The purpose of this paper is to consider diffusion problems in time-space. It also aims at illustrating the usefulness of spatial analysis in terms of metric transformation. Previous works suggest that another type of distance such as time-distance or social distance is more relevant to analyze diffusion process rather than physical distance, but no positive work seems to exist. So, in this paper, inter-urban diffusion of the 1957 epidemic of Asian influenza in Nagoya and its environs is studied by recovering time-space. This study is concerned with 21 cities with more than fifty thousands population as of 1960. The epidemic hit 20 cities except Nishio, among which the first outbreaks were reported in Nagoya and Kasugai on May 24 (Fig. 2).
First of all, we must present time-space before examining the diffusion process. Timespace and physical space, by definition, are those recovered by application of M-D-SCAL, an algorithm of non-metric multi-dimensional scaling, to matrices of railway and/or bus travel times and straight distances between every pair of 21 cities respectively. In recovering each space, the following assumptions were made
(1) Two-dimensional solution is sought.
(2) Interpoint distances are measured by Euclidean distance.
(3) The initial value of step-size is set at 0.2.
(4) Seven starting coordinates are used, that is, five random coordinates, the coordi nates based on the metric method of Torgerson and the ordinary map coordinates.
(5) In order to get the configuration from which no further improvement is possible, 100 iterations are equally made for each case.
(6) The minimum-stress configuration derived from seven starting coordinates is re garded as fitting the data best, and used for the later analysis.
Table 2 shows both stresses of physical space and time-space derived from seven starting coordinates. For physical space, the starting coordinates based on the metric method of Torgerson yielded the minimum stress, nearly 0. 0. Its corresponding two-dimensional configuration is shown in Fig. 3. The first dimension could be interpreted roughly as representing SE-NW direction and the second dimension as representing NE-SW direction. Fig. 4 represents the two-dimensional configuration of time-space with the minimum stress (14. 907%), whose starting coordinates are the second random one. Locations of cities by quadrants characterize the derived two-dimensional time-space: Gifu, Ogaki, Ichinomiya and so on in the western Owari and the Seino districts are located in the top right-hand quadrant; cities in the eastern Owari and the Tono districts as well as Nagoya are located in the top left-hand quadrant; Handa and Tokoname in the Owari-Chita district are located in the bottom left-hand quadrant; Toyohashi, Okazaki and so on in the Mikawa district are located in the bottom right-hand quadrant. Compared Fig. 4 with Fig. 2 and Fig. 3, however, not only locations of cities but also the relative locations of districts don't coincide with those in the ordinary map. This is typically indicated by the fact that the locational relationship between the western Owari and the Seino districts, and the eastern Owari and the Tono districts is reversed. Such difference between time-space and ordinary map would be ascribed to the imperfectness of inter-district railways systems in the study area. For example, there exists no railway directly connecting the Tono district and the Mikawa district.

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Models based on the diffusion equation have been developed by Hirano (1975, 1976). The purpose of this paper is to test one of these models by computing hillslope form changes over time in an area in Hokkaido for which tephro-chronological data are available. Therefore computed profiles can be compared with actual profiles whose absolute ages are known. The equation used is
_??_(2)
in which Ut=rate of change of height with respect to time at a given point X; a is a coefficient which controls the rate of slope degradation; and Uxx is slope curvature.
The analytical solution of the differential equation (2), with the initial value U (0, X)=0, (X < 0); U(0, X)=U0, (X > 0), is given by the following integral:
_??_(5) 2
Elevation of a given point X with time t is obtained from the integral of the normal probability density function with mean p=0, and variance a2 =2 a t. Least-squares method is used to obtain a “best” value for coefficient a which maximizes the level of agreement between actual profile and computed profile. Time t is considered to be equal to the age of terrace surface at the cliff base.
A value of the standard deviation a can be also read off a slope profile plotted on normal probability paper. The value a at time t is calculated as follows:
_??_(7)
This method produces a value of coefficient a which is sufficiently accurate in the present context.
All the sampled slopes in unconsoliated fan gravel faced south but they differed in cliff height and in the period of time elapsed since the cessation of river erosion (Nogami, 1977). But they gave us almost the same results.
A value of coefficient a, equal to 5.9 × 10^-3 m²/yr, with standard deviation 1.5 × 10-s, was obtained from analytical solution and least squares method, as an average value for the 16 profiles. And the values ranged between 3.5 × 10^-3 and 2.2 × 10^-3.
Agreement between theoretical profiles and measured profiles is generally good, with the exception of profiles deformed by slumping. Therefore, the preliminary results suggest that the model can be used to simulate slope-form changes over time according to equation (2). There are, however, two unresolved problems concerning the use of this model. Firstly, the agreement between the model predictions and actual slope forms becomes progressively weaker as the elapsed time approaches 105 years. This can be attributed to the increase in the length of the concave segment over time in actual slopes when compared with model slopes. Secondly, the value of coefficient a depends not only on climate and rock properties but also on the initial condition Uo (original cliff height).
These problems should, be taken into account in future attempts to model slope form, but their solution is beyond the scope of this paper.

