Open AccessArticle

Functional Method for Analyzing Open-Space Ratios around Individual Buildings and Its Implementation with GIS

Kayo Okabe

¹ and

Atsuyuki Okabe

^2,*

LatLng Co., Tokyo 184-0005, Japan

Center for Spatial Information Science, University of Tokyo, Kashiwa 277-8568, Japan

Author to whom correspondence should be addressed.

ISPRS Int. J. Geo-Inf. 2024, 13(3), 70; https://doi.org/10.3390/ijgi13030070

Submission received: 12 October 2023 / Revised: 22 February 2024 / Accepted: 23 February 2024 / Published: 25 February 2024

Download

Browse Figures

Versions Notes

Abstract

An open-space ratio is often used as a first basic metric to examine the distribution of open space in urbanized areas. Originally, the open-space ratio was defined as the ratio of the area of open space (unbuilt area) to the area of its building site. In recent years, residents have become more concerned with the open-space ratios in the broader neighborhoods of their individual buildings than with their own building sites. To address this concern, this paper proposes a method for dealing with the open-space ratio in the variable

x

-meter buffer zone around each building, called the open-space ratio function, and implements it using standard GIS operators. The function and its implemented analytical tool can answer the following questions. First, this function shows how the ratio varies with respect to the bandwidth to discuss the modifiable area unit problem. Second, as the ratio changes, the function shows in which bandwidth zone the ratio is the highest, indicating the best open-space environment zone. Third, in the pairwise comparison for housing selection, the function shows in which bandwidth zone a specific house is better than another. Fourth, the function shows in which bandwidth zone the variance among all buildings in a region is the greatest. Fifth, in this zone, buildings are clustered in terms of open-space ratio. The resulting clusters are the most distinct. Sixth, to examine the open-space ratio around a clump of buildings (such as a housing complex), the function shows how to obtain clumps. Seventh, it is shown how the open-space function provides a wide range of applicability without changing the mathematical formulation. Finally, this paper shows how to implement the function in a simple computational method using operators and a processing modeler provided by the standard GIS without additional software.

Keywords:

open space ratio; open-space ratio function; local and global; GIS implementation; processing modeler

1. Introduction

An open-space ratio is often used in the literature (e.g., urban geography [1], urban analysis [2], environmental studies [3], land use studies [4], public health studies [5], city planning [6], building studies [7], and urban morphology [8]) as a first basic metric to examine the distribution of open space in urbanized areas. Originally, the open-space ratio was defined in the 1960s (Felt [9]; Tough [10]) as the ratio of the area of open space (unbuilt area) to the area of its building site. In recent years, residents have become more concerned with the open-space ratios in the broader neighborhoods of their individual buildings than with their building sites in general. To address this concern, this paper proposes a method for dealing with the open-space ratio in the variable buffer zone around each building, called the open-space ratio function, and implements it using standard GIS operators.

Regarding the extension of the area in which the open space ratio is defined, several attempts can be found in the literature. For example, Hamaina et al. [11] and Schinmer et al. [12] used the area Voronoi diagram. Some researchers use multiple bandwidth zones around buildings, such as Steiniger et al. [13], Pont et al., and Boffet et al. [14]. Obviously, the open-space ratio varies with the choice of the bandwidth zones, known as the notorious modifiable area unit problem (MAUP) (Openshaw [15]). The results would have changed if different buffer widths had been used. To observe how the open-space ratio varies with bandwidth and to determine which bandwidth is appropriate, an open-space ratio with respect to a continuous bandwidth, that is, a function of an open-space ratio with respect to a continuous bandwidth, is very useful.

The open-space function can answer the following questions. Q1: How does the open-space ratio continuously vary across small to large bandwidth zones? The answer indicates whether the notorious MAUP exists. If the ratio changes, we must keep in mind that the results may change depending on the choice of a zone width. If the ratio changes, we can ask the following question. Q2: Which bandwidth buffer zone has the highest or lowest open-space ratio? The answer tells us which zones have the “best” or “worst” open space environments, as measured by the open-space ratio. When comparing two buildings, we can ask the following question. Q3: In which bandwidth buffer zone is the difference in open-space ratio the largest or smallest? The answer would be useful for a family who are looking for a rental house in order to choose the best house by comparing the current house with an alternative house. As for the open-space ratios of all buildings in a region, we can ask the following question. Q4: In what bandwidth zone is the variance of open-space ratios at its maximum in the region? The answer shows in which bandwidth zones the open space environment of the buildings in the region are very different. When we want to see the distribution of buildings with respect to different open-space ratios on a map, we want to classify buildings into several classes that are distinctively different from each other. Q5: In what bandwidth zone can we clearly see the distribution of different classes of buildings on a map? This map is useful for discussing the open space environment created by the mix of buildings with different open-space ratios in a district. If we want to examine the open-space ratio not only around individual buildings but also around a clump of closely spaced buildings, such as a housing complex, we can ask the following question. Q6: How do we find clumps of buildings? The answer is useful for evaluating the open-space environment around a clump of buildings in terms of the open-space ratio. Q7: How does the open-space ratio function provide a wide range applicability without changing the mathematical formulation?

This paper consists of four sections and Appendix A. Section 2 deals with the methods and their implementations. The first method is the open-space ratio function, which has four types. We formulate the local open-space ratio function as the open-space ratio in a buffer zone around a single building where the width of the buffer zone varies continuously. Next, we formulate the global open-space function, which is obtained by averaging the local open-space ratio functions of all buildings in a region. To observe the distribution of open space within a buffer zone, we formulate the incremental local open-space ratio function (the concept of “incremental” is also used in [16]). Explicitly, we decompose a buffer zone into buffer rings. The incremental local open-space function is formulated as the open-space ratio in each buffer ring surrounding a given building. The incremental global open-space function is obtained by averaging the incremental local open-space ratio functions of all buildings in a region. We show how to implement these functions using standard GIS operators. Once these functions are estimated from the data, the functions answer the questions Q1, Q2, and Q3 above.

