Notes on classifying and scaling budget variables
Ensenada skunkworks
May 2, 2001
“The scaledown problem” –How do we estimate variables at a particular (budget) site from data gathered at relatively coarse granularity (typology dataset)? For simple linear or proportional relationships, we might expect to be able to use coarsegrained variables to estimate finegrained variables by simply multiply by an appropriate scaling factor. e.g. to estimate runoff (m3/yr) for a subregion of a grid cell, we might multiply the runoff associated with the grid cell by the ratio of the drainage area associated with the subregion to the drainage area of the entire gridcell. However, due to the difference in resolution between the two scales, we might expect these simple proportional relationships to break down. To test this, we evaluated the relationship between some variables estimated independently in both datasets: system area (water surface area of a budget site) vs water area of the associated typology cell, Vq (riverine flow, m^{3}/yr) vs corresponding freshwater flow to the associated typology cell, drainage area of the budget site vs drainagarea of the corresponding typology cell. To test whether a simple proportional relationship between variables across scales would hold, we examined a relationship of the following form using linear regression:
Ln( X_{Qbudget})= a + b Ln(X_{Qtypologycell} )
or
Ln( Y_{Qbudget}/Y_{Qtypologycell} )= a + b Ln( X_{Qbudget}/X_{Qtypologycell} )
Where the X and Y refer to the above variables in either the budget or typology dataset.
These are equivalent to
X_{Qbudget} = a [X_{Qtypologycell l}]^{b}
And
Y_{Qbudget}/Y_{Qtypologycell} = a [X_{Qbudget}/X_{Qtypologycell}]^{b}
Table 1 Parameters for power law relationships between budget and typology variables (all are statistically significant at the 0.05 level unless otherwise indicated) 

a) using all budget sites (including seasonal budgets) except Manila bay 

Y 
X 
a (se) 
b (se) 
R2 
runoff _{budget}/ runoff_{ typologycell} 
Drainage area_{budget}/ Drainage area _{typologycell} 
1.18(0.27) 
0.51(0.12) 
0.28 
runoff _{budget} 
runoff_{ typologycell} 
1.84(0.8) 
0.81(0.11) 
0.36 
Drainage area_{budget} 
Drainage area _{typologycell} 
8.0(0.62) 
0.13(0.07) 
0.06 
system area_{budget} 
Water area in cell_{typologycell} 
2.19(0.99, NS) 
0.32(0.12) 
0.05 





b ) using only annual budget sites except Manila bay 

Y 
X 
a (se) 
b (se) 
R2 
runoff _{budget}/ runoff_{ typologycell} 
Drainage area_{budget}/ Drainage area _{typologycell} 
1.07(0.29) 
0.68(0.13) 
0.43 
runoff _{budget} 
runoff_{ typologycell} 
2.46(1.08) 
0.72(0.15) 
0.24 
Drainage area_{budget} 
Drainage area _{typologycell} 
2.19(2.71, NS) 
0.72(0.3) 
0.11 















The results of Table 1 show that for the variables examine, the proportional relationship across scales breaks down. We hope the relationships developed above will show a reasonable route to bridging the gap across scales.