2UP

The 2Up model is a spatially explicit global urbanisation and population growth model.

=Indices=
 * \( i \): Land Unit, a 30" x 30" lat long part.
 * \( c \): Country
 * \( t \): Time step: {2010, 2020, 2030, 2040, 2050, 2060, 2070, 2080}

=Urban Allocation=

Static variables
 * \( I^i_c \): Incidence matrix; for each \(i\): \(\sum\limits_c I^i_c = 1\)
 * \( A_i \): Area of land unit \( i \)

Dynamic variables
 * \( \sideset{^t}{_i}S \): Suitability, a function or combination of spatially explicit suitability factors
 * \( \sideset{^t}{^c}P \): Projected population, source: SSP database - Institute of Applied Systems Analysis (IIASA) -
 * \( \sideset{^t}{^c}D \): Hyde population density index, source: History Database of the Global Environment - Klein Goldewijk, Dr. ir. C.G.M. (Utrecht University) (2017): Anthropogenic land-use estimates for the Holocene; HYDE 3.2. DANS.
 * \( \sideset{^t}{^c}C \): Urban Claim

Urban Claim
 * \( \sideset{^t}{^c}U := \sideset{^{2010}}{^c}U \cdot {\sideset{^t}{^c}P \over \sideset{^{2010}}{^c}P} \cdot {\sideset{^{2010}}{^c}D \over \sideset{^t}{^c}D} \)
 * \( \sideset{^t}{^c}C := \min(\sideset{^t}{^c}U, \sum\limits_i{A_i \cdot I^i_c})\) Claim is U, but no more than the available land

Urban Spatial Allocation This allocation is done by taking the \( C^c \over \sum\limits_i A_i \cdot I^i_c \) th percentile of the land units of \( c \), descendingly ordered by \( S_i \) and weighted by \( A_i \).
 * \( X_i \in \{0, 1\} \) such that \( \sum\limits S_i \cdot X_i \) is maximal and that for each \( c \): \(\sum\limits_i X_i \cdot A_i \cdot I^i_c = C^c \) where \( X_i = 1 \) represents projected urban land use.

=Population Projection= Projected population growth \( \sideset{^t}{^c}{EP} := \sideset{^t}{^c}P - \sideset{^{t-1}}{^c}P\) is spatially allocated according to the same suitability \( \sideset{^t}{_i}S \), taking into account a spatially explicit maximum population density \( \sideset{^t}{_i}{MD} \).
 * \( \sideset{^t}{_i}{MD} := \max\left(\sqrt{\sum\limits_{j \in W(i)} \left( \sideset{^{t-1}}{_i}P \over A_i \right)^2 \cdot \sideset{^t}{_i}X \cdot {dd}_{ij} \over \sum\limits_{j \in W(i)} \sideset{^t}{_i}X \cdot {dd}_{ij}}, \sum\limits_c { I^i_c \cdot {\sum\limits_i \sideset{^{t-1}}{_i}P \cdot \sideset{^t}{_i}X \cdot I^i_c \over \sum\limits_i A_i \cdot \sideset{^t}{_i}X \cdot I^i_c} } \right) \)
 * \( {dd}_{ij} \): distance decay function.
 * \( W(i) \): A window around land unit \( i \) such that \( {dd}_{ij} \ne 0 \implies j \in W(i) \).

If \( \sideset{^t}{^c}{EP} > 0 \)
 * \( \sideset{^t}{_i}\Delta := \sum\limits_c I^i_c \cdot \sideset{^t}{^c}{EP} \cdot { \sideset{^t}{_i}S \cdot A_i \cdot \sideset{^t}{_i}X \cdot (\sideset{^{t-1}}{_i}P < \sideset{^{t-1}}{_i}{MD} * A_i ) \over \sum\limits_i I^i_c \cdot \sideset{^t}{_i}S \cdot A_i \cdot \sideset{^t}{_i}X \cdot (\sideset{^{t-1}}{_i}P < \sideset{^{t-1}}{_i}{MD} * A_i ) }\)