The second method is how to answer the questions Q4 and Q5 using the local and global open-space ratio functions. We can illustrate the distribution with distinctly different open-space ratio classes of buildings on a map. The third method is how to answer Q6, i.e., how to find clumps of buildings to obtain the open-space ratio functions of a set of closely spaced buildings. Since the exact computation of this method is complex, we alternatively show a practical implementation using the local open-space ratio function. In Section 3, we first discuss a method of how the open-space function provides a wide range of applicability without changing the mathematical formulation. Second, we discuss the limitations of the method and how to overcome them. Finally, in Section 4, we summarize our main results and point out possible extensions.

2. Methods and Implementation

This section explicitly formulates the open-space ratio functions outlined in the introduction. For illustrative purposes, we use a small empirical example and assume that the open space is the area not covered by building footprints. The study area is located within the Shibuya ward in Tokyo. The area is almost a rectangle (364 m × 268 m) surrounded by streets, and its area is 9.5 hectares (the thick lines in Figure 1). The building footprints in the study area are indicated by the gray polygons. The white polygons outside the study area are used for edge correction [17,18]. Note that the gray and white buildings are mainly for residential use, and the total number of buildings is 406.

Let S be a study area (indicated by the bold polygon in Figure 1); let

B_{1}, \dots, B_{n}

(the gray polygons) be building footprints in S; let

B_{n + 1}, \dots, B_{n *}

(the white polygons) be building footprints outside the study area S used for edge correction; and let S* be the extended study area including

B_{1}, \dots, B_{n}

and

B_{n + 1}, \dots, B_{n *}

(the area including all the gray and white polygons). The crude open space O in S* is obtained by excluding buildings

B_{1}, \dots, B_{n *}

from S*, i.e., O = S*

∖ ⋃_{j = 1}^{n^{*}} B_{j}

(the white area in S* in Figure 1).

2.1. Local Open-Space Ratio Function

We first consider the open-space ratio around a building,

B_{i}

. Let

B_{f} (x | B_{i})

be the buffer zone around building

B_{i}

with a width of x. An example,

B_{f} (50 m | B_{21})

, is illustrated by the area enclosed by the circular hairline centered at

B_{21}

in Figure 2a. Note that

B_{f} (x | B_{i})

includes

B_{i}

. To obtain the open space in

B_{f} (50 m | B_{21})

, we defined the buildings in

B_{f} (50 m | B_{21})

, indicated by the gray polygons in Figure 2b. Then, the open space is obtained by excluding the buildings

⋃_{j = 1}^{n^{*}} B_{j}

from the buffer zone

B_{f} (x | B_{i})

, i.e.,

O (x| B_{i}) = B_{f} (x| B_{i}) ∖ ⋃_{j = 1}^{n^{*}} B_{j}

as in Figure 2c. We define the local open-space ratio function for the x-width buffer zone around Building

B_{i}

as the ratio of the area of open space (e.g., the white area within the hairline circular area in Figure 2a) to the area of the x-width buffer zone around Building

B_{i}

(e.g., the gray area in Figure 2c). Mathematically, the local open-space ratio function of building

B_{i}

is written as:

L_{C} (x | B_{i}) = [O (x| B_{i})] / [B_{f} (x| B_{i})], 0 \leq x {\leq x}_{m a x},

(1)

where the symbol

[p o l y g o n]

is the operator of measuring the area of the polygon, and

x_{m a x}

is the maximum distance. This function is the function of x-width. Therefore, we can continuously observe open-space ratios across a narrow zone to a wide zone.

It should be noted that the open space in the local open-space ratio function defined by Equation (1) is cumulative concerning x-width. Therefore, we cannot directly determine in which part of the x-width buffer zone the open-space ratio is high or low. More explicitly, we consider buffer rings surrounding Building

B_{i}

. For example, in Figure 1, the circular hairlines indicate the buffer rings with 10 m intervals surrounding

B_{130}

in the 50m width buffer zone. The j-th ring,

R (x_{j}, x_{j + 1} | B_{i})

, is obtained by excluding the

x_{j}

-width buffer zone (e.g.,

B_{f} (x_{4}| B_{i})

in Figure 3a) from the

x_{j + 1}

-width buffer zone (e.g.,

B_{f} (x_{3}| B_{i})

in Figure 3b), i.e.,

R (x_{j}, x_{j + 1} | B_{i})

B_{f} (x_{j + 1}| B_{i}) ∖ B_{f} (x_{j}| B_{i})

j = 0,1, \dots, m

, where

x_{0} = 0

, and

x_{m} = x_{m a x}

(e.g.,

R (x_{3}, x_{4} | B_{130})

in Figure 3c is obtained by substracting

B_{f} (x_{4}| B_{130})

in Figure 3a from

B_{f} (x_{3}| B_{i})

in Figure 3b). Similarly, the open space,

O (x_{j}, x_{j + 1}| B_{i})

, in a ring

R (x_{j}, x_{j + 1} | B_{i})

(e.g.,

O (x_{3}, x_{4}| B_{130})

in Figure 3f) is obtained by excluding the open space in the

x_{j}

-width buffer zone (e.g.,

O (x_{3}| B_{130})

in Figure 3e) from that in the

x_{j + 1}

-width buffer zone (e.g.,

O (x_{4}| B_{130})

in Figure 3d), i.e.,

O (x_{j}, x_{j + 1} | B_{i})

O (x_{j + 1}| B_{i}) ∖ O (x_{j}| B_{i})

. In these terms, the open-space ratio in a ring

R (x_{j}, x_{j + 1} | B_{i})

is written as:

L_{I} (x_{j} | B_{i}) = [O (x_{j}, x_{j + 1}| B_{i})] / [R (x_{j}, x_{j + 1} | B_{i})], j = 0, 1, \dots, m .