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For unconfined aquifers, the storage coefficient can be called specific yield, which is defined as a ratio of the volume of water which a rock or soil, after being saturated, will yield by gravity drainage to that of the rock or soil. Unconfined aquifers yield water to wells or other collection facilities because of drainage of pore space and air replacing water in the dewatered zone as the drop of the water table. Accordingly, if the height of the water table in an unconfined aquifer changes by an amount of 4h, the volume change in water storage per unit area, 4S, is given by,
_??_
where S_y, is the specific yield.
Because of this important relationship, many efforts have been made to determine the specific yield. Under natural conditions, the specific yield of an unconfined aquifer is often obtained by a pumping test, but this is an expensive procedure. Furthermore, a pumping test of short duration may not produce a full specific yield, because delayed release of pore water from a pumped unconfined aquifer usually occurs. Other methods to determine the specific yield are laboratory measurements on representative samples of the aquifer material, but such determinations are difficult. It is more reasonable to determine the specific yield by plotting the equilibrium volumetric water contents above the water table at the beginning and at the end of a certain drop of the water table. Estimates of the specfic yield can also be obtained by determining the difference between the porosity and the specific retention.
In the present paper, the specific yield of Kan to Loam which is a volcanic ash layer covering diluvial uplands in Kanto district is evaluated from the relation between a certain drop of the water table and the volume change of water per unit area drained by a falling of the water table. In addition to this purpose, the author also investigated the relations between the soil water behavior taking place in Kanto Loam and the soil water characteristics of that soil. The study area is located at Imaichi dissected fan, Tochigi Prefecture (Fig. 1). Two test spots surrounded with a paddy field were selected as the representative fields of the fan. The study spots are covered with Kanto Loam (Fig. 2), which has physical properties as shown in Tables 1 and 2 and soil water characteristics as shown in Fig. 4. Water contents of the investigated soil were measured by the neutron method every 7 or 10 days during a ten-month period from April in 1973 to January in 1974. In each measurement, soil water contents were obtained at intervals of 0.2 m down to a depth of 7. 2 meters. Wet bulk densities were also measured by the gamma density meter at the same intervals of that neutron method.
The three-phase distribution profiles in each month at the study spots are shown in Figs. 5 and 6. These figures offer some informations suggesting the behavior of the soil water taking place in Kanto Loam. Based on these data and the soil water characteristics of the investigated soil, the author calculated some significant parameters related to soil water characteristics of Ka.nto Loam, such as porosity, field capacity, specific retention and specific yield. These parameters are summarized in Table 3. The excess water retention above the field capacity in each depth seasona lly varied according to the water management in irrigation for the paddy field as shown in Figs. 7 and 8. As can be seen from Fig. 10, the specific yield calculated as the volume change of water by the drop of the water table showed some scatter but was essentially constant at a value of 16% for the spot 1 and 18% for the spot 2. Figure 11 shows the schematic relationship between profiles of the soil water content and the soil water characteristics for Kanto Loam, which was arranged from the data obtained through the present study.
The results are summarized as follows.

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Response of groundwater table to rainfall in a diluvial upland is analyzed by the weighted mean method developed by Tsuboi (1941). This method employs the Fourier series based on the assumption of periodic waves of the input and output. The groundwater level has been used as an output for analysis in previous studies (e. g. Hirata, 1971). The authors employ the amount of groundwater :recharge as :the output because it is the amount of groundwater recharge that directly causes changes in groundwater level. The analysis is also carried out using the groundwater level as the output and the results are compared with that using the amount of groundwater recharge.
Results obtained are as follows:
1. The time lag between the rainfall and the amount of groundwater recharge is about 10 days for wells whose groundwater levels are about 70 to 80 meters below the surface.
2. The groundwater level is affected by rainfall with a time lag of about 3 months. This coincides with a theoretically derived phase difference of π/2.
3. It may be said that results mentioned above give support to the pressure transmission theory as one of the important groundwater recharge mechanisms.

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