5.2.4 Mathematical Definition of ULVF

5.2.4 Mathematical Definition of ULVF

The conceptual evolution presented in the previous sections naturally leads to the mathematical formulation of the Urban Land Value Field (ULVF).

Unlike conventional valuation approaches that estimate the market price of individual land parcels, ULVF represents urban land value as a continuous scalar field distributed across geographic space and evolving through time.

Formal Definition

Let denote the urban spatial domain.

The Urban Land Value Field (ULVF) is defined as the continuous scalar function

or equivalently,

where

• denote spatial coordinates;

• denotes time;

• represents the urban land value at location and time .

Unlike discrete parcel prices, this formulation assigns a value to every location within the urban space, enabling continuous spatial analysis and dynamic temporal modeling.

Fundamental Properties

The Urban Land Value Field possesses several fundamental characteristics:

Spatial Continuity

Urban value varies continuously across geographic space. or, more generally, when spatial derivatives are required.

Temporal Evolution

Urban value changes continuously through time.

Spatial Interaction

The value at one location is influenced by surrounding urban elements.

where represents spatial interaction weights.

Multi-factor Dependence

Urban value depends on multiple interacting urban variables,

where the variables may represent accessibility, infrastructure, land use, economic activity, environmental quality, institutional conditions, and other urban factors.

Interpretation

Within ULVF, individual land prices are interpreted as discrete observations sampled from an underlying continuous urban value field. The objective of urban value analysis therefore shifts from estimating isolated parcel prices toward reconstructing, analyzing, and predicting the continuous field that governs value formation throughout the city.

This mathematical definition establishes the theoretical foundation upon which the subsequent methodological components of ULVF—including the Dual-Scale Variable Structure (DSVS), the Value Correlation Matrix (VCM), the Value Relation Matrix (VRM), GIS integration, and AI-based urban value modeling—are developed.

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Definition 1 (Urban Land Value Field).

The Urban Land Value Field (ULVF) is a continuous scalar field

defined over geographic space and time, representing the spatial and temporal distribution of urban land value generated by the interaction of physical, economic, social, institutional, and environmental processes.

Conclusion

The intellectual journey from land as a special asset, through land rent theory, land price capitalization, and urban value, demonstrates the progressive expansion of our understanding of urban land.

Classical land economics established the fundamental mechanisms by which land generates economic rent and how this rent is capitalized into market prices. Urban economics further emphasized the importance of location, accessibility, infrastructure, and spatial competition in shaping land value. However, the increasing complexity of contemporary cities reveals that urban value cannot be fully explained by market prices alone.

Urban value emerges from the continuous interaction of physical infrastructure, economic activities, land-use transformation, institutional policies, environmental quality, and social dynamics. These interacting processes extend beyond individual land parcels and evolve continuously across space and time.

This theoretical evolution naturally leads to the Urban Land Value Field (ULVF). Rather than viewing urban land value as isolated market prices, ULVF conceptualizes it as a continuous spatial phenomenon generated by interconnected urban processes. In this perspective, land prices become observable manifestations of an underlying urban value field rather than independent economic variables.

The transition from land rent to urban land value field therefore represents not a replacement of classical land economics, but its natural extension within the context of modern urban science, Geographic Information Systems (GIS), spatial analytics, and artificial intelligence.

The following section develops this concept further by introducing the theoretical foundation of urban land value as a continuous field distributed across geographic space and evolving through time.