(2)

With this function, we can observe the open-space ratio in every ring in detail in the

x_{j + 1}

-width buffer zone. Because the ring

R (x_{j}, x_{j + 1} | B_{i})

B_{f} (x_{j + 1}| B_{i}) ∖ B_{f} (x_{j}| B_{i})

is the increased area from the

x_{j}

-width buffer zone to the

x_{j + 1}

-width buffer zone, we call the function defined by Equation (2) the local incremental open-space ratio function. To distinguish the local open-space ratio function given by Equation (1) from this function, we call the former function the local cumulative open-space ratio function. The terms cumulative and incremental are the same use as in the K function [16]. It is noted that the cumulative open-space ratio function deals with open space in a buffer zone of

B_{i}

that includes Building

B_{i}

, whereas the incremental open-space ratio function deals with open space in a buffer ring surrounding Building

B_{i}

which does not include Building

B_{i}

. These two functions show different aspects of the open-space ratio, and they are complementary.

2.2. Global Open-Space Ratio Function

In the preceding subsection, we formulated the local open-space ratio functions in terms of the open-space ratio around a specific building in a study area. In this subsection, we consider a function for examining open-space ratios of all buildings in the whole study area. We define it as the average of local open-space ratios in the whole study area. The average is obtained by averaging the local cumulative and incremental open-space ratio functions given by Equations (1) and (2). In mathematical terms, they are written as:

G_{C} (x| B_{1}, \dots, B_{n}) = \frac{1}{n} \sum_{i = 1}^{n} [O (x| B_{i})] / [B_{f} (x| B_{i})], 0 \leq x \leq x_{m a x},

(3)

G_{I} (x_{j}| B_{1}, \dots, B_{n}) = \frac{1}{n} \sum_{i = 1}^{n} [O (x_{j}, x_{j + 1}| B_{i})] / [R (x_{j}, x_{j + 1}| B_{i})], j = 0,1, \dots, m .

(4)

We call

G_{C} (x| B_{1}, \dots, B_{n})

the global cumulative open-space ratio function and

G_{I} (x_{j}| B_{1}, \dots, B_{n})

the global incremental open-space ratio function, respectively. Note that the terms local and global are also used in the K function [19,20]. Because these global functions are obtained from the sum of local functions (variables), we can obtain their standard deviations as a function of the x-width buffer zone or ring. These functions are useful to determine which width zone or in which ring the difference among local open-space ratios of buildings is small or large.

The implementation of the local and global open-space ratio functions is illustrated by the flows in Figure 2 and Figure 3. These figures illustrate the calculation of a single building. In practice, we deal with hundreds of buildings. Such a calculation is very difficult to do by hand. Fortunately, the standard GIS provides a processing modeler, such as the graphic modeler of QGIS and the model builder of ArcGIS. The steps are shown in Appendix A. Using this computational procedure with the empirical data shown in Figure 1, we test whether the open-space ratio functions are obtained in practice and consider the implications of the results with respect to the questions Q1, Q2, and Q3.

The thick black curve in Figure 4a indicates the cumulative local open-space ratio function of Building

B_{21}

in Figure 1, and the thick gray curve indicates that of Building

B_{130}

. The vertical axis in Figure 4a shows the open-space ratio, and the horizontal axis shows the x-width (meter) of a buffer zone. The 0-width buffer zone means a building itself, so the 0-width buffer zone does not include any open space. Therefore, every local open-space ratio function starts at the origin (0, 0). The upper dotted horizontal straight line in Figure 4a shows an average open-space ratio of 0.72 in the study area.

As x-width becomes larger than 0, the local open-space ratio functions of

B_{21}

and

B_{130}

show different curves. However, it is interesting to note that both curves approach the average ratio of 0.72 when the width of their buffer zones is 50 m. As discussed in the following section, this phenomenon is the same for almost all buildings in this study area. It is a little bit surprising to know that in this study area, the open-space environment around each building measured by the open-space ratio is almost the same in the x > 50 m wide buffer zone. Such a critical bandwidth c exists in any region although the value of c varies from region to region. In the buffer zone where x > c, MAUProblem [15] does not exist. To examine the difference in the open-space ratio between buildings, we have to focus on the x < 50 m wide buffer zone. In this zone, the open-space ratio varies with respect to x. We should keep in mind that the result obtained from the analysis with a specific zone width varies according to the width (MAUProblem). The thick gray curve in Figure 4a shows that the open-space ratio of Building

B_{130}

is the highest 0.88 in the 13 m bandwidth zone, which is much higher than the average open-space ratio. On the other hand, the thick black curve in Figure 4a shows that the open-space ratio of Building

B_{21}

is always below the average open space. The buildings in this study area are almost all residential houses. Therefore, we may evaluate the open-space ratio viewing from a residential open-space control of the Building Standard Act. In this study area, the lower bound of the open-space ratio is 50%. Comparing the lower horizontal straight dotted line in Figure 4a with the open-space ratio functions of Buildings

B_{21}

and

B_{130}

, both buildings satisfy this condition in the x > 3.5 m wide buffer zone. The open-space environment of both houses is fairly good in terms of the open-space ratio. In examining open space, we often want to compare differences in open-space ratio between paired buildings. For example, a family who are looking for a rental house chooses the best house by comparing their current house with an alternative house. For this purpose, Figure 4b is useful. The thick curved line indicates the difference in the open-space ratio between Buildings

B_{21}

and

B_{130}

, and the horizontal dotted straight line indicates that the difference is 0.1. From this figure, a distinct difference in the open-space ratio is between the 3 m to 31 m bandwidth zones. The difference is negligibly small outside this zone.

Up to here, we discussed empirical findings obtained from the cumulative open-space ratio function defined by Equation (1). This function has a macroscopic perspective in the sense that it deals with the open-space ratio within a buffer zone around a building, but it does not show how open space is distributed over that zone. Alternatively, the incremental open-space ratio function defined by Equation (2) has a microscopic perspective in the sense that it deals with the open-space ratios within respective 1 m wide buffer rings constituting the 5 m wide buffer zone in Figure 1 and Figure 3. Figure 5 shows the incremental open-space ratio functions of Buildings

B_{21}

and

B_{130}

. Comparing Figure 4 and Figure 5, we can visually see that the difference between the cumulative open-space ratio function and the incremental open-space ratio function is large, implying that these functions show different complementary aspects of open-space ratios.

