3.2.4 Mathematical Formulation of the Urban Value Field

Purpose

The conceptual principles presented in the previous section describe the qualitative behavior of the Urban Value Field. To enable scientific analysis, computational modeling, and practical applications, these principles must be expressed in mathematical form.

The objective of this section is not merely to introduce an equation but to establish a mathematical representation that is consistent with the theoretical foundations of ULVF.

The proposed formulation is designed to describe how urban value is generated, propagated, attenuated, and integrated across complex urban systems.

1. General Representation

Let

V(x,t)V(x,t)V(x,t)

denote the intensity of urban value at spatial location xxx and time ttt.

Unlike conventional parcel-based valuation, V(x,t)V(x,t)V(x,t) represents a continuous spatial field.

2. Value Sources

Urban value originates from multiple sources.

Suppose the city contains

nnn

major value sources.

Each source generates an influence

SiS_iSi​

depending on its characteristics.

Examples include

• CBD

• Metro stations

• Airports

• Universities

• Hospitals

• Commercial centers

• Industrial parks

3. Spatial Propagation

The influence generated by each source propagates through the urban system.

Propagation depends on

• accessibility;

• transportation networks;

• travel impedance;

• urban morphology;

• functional connectivity.

This process is represented by a propagation function

Pi(x)P_i(x)Pi​(x)

which describes how value spreads from each source.

4. Spatial Attenuation

Value influence decreases with distance or spatial resistance.

Therefore

Pi(x)P_i(x)Pi​(x)

is not constant but declines according to an attenuation function

Di(x)D_i(x)Di​(x)

whose exact mathematical form depends on the urban context.

5. Interaction Among Sources

Urban environments contain multiple overlapping value sources.

The resulting Urban Value Field is therefore not determined by a single source but by the combined interaction among all sources.

The total value field may therefore be expressed conceptually as

V(x,t)=F(S1,S2,…,Sn;P;D;I;T)V(x,t) = F\left( S_1,S_2,\ldots,S_n; P; D; I; T \right)V(x,t)=F(S1​,S2​,…,Sn​;P;D;I;T)

where

• SSS represents value sources,

• PPP propagation,

• DDD attenuation,

• III interaction,

• TTT temporal evolution.

This equation represents the general structure of ULVF rather than a specific computational model.

6. Temporal Evolution

Urban value changes continuously through time.

Consequently,

V(x,t)V(x,t)V(x,t)

is time dependent.

Infrastructure investment, land-use change, policy interventions, economic development, and demographic change continuously modify the value field.

Future sections introduce mathematical models describing these temporal dynamics.

General ULVF Equation

The Urban Value Field may therefore be represented conceptually as

V(x,t)=∑i=1nSi⋅Pi(x)⋅Di(x)+I(x,t)V(x,t) = \sum_{i=1}^{n} S_i \cdot P_i(x) \cdot D_i(x) + I(x,t)V(x,t)=i=1∑n​Si​⋅Pi​(x)⋅Di​(x)+I(x,t)

where

• SiS_iSi​ denotes the strength of value source iii;

• Pi(x)P_i(x)Pi​(x) describes spatial propagation;

• Di(x)D_i(x)Di​(x) represents attenuation;

• I(x,t)I(x,t)I(x,t) represents interactions among multiple value fields.

This formulation provides the theoretical foundation upon which more specialized mathematical models can be developed.

Scientific Meaning

The proposed equation should not be interpreted as a fixed valuation formula.

Instead, it represents the mathematical structure of the Urban Value Field.

Different applications may employ different propagation functions, attenuation functions, interaction models, or computational algorithms while remaining consistent with the theoretical principles of ULVF.

Consequently, ULVF serves as a mathematical framework rather than a single predictive equation.

Conclusion

The mathematical formulation presented here establishes a bridge between the conceptual principles of ULVF and subsequent computational implementations.

By representing urban value as a continuous spatial field generated by multiple interacting sources, ULVF provides a flexible theoretical foundation for spatial analysis, numerical simulation, artificial intelligence, land valuation, and urban planning.