Target period, buildings, and statistical data
The target period was 1964 to 2020, a period during which statistical data [30-32] on buildings in Japan was available for compilation. In addition, the target buildings were divided into wooden and non-wooden buildings throughout Japan.
Statistical data on existing building floor area [30, 31], which represents the amount of stock, were divided into taxable and non-taxable. While there were data available on taxable buildings in terms of whether they were wooden or non-wooden, no structural data on non-taxable buildings could be obtained. Therefore, we used the percentages of wooden and non-wooden taxable building floor areas for each year and proportionally divided the non-taxable building floor area between wooden and non-wooden. The percentage of all building floor areas during the target period that was non-taxable was 4 %–9 %, and thus more than 90 % was taxable. In addition, values that were recorded for the non-taxable building floor area in 1973 were approximately double of those in the years before and after that year, which was deemed as an abnormality in terms of statistical data. Therefore, we used the average value of the years before and after 1973. Supplementary Table S6 and Fig. S1(a) show the statistical data for the existing building floor area during the target period for wooden and non-wooden buildings.
Statistical data on the new building floor area , which represents the amount of inflow, was acquired for wooden and non-wooden buildings. Furthermore, data were available on detailed structural categories of non-wooden buildings in the same statistics, including reinforced concrete construction, steel-framed reinforced concrete construction, and steel-framed construction. However, statistical data [30, 31] on detailed categories were not obtained for the building stock during the target period, and little difference was found when comparing the amount of wood used in non-wooden buildings to wooden buildings per these detailed categories [28, 29]. Therefore, we used statistical data batching together with non-wooden buildings to determine the amount of building inflow. Supplementary Table S6 and Fig. S1(b) show the statistical data on the new building floor area during the target period for wooden and non-wooden buildings.
Method of estimating amount of stock
We decided to determine the amount of building stock using equation (1). In addition, the amount of carbon stock in wood used in buildings can be determined by equation (2).
Here, Si (t) indicates the existing building floor area (m2) built in year i at the start of year t, S (t0) is the existing building floor area (m2) at the start of the initial year t0, Fi indicates the new building floor area (m2/yr) built during year i, R0 (t–1–t0) indicates the building lifetime function (cumulative distribution function of lifetime distribution) for the years elapsed from the initial year t0 to year t-1, and R (t–1–i) indicates the lifetime function of a building for the years elapsed from year i to year t-1. CSi (t) indicates the amount of carbon stock (Mg-C) in wood used in existing buildings built in year i at the start of year t, Wi indicates the amount of wood used per unit floor area (m3/m2) in buildings built during year i, D indicates the density of wood (oven dry mass over air dry volume) (Mg/m3), and C indicates the carbon content of oven-dried wood (Mg-C/Mg). Furthermore, the initial year t0 is 1964, and year t is each year until 2020.
The statistical data (B (t)) [30, 31] on the existing building floor area during the target period is not published for each new construction building year; thus, Si (t) on the left side of equation (1) cannot be directly ascertained. On the other hand, because the amount of wood used in buildings fluctuates according to the building year, the existing building floor area per building year (Si (t)) and the amount of wood used per building year (Wi) must be used according to the right side of equation (2) to make a precise estimation. Hence, we substituted statistical data for the existing building floor area in the initial year of 1964 (B (1964))  into S (t0) on the right side of equation (1), and determined the parameters of the lifetime functions (R0, R), such that the estimated value Si (t) determined by conferring statistical data on the new building floor area  on Fi fits the statistical value B (t). In addition, because the existing buildings (S (t0)) in the initial year (1964) is an accumulation of all buildings that were constructed beforehand, they decrease with the passage of years. However, because it is not possible to ascertain the existing buildings per building year from statistical data, we could not craft a model based on new buildings and the lifetime function indicated by the second term on the right side of equation (1). Therefore, we decided to regard S (t0) as a single building group and established a model that decreased according to the lifetime function R0, indicated by the first term on the right side of equation (1).
If the lifetime functions (R0, R) of equation (1) can be identified by the above method, then it is also possible to make future predictions of the amount of building stock and the amount of wood carbon stock based upon these lifetime functions by conferring future scenarios on the amount of future building inflow (Fi) and lifetime (R).
The amount of wood used per unit floor area (Wi) in equation (2) is applied to products containing roundwood, sawn wood, plywood, and wood board. For roundwood, sawn wood, and plywood, we cited statistical data  for wooden and non-wooden buildings surveyed every two–three years during the target period from 1976 to 2017. For years lacking in statistical values, we decided to use statistical values from 1976 for the years prior to 1975, and henceforth statistical values from the immediately preceding year. On the other hand, for wood boards (particle board, hardboard, medium-density fiberboard, insulation board), although we cited values from previous research  used every three years for the period 1976–2012, it was assumed that there were no structural differences between wooden and non-wooden buildings as it was not possible to ascertain the amount of use per wooden and non-wooden buildings. Years lacking in reported values were handled in a manner similar to the statistical values for roundwood, sawn wood, and plywood described above. We used the numerical values in Table 12.1 and 12.2 from the IPCC 2019 Refinement  per wood product for carbon conversion factors D and C.
Previous research on lifetime functions (R0, R) mainly examined exponential distribution (FOD) [3, 10, 13, 33-35], logistic distribution [4, 5, 28], normal distribution [1, 2, 12, 14, 16, 21-23, 26], log-normal distribution [9, 12, 21-23, 26, 41, 42], Weibull distribution [9, 12, 14, 17-20, 21-23, 42], and Gamma distribution [3, 9, 10, 12], and these six types of functions were also studied. However, for the logistic and normal distributions, we used a function normalized by setting the remaining ratio (1 − R (t−1−i)) of zero elapsed years (t−1 = i) to 1.
We determined the parameter for minimizing the residual sum of squares (E) between the statistical value B (t) and the estimated value Si (t) of the existing building floor area for each lifetime function according to equation (3).
We defined the lifetime function parameter of half-life as the number of years elapsed at which the remaining fraction (1 − R (t−1−i)) reached 0.5. We then examined both cases: we set a single half-life (and other parameters) assuming that the building half-life (and other parameters) did not change throughout the entire target period, and cases in which we set multiple half-lives (and other parameters) assuming that half-life (and other parameters) were changing during the target period. The former refers to cases in which the lifetime functions R0 and R on the right side of equation (1) represent the same case. On the other hand, in the latter cases, considering the ratio of building stock (existing building floor area) to building inflow (new building floor area) (Supplementary Table S6 and Fig. S1(c)), wooden buildings increased from 1997, and non-wooden buildings increased from 1991, and thus the amount of stock increased relative to inflow. Hence, assuming that building lifetimes clearly fluctuated more than before from 1997 for wooden buildings and from 1991 for non-wooden buildings, we decided to set different half-lives (and other parameters) before and after these years. However, since the initial-year building stock consisted of buildings constructed prior to 1964, whether its half-life was the same as that of building stock from 1965 onwards is unknown. Therefore, considering the possibility that initial-year building stock half-lives were different from those of subsequent building stock, we set different half-lives. Based on the above, we categorized wooden buildings into three target periods of 1964, 1965 to 1996, and 1997 to 2020, and non-wooden buildings into three target periods of 1964, 1965 to 1990, and 1991 to 2020, and then examined the half-lives of existing buildings in each period. This means that we divided the lifetime function R, the second term on the right side of equation (1), into two periods, and then set three different half-lives (and other parameters) together with the lifetime function R0, which is the first term on the right side of equation (1).