Both curves in Figure 5a start at 1.0. This means that both buildings have no other buildings within the 1 m area surrounding Buildings

B_{21}

and

B_{130}

. A distinct difference, which we cannot clearly see from the cumulative function in Figure 4, appears in the 2nd to 10th rings. The gray curve remains at 1.0 in these rings, indicating that there are no buildings in the 10 m wide zone around Building

B_{130}

. The open-space ratio of Building

B_{21}

is lower than 50% in the 8th to 12th rings. This implies that the open-space environment in these rings is not good enough. In addition, Figure 5a shows that the gray and black curves are not so different in the 17th to 49th rings. In fact, Figure 5b shows that the difference is less than 0.1 in those rings. The distribution of open space is almost the same in the area farther than 49 m from these houses. Compared with the cumulative open-space function shown in Figure 4b, the graph in Figure 5b is useful for a family with a 5-year-old child who lives in House

B_{130}

and is looking for a new house by paying close attention to open space around their current house.

2.3. Distinctly Different Classes of Open-Space Ratios

The open-space ratio varies from building to building in a region. When we study the open space environment created by the mix of buildings with different open-space ratios in a region, we classify the buildings into similar open-space ratios and observe the distribution of buildings belonging to the same class on a map. In this case, we want to have distinctly different classes of open-space ratios. In this subsection, we present a method for finding such classes.

Figure 6a shows the global cumulative open-space ratio function defined by Equation (3), i.e., the average of local cumulative open-space ratios of all buildings in the whole study area. If the standard deviation is less than 0.05, the difference in open-space ratios is negligibly small. The dotted horizontal line in Figure 6b shows that the open-space ratios in the x > 42 m wide buffer zones are almost the same as 0.72. Therefore, to examine the difference in the open-space ratio among buildings, the x < 42 m wide buffer zones must be examined. Figure 6b shows that the maximum standard deviation is achieved at the 15 m wide buffer zone. This means that diverse open-space ratios of buildings are observable in the 15 m wide buffer zone. Therefore, if buildings are classified by their open-space ratios in a 15 m wide buffer zone, the resulting classes are well distinguishable.

Given the open-space ratios in the 15 m wide buffer zone and the number of classes is four, we used the Ward hierarchical cluster method [21]. The resulting classified buildings are indicated by color in Figure 7. Buildings with the same color share similar local open-space ratios. The buildings colored red are those with the lowest open-space ratio and the buildings colored green are those with the highest open-space ratio. The map in Figure 7 is useful for discussing the open space environment created by the mix of buildings with different open-space ratios in a district.

2.4. Open-Space Ratio Function of a Clump of Buildings

Until now, we have considered the open-space ratio function of a single building. In this section, we show a method for obtaining the open-space ratio function of a clump of buildings, such as a housing complex represented by the gray polygons in Figure 1. The first step is to obtain the clumps of buildings. The exact method is as follows: first, to construct the area Voronoi diagram (Section 3.6 in [22]); second, to construct the area Delaunay diagram; third, to construct the MPT for polygons. Currently, the first computation cannot be carried out by the operators of standard GIS, although it can be performed by an open-source program [23]; however, there are very few easy-to-access software that are able to perform the second and third computations. Alternatively, we propose a simple method using a function of the open-space ratio function. We generate the buffer zone with

x

-width for

x = x_{1}, x_{2}, \dots

with equal intervals. The choice of the interval depends on the precision level. An example is illustrated in Figure 8. As the width increases (Figure 8a–h), the number of clumps decreases. In the end, all buildings form one clump (Figure 8f). This procedure results in hierarchical clumps. It is time-consuming work to generate

x = x_{1}, x_{2}, \dots

by hand but can be easily performed by a processing modeler provided by standard GIS. The outline of this procedure is shown in Appendix A. Using this method, we obtained the clumps of buildings in the case of 6 m wide buffer zones in Figure 9. The incremental open-space function can be obtained from Equation (2) by just replacing

B_{i}

with the union of the buildings forming a clump. The result is illustrated in Figure 10. As expected, the open-space ratio is high (1.0) within 10 m around the clump.

3. Discussion

In this section, we discuss the board applicability of the open-space ratio functions. We also discuss how to overcome the limitations of the open-space ratio functions. In the previous sections, we have explained the methods of the open-space ratio functions, assuming that the open space is defined as the area not covered by building footprints, called the crude open space. The crude open space consists of many types of open spaces, and specific open-space ratios are defined. Examples include green open-space ratio [24,25], green coverage ratio [26,27], public open-space ratio [28,29], space ratio [30], park ratio [31,32], water surface ratio [33,34], river ratio [35,36], and road ratio [37,38]. These open-space ratio functions are easily computed using the crude open-space ratio functions formulated in the previous section. Let O_k,

k = 1, \dots, n

be a specific type of open space, such as roads O₁, parking lots O₂, parks O₃, playgrounds O₄, groves O₅, lakes O₆, rivers O₇, and so on. Suppose that the k-th type open space consists

m_{k}

polygons O_k =

⋃_{j = 1}^{m_{k}} O_{k j}

, e.g., the park open space consists of

m_{k}

parks. Then, the k-th type open-space ratio functions

L_{C k} (x| B_{i}),

k = 1, \dots, n

are obtained by simply replacing O = S*

∖ ⋃_{j = 1}^{n^{*}} B_{j}

with O_k =

⋃_{j = 1}^{m_{k}} O_{k j}

in Equations (1)–(4). The computational method is the same.

The open-space function is a univariate function for a certain type of open space. To comprehensively analyze the open-space environment, we should consider the open space of several types and examine their interactions. For this analysis, we can readily formulate a multi-variate functions of n_k types

L_{C} (x | B_{i}

) = (

L_{C 1} (x| B_{i}), \dots, L_{C n_{k}} (x | B_{i}))

. Using this function, open-space environments can be discussed more comprehensively through its covariance matrix.

