The travel time table which was proposed by Ichikawa and Mochizuki (1971) and in use at the Japan Meteorological Agency for hypocenter determination since 1973 was revised. The table was primarily based on the observational results of explosion seismic experiments carried out in and around Japan. It consists of two parts, namely one part is the travel time of surface focus which was derived by averaging observational results of explosion seismic experiments and other parts are travel times for other depths which were computed from the velocity structure which was primarily derived from the results of explosion seismic experiments. However, investigation revealed that the velocity structure was inconsistent with the travel time from the surface focus. This inconsistency might be due to a process in which the velocity structure was created using the Herglotz-Wiechert method. Some errors in computation were also seen in the table. These problems frequently caused unstable hypocenter solution and minus depth solution for shallow focus earthquakes. Considering heteorogeneous crustal structure in and near the Japanese Islands and anisotropic P wave velocity distribution reported by explosion seismic experiment, it doesn't seem to be always meaningful to make a total revision of the travel time tables in the present circumstances. Considering several velocity structures reported as a standard model for the crustal structure of the Japanese Islands, the velocity structure of the table in use doesn't seem to be unusual, although velocity around the Mohorovicic discontinuity seems to be slightly small. Therefore, the revised travel time tables given in unit of 0.01 second and called 83A were computed from the same velocity structure. An attempt to make new travel time tables which are more convenient for hypocenter determination was also made. On the basis of the travel time curve for surface focus of Ichikawa and Mochizuki (1971)'s table (I-M table), a new velocity model for crustal layer was computed. The new structure was combined with the velocity structure of Jeffreys and Bullen (1958)'s table (J-B table) by three different ways to compute new tables (83B, 83C and 83D). Applications of these new tables to hypocenter determination showed that some of these tables are generally more convenient than the revised table 83A for hypocenter determination. For the sake of continuity in the routine work, however, the author concluded that the revised travel time table 83A should be used for a while.