In addition to many kinds of the open-space ratio functions, our method can also be directly applied to the building-coverage ratio, which is often used in building studies. Since [building-coverage ratio] = 1 − [open-space ratio], the building-coverage ratio function is formulated by simply replacing a given open space with buildings. This function allows us to analyze the distribution of the building-coverage area around individual buildings. Buildings have many uses such as offices, retail, and public facilities. The valuation of open space varies from use to use. Our method can directly analyze the distribution of parking lots (a specific open space) around buildings of a certain use by simply restricting buildings in terms of their use, such as the parking space around supermarkets and hospitals. Furthermore, the base polygons

B_{j}, j = 1, \dots, n

are not necessarily buildings but also other facilities. For example, the buildings

B_{j}, j = 1, \dots, n

in the original function can be replaced by roads (long narrow polygons) without changing the mathematical equations of the original function. With this function, we can analyze the distribution of open space on the side of roads. Also, by replacing a certain type of open space with a certain type of building, we can examine the area covered by retail stores around each house.

In recent years, residents have been concerned about walkability to open space, and many studies analyze walkability. For example, Yen et al. [39] analyzed open spaces for sports within walking distance. Khatavkar et al. [40] examined walking access to parks, playgrounds, and streets as play areas near their homes. Yang et al. [41] surveyed the distribution of parks and green space around a community of elderly people. Kesarovski et al. [42] examined 10 min isochrones for walking and cycling centered on homes. These studies do not use the Euclidean distance but the shortest-path distance. Since our method assumes the Euclidean distance, the method cannot be directly applied to these studies. It is expected that the open-space ratio function will be developed assuming the shortest-path distance using the method developed by Okabe et al. [43].

Our method considers open space in the two-dimensional space. To discuss the open space environment in built-up areas, three-dimensional open space surrounding buildings is an important factor, as shown in Cherian et al. [6], Gamero-Salinas et al. [7], and Usui [44]. A challenging attempt is to extend the two-dimensional open-space function to the three-dimensional open-space function.

4. Conclusions

In the studies of open space in urbanized areas, an open-space ratio is often used as a first basic metric to examine the distribution of open space. Originally, the open-space ratio was defined as the ratio of the area of open space (unbuilt area) to the area of its corresponding building. In recent years, residents have become more concerned with the open-space ratio in the broader neighborhoods of their individual buildings than with the open-space ratio in their own building sites. To address this concern, we have formulated the local and global open-space ratio functions and implemented it using standard GIS operators. The local and global open-space ratio functions analyze the following problems.

First, because this function shows the change in the open-space ratio across a zone from small to large width, we can check whether the notorious modifiable area unit problem [15] exists. If the ratio changes, we must keep in mind that the results may change depending on the choice of a zone width. Second, as the ratio changes, the function shows in which bandwidth zone the ratio is the highest or lowest. This tells us which zones have the best or worst open space environment as measured by the open-space ratio. Third, when comparing between two buildings, the function shows in which bandwidth zone the difference is the largest or smallest. We can discuss the difference in open space between buildings in detail. This is useful for a family who are looking for a rental house as they can choose the best house by comparing their current house with an alternative house. Fourth, the function shows in which bandwidth zone the variance of the open-space ratios among all buildings in a region is the largest. This result shows us in which bandwidth zone the open space environment differs most among buildings in a region. Fifth, in the resulting bandwidth zone, we classify buildings into several classes according to the similarity of their open-space ratios and show their spatial distribution on a map. This map is useful for discussing the open space environment created by the mix of buildings with different open-space ratios in a district. Sixth, our method can deal with the open-space ratio not only around each building but also the around a clump of closely placed buildings, such as a housing complex. We can evaluate the open space environment around a housing complex in terms of the open-space ratio. Seventh, the open-space ratio function has broad applicability. It can be applied to specific open spaces, such as green spaces, public open spaces, parks, water surfaces, roads, and so on using the same computational method.

The open-space ratio function has limitations. Since the function assumes a Euclidean distance, it cannot analyze the walking access to open space. Since the function assumes two-dimensional open space, it cannot analyze the volumetric open space around a building in high-rise areas. We expect these limitation to be overcome in the future.

Author Contributions

Kayo Okabe conceived the idea of the open-space ratio function. Atsuyuki Okabe developed the idea mathematically and formulated an analytical method. Kayo Okabe implemented the analytical method in a GIS tool. Both authors applied the GIS tool to an actual empirical example and improved the method and the GIS tool. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The building footprint data used this empirical study were downloaded from the site https://fgd.gsi.go.jp/download/menu.php (accessed on 22 February 2024).

Acknowledgments

We acknowledge the support of the Joint Research Program No. 943 at the Center for Spatial Information Science, the University of Tokyo.

Conflicts of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be constructed as a potential conflict of interest.

Appendix A. Implementation of the Open-Space Functions with GIS

Computation of the open-space functions by hand is heavy work. Fortunately, this workload is greatly decreased by GIS operations. The following GIS operations assume operators in the standard GIS, such as QGIS and ArcGIS.

The computational procedure of the local incremental open-space ratio function given by Equation (1) is as follows. The workflow of the graphic modeler of QGIS is illustrated in Figure A1.

Step 1: Compute the buffer zone of building $B_{i}$ , i.e., $B_{f} (x| B_{i})$ in Equation (1). This computation is performed by a GIS buffering operator (Figure 2a).
Step 2: Compute the area occupied by buildings $B_{1}, \dots, B_{n}$ , $B_{n + 1}, \dots, B_{n *}$ , i.e., $⋃_{j = 1}^{n^{*}} B_{j}$ in Equation (1). This computation is performed by GIS union and dissolve operators (Figure 2b).
Step 3: Compute the open space in the buffer zone of $B_{i}$ by excluding the union of all buildings obtained in Step 2 from the buffer zone obtained in Step 1, i.e., $O (x | B_{i}) = B_{f} (x| B_{i}) ∖ ⋃_{j = 1}^{n^{*}} B_{j}$ in Equation (1). This computation is performed by a GIS difference operator (Figure 2c).
Step 4: Compute the local cumulative open-space ratio function by substituting these terms in Equation (1).
Step 5: Repeat the presented procedure for $x_{j}$ , j = 1, …, m and $B_{i}$ , i = 1,…, n using a GIS processing modeler, such as the graphic modeler of QGIS or the model builder of ArcGIS.

The computation of the local incremental open-space function given by Equation (2) is almost the same because the (j + 1)-th ring is obtained from the difference between the (j + 1)-th buffer zone and j-th buffer zone

({i . e ., B}_{f} (x_{j + 1}| B_{i}) ∖ (B_{f} (x_{j + 1}| B_{i})

; the open space in the (j + 1)-th ring is the difference between the open space in the (j + 1)-th buffer zone and that in the j-th buffer zone

(i . e ., O (x_{j + 1}| B_{i}) ∖ (O (x_{j + 1}| B_{i})

The global cumulative and incremental open-space ratio functions given by Equations (3) and (4) are algebraically obtained from the local cumulative and incremental open-space ratio functions obtained in the presented computation. Regarding the computational implementation of the open-space ratio function for a specific type open space discussed in Section 3, Okabe et al. [45] is helpful.

Figure A1. The graphic modeler (QGIS) of computing the open-space ratio function.

References

Solecki, W.D.; Mason, R.J.; Martin, S. The Geography of Support for Open-Space Initiatives: A Case Study of New Jersey’s 1998 Ballot Measure. Soc. Sci. Q. 2004, 85, 624–639. [Google Scholar] [CrossRef]
Wijayawardana, N.; Abenayake, C.; Jayasinghe, A.; Dias, N. An Urban Density-Based Runoff Simulation Framework to Envisage Flood Resilience of Cities. Urban Sci. 2023, 7, 17. [Google Scholar] [CrossRef]
Sharifi, E.; Lehmann, S. Correlation Analysis of Surface Temperature of Rooftops, Streetscapes and Urban Heat Island Effect: Case Study of Central Sydney. J. Urban Environ. Eng. 2015, 9, 3–11. [Google Scholar] [CrossRef]
Chotchaiwong, P.; Wijitkosum, S. Predicting Urban Expansion and Urban Land Use Changes in Nakhon Ratchasima City Using a CA-Markov Model under Two Different Scenarios. Land 2019, 8, 140. [Google Scholar] [CrossRef]
Cherian, N.C.; Subasinghe, C. Sun-Safe Zones: Investigating Integrated Shading Strategies for Children’s Play Areas in Urban Parks. Int. J. Environ. Res. Public Health 2023, 20, 114. [Google Scholar] [CrossRef]
Catharine Ward Thompson, C.W. Urban Open Space in the 21st Century. Landsc. Urban Plan. 2002, 60, 59–72. [Google Scholar] [CrossRef]
Gamero-Salinas, J.; Kishnani, N.; Monge-Barrio, A.; L’opez-Fidalgo, J.; Ana S’anchez-Ostiz, A. Evaluation of Thermal Comfort and Building Form Attributes in Different Semi-outdoor Environments in a High-density Tropical Setting. Build. Environ. 2021, 205, 108255. [Google Scholar] [CrossRef]
Whitehand, J.W.R. Green Space in Urban Morphology: A Historico-geographical Approach. Urban Morphol. 2019, 23, 5–15. [Google Scholar] [CrossRef]
Felt, J. Modern Zoning and Planning Progress in New York. Fordham Law Rev. 1961, 29, 681–692. [Google Scholar]
Tough, R.; MacDonald, G. The New Zoning and New York City’s New Look. Land Econ. 1965, 41, 41–48. [Google Scholar] [CrossRef]
Hamaina, R.; Leduc, T.; Moreau, G. A New Method to Characterize Density Adapted to a Coarse City Model. In Information Usion and Geographic Information Systems (IF&GIS 2013); Popovich, V., Claramunt, C., Schrenk, M., Korolenko, K., Eds.; Lecture Notes in Geoinformation and Cartography; Springer: Berlin/Heidelberg, Germany, 2014; pp. 249–263. [Google Scholar]
Schirmer, P.M.; Axhausen, K.W. A Multiple Clustering of the Urban Morphology for Use in Quantitative Models. In The Mathematics of Urban Morphology; D’Acci, L., Ed.; Birkhauser: Basel, Switzerland, 2019; pp. 355–382. [Google Scholar]
Steiniger, S.; Lange, T.; Burghardt, D.; Weibel, R. An Approach for the Classification of Urban Building Structure Based on Discriminat Analysis Techniques. Trans. GIS 2008, 12, 31–59. [Google Scholar] [CrossRef]
Boffet, A.; Rocca Serra, S. Identification of Spatial Structures within Urban Blocks for Town. Algorithmica 2001, 30, 312–333. [Google Scholar]
Openshaw, S. The Modifiable Areal Unit Problem. 1984 Concepts and Techniques in Modern Geography No. 38. Norwich: Geobooks.
Yamada, I.; Thill, J. Comparison of Planar and Network K-functions in Traffic Accident Analysis. J. Transp. Geogr. 2004, 12, 149–158. [Google Scholar] [CrossRef]
Ripely, B. Spatial Statistics; John Wiley: Hoboken, NJ, USA, 1981. [Google Scholar]
Yamada, I.; Rogerson, P. An Empirical Comparison of Edge Effect Correction Methods Applied to K-function Analysis. Geogr. Anal. 2003, 35, 97–109. [Google Scholar]
Boots, B.; Okabe, A. Local Statistical Spatial Analysis: Inventory and Prospect. Int. J. Geogr. Inf. Sci. 2007, 21, 355–375. [Google Scholar] [CrossRef]
Okabe, A.; Boots, B.; Satoh, T. A Class of Local and Global K Functions and Their Exact Statistical Methods. In Perspectives on Spatial Data Analysis; Anselin, L., Rey, S., Eds.; Advances in Spatial Science; Springer: Berlin/Heidelberg, Germany, 2010; pp. 101–112. [Google Scholar]
Ward, H. Hierarchical Grouping to Optimize an Objective Function. J. Am. Stat. Assoc. 1963, 58, 236–244. [Google Scholar] [CrossRef]
Okabe, A.; Boots, B.; Sugihara, K.; Chiu, S.N. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams; Jon Wiley: Chicester, UK, 2000. [Google Scholar]
Usui, H.; Teraki, A.; Okunuki, K.; Satoh, T. A Comparison of Neighbourhood Relations based on Ordinary Delaunay Diagrams and area Delaunay Diagrams: An Application to Define the Neighbourhood Relations of Buildings. Int. J. Geogr. Inf. Sci. 2020, 34, 2177–2203. [Google Scholar] [CrossRef]
Kristiadi, Y.; Sari, R.F.; Herdiansyah, H.; Hasibuan, H.S.; Lim, T.H. Developing DPSIR Framework for Managing Climate Change in Urban Areas: A Case Study in Jakarta, Indonesia. Sustainability 2022, 14, 15773. [Google Scholar] [CrossRef]
Choi, S.; Jiao, J.; Lee, H.K.; Farahi, A. Combatting the Mismatch: Modeling bike-sharing Rental and Return Machine Learning Classification Forecast in Seoul, South Korea. J. Transp. Geogr. 2023, 109, 103587. [Google Scholar] [CrossRef]
Sotoma, M.; Miyazaki, H.; Kyakuno, T.; Moriyama, M. Analysis of Land Use Zoning Regulations and Green Coverage Ratio. J. Asian Archit. Build. Eng. 2003, 2, b29–b34. [Google Scholar] [CrossRef]
Mihara, K.; Hii, D.J.C.; Takasuna, H.; Sakata, K. How does green coverage ratio and spaciousness affect self-reported performance and mood? Build. Environ. 2023, 245, 1. [Google Scholar] [CrossRef]
Loo, B.P.Y.; Lam, W.W.Y.; Mahendran, R.; Katagiri, K. How Is the Neighborhood Environment Related to the Health of Seniors Living in Hong Kong, Singapore, and Tokyo? Some Insights for Promoting Aging in Place. Ann. Am. Assoc. Geogr. 2017, 107, 812–828. [Google Scholar] [CrossRef]
Ariffin, A.; Mukhelas, H.K.; Iman, A.H.M.; Desa, G.; Mohammad, I.S. Spatial-Based Sustainability Assessment of Urban Neighbourhoods: A Case Study of Johor Bahru City Council, Malaysia. In Geoinformation for Informed Decisions; Rahman, A., Boguslawski, P., Anton, F., Said, M., Omar, K., Eds.; Lecture Notes in Geoinformation and Cartography; Springer: Cham, Switzerland, 2014. [Google Scholar]
Bolton, L.T. Space Ratio: A measure of Density Potentials in the Built Environment. Sustain. Cities Soc. 2021, 75, 103356. [Google Scholar] [CrossRef]
Fasihi, H. Urban Parks and Their Accessibility in Tehran, Iran. Environ. Justice 2019, 12, 242–249. [Google Scholar] [CrossRef]
Larson1, L.R.; Zhang, Z.; Oh, J.I.; Beam, W.; Ogletree, S.S.; Bocarro, J.N.; Lee, K.J.J.; Casper, J.; Stevenson, K.T.; Hipp, J.A.; et al. Urban Park Use during the COVID-19 Pandemic: Are Socially Vulnerable Communities Disproportionately Impacted? Front. Sustain. Cities 2021, 3, 710243. [Google Scholar] [CrossRef]
Cheng Gao, C.; Liu, J.; Cui, H.; Wang, Z.; He, S. Optimized Water Surface Ratio and Pervious Surface Proportion in Urbanized Riverside Areas. Environ. Earth Sci. 2014, 72, 569–576. [Google Scholar]
Cheng Gao, C.; Liu, J.; Liu, X.; Zhang, H. Compensation Mechanism of Water Surface Ratio and Pervious Surface Proportion for Flood Mitigation in Urban Areas. Disaster Adv. 2012, 5, 1294–1297. [Google Scholar]
Zhaoa, Y.; Zeng, L.; Weic, Y.; Liud, J.; Dengd, J.; Qucheng Dengc, Q.; Tonga, K.; Lib, J. An Indicator System for Assessing the Impact of Human Activities on River Structure. J. Hydrol. 2020, 582, 1245. [Google Scholar] [CrossRef]
Yang, B.; Kim, C.; Lee, H.; Lim, W. A Study on the Distribution Characteristics of NATURALIZED PLANTS: For Gyeongsangbuk-do Area. Int. J. Hum. Disaster 2012, 6, 1–10. [Google Scholar] [CrossRef]
Soehodho, S. Public transportation development and traffic accident prevention in Indonesia. IATSS Res. 2017, 40, 76–80. [Google Scholar] [CrossRef]
Parka, S.; Jeonb, S.; Shinyup Kimc, S.; Choi, C. Prediction and Comparison of Urban Growth by Land Suitability Index Mapping Using GIS and RS in South Korea Soyoung Parka. Landsc. Urban Plan. 2011, 99, 104–114. [Google Scholar] [CrossRef]
Yen, H.; Li, C. An Application of Open Government Data: An Evidence on Physical Activity Environment and Diseases. Int. J. Comput. Theory Eng. 2017, 9, 308–312. [Google Scholar] [CrossRef]
Khatavkar, P. Nurturing Children’s Health Through Neighbourhood Morphology. Creat. Space 2018, 6, 11–22. [Google Scholar] [CrossRef]
Yang, X.; Fan, Y.; Xia, D.; Zou, Y.; Deng, Y. Uses of and Preferences for Community Outdoor Spaces during Heat Periods. Sustainability 2023, 15, 11264. [Google Scholar] [CrossRef]
Kesarovski, T.; Hern’andez-Palacio, F. Time, the Other Dimension of Urban Form: Measuring the Relationship between Urban Density and Accessibility to Grocery Shops in the 10-minute City. Urban Anal. City Sci. 2023, 50, 44–59. [Google Scholar] [CrossRef]
Okabe, A.; Sugihara, K. Spatial Analysis along Networks; John Wiley: Chichester, UK, 2012. [Google Scholar]
Usui, H. How to Harmonise Variations in Streetscape Skeletons under Zoning Regulations: Considering their External Diseconomies. Environ. Plan. B Urban Anal. City Sci. 2023, 50, 434–452. [Google Scholar] [CrossRef]
Okabe, A.; Okabe, K. An Extended K Function Method for Analyzing Distributions of Polygons with GIS. Geogr. Anal. 2023, 55, 268–279. [Google Scholar] [CrossRef]

Figure 1. The building footprints in the study area in the Shibuya ward in Tokyo. The gray polygons are buildings in the study area. The white polygons are buildings in the peripheral area surrounding the study area and are used for edge correction. The circular curve centered at B₂₁ is the 50 m wide buffer zone around B₂₁. The circular rings centered at B₁₃₀ are 10 m with buffer rings surrounding B₁₃₀. The broken circular curve around the clump of the dark gray buildings is the 50 m wide buffer zone around the clump.

Figure 2. (a) The buffer zone

B_{f} (50 m| B_{21})

around Building

B_{21}

. (b) The buildings in the buffer zone. (c) The open space in the buffer zone

O (x| B_{21})

Figure 2. (a) The buffer zone

B_{f} (50 m| B_{21})

around Building

B_{21}

. (b) The buildings in the buffer zone. (c) The open space in the buffer zone

O (x| B_{21})

Figure 3. (a) The buffer zone of 40 m in width around

B_{130}

. (b) The buffer zone of 30 m in width around

B_{130}

. (c) The fourth ring area surrounding

B_{130}

. (d) The open space in the buffer zone

B_{f} (x_{4}| B_{130})

. (e) The open space in the buffer zone

B_{f} (x_{3}| B_{130})

. (f) The open space in the fourth ring.

Figure 3. (a) The buffer zone of 40 m in width around

B_{130}

. (b) The buffer zone of 30 m in width around

B_{130}

. (c) The fourth ring area surrounding

B_{130}

. (d) The open space in the buffer zone

B_{f} (x_{4}| B_{130})

. (e) The open space in the buffer zone

B_{f} (x_{3}| B_{130})

. (f) The open space in the fourth ring.

Figure 4. (a) Cumulative local open-space ratio function of Building

B_{21}

indicated by the thick black curve and that of Building

B_{130}

indicated by the thick gray curve. The upper dotted horizontal straight line shows the average open-space ratio in the whole study area. (b) The difference in open-space ratio between the two open-space ratio functions. The dotted horizontal line indicates that the difference in the open-space ratio is 0.1.

Figure 4. (a) Cumulative local open-space ratio function of Building

B_{21}

indicated by the thick black curve and that of Building

B_{130}

Figure 5. (a) Incremental local open-space ratio function of Building

B_{21}

indicated by the thick black curve and that of Building

B_{130}

indicated by the thick gray curve. The upper dotted horizontal straight line shows the average open-space ratio in the whole study area. The dotted horizontal line indicates the average open-space ratio in the whole study area. (b) The difference in open-space ratios between the two open-space ratio functions. The

x

-th ring means the ring area between the

x

m buffer circular line and the

x

− 1 m buffer circular line. The dotted horizontal lines indicate that the difference in the open-space ratio is between 0.1 and −0.1.

Figure 5. (a) Incremental local open-space ratio function of Building

B_{21}

indicated by the thick black curve and that of Building

B_{130}

x

-th ring means the ring area between the

x

m buffer circular line and the

x

− 1 m buffer circular line. The dotted horizontal lines indicate that the difference in the open-space ratio is between 0.1 and −0.1.

Figure 6. (a) The global cumulative open-space ratio function, i.e., the average of local cumulative open-space ratios of all buildings in the study area. The dotted horizontal line indicates the average open-space ratio in the whole study area. (b) The standard deviation of the cumulative local open-space ratios of all buildings in the whole study area. The dotted horizontal line indicates that the standard deviation is 0.05.

Figure 7. Four classes of buildings with similar open-space ratios in a 15 m wide buffer zone around each building. The open-space ratio of buildings decreases from green (highest), yellow, orange and red (lowest) in this order.

Figure 8. The process of clumping polygons using the buffer zone operator with the graphic modeler of QGIS. The number of clumps decreases from (a) nine, (b) eight, (c) seven, (d) six, (e) five, (f) four, (g) three, (h) two, (i) one.

Figure 9. The clumps of buildings in the case of 6 m wide buffer zones. Buildings belonging to the same clump are surrounded by the same color.

Figure 10. The incremental local open-space function of the clump of buildings indicated by the black curve in Figure 9. The width of a ring is 1 m. The dotted horizontal line indicates the average open-space ratio in the whole study area.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Okabe, K.; Okabe, A. Functional Method for Analyzing Open-Space Ratios around Individual Buildings and Its Implementation with GIS. ISPRS Int. J. Geo-Inf. 2024, 13, 70. https://doi.org/10.3390/ijgi13030070

AMA Style

Okabe K, Okabe A. Functional Method for Analyzing Open-Space Ratios around Individual Buildings and Its Implementation with GIS. ISPRS International Journal of Geo-Information. 2024; 13(3):70. https://doi.org/10.3390/ijgi13030070

Chicago/Turabian Style

Okabe, Kayo, and Atsuyuki Okabe. 2024. "Functional Method for Analyzing Open-Space Ratios around Individual Buildings and Its Implementation with GIS" ISPRS International Journal of Geo-Information 13, no. 3: 70. https://doi.org/10.3390/ijgi13030070

APA Style

Okabe, K., & Okabe, A. (2024). Functional Method for Analyzing Open-Space Ratios around Individual Buildings and Its Implementation with GIS. ISPRS International Journal of Geo-Information, 13(3), 70. https://doi.org/10.3390/ijgi13030070

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

Functional Method for Analyzing Open-Space Ratios around Individual Buildings and Its Implementation with GIS

Abstract

1. Introduction

2. Methods and Implementation

2.1. Local Open-Space Ratio Function

2.2. Global Open-Space Ratio Function

2.3. Distinctly Different Classes of Open-Space Ratios

2.4. Open-Space Ratio Function of a Clump of Buildings

3. Discussion

4. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

Appendix A. Implementation of the Open-Space Functions with GIS

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI