Aggregation criteria for surface heat balances in a heterogeneous area based on a linear model Tosiyuki Nakaegawa a, * , Taikan Oki b , Katumi Musiake b a Climate Research Department, Meteorological Research Institute, Japan Meteorological Agency, Tsukuba, Ibaraki, 305-0052, Japan b Institute of Industrial Science, University of Tokyo, Tokyo, 153-8505, Japan Received 11 July 2000; received in revised form 9 January 2001; accepted 3 April 2001 Abstract The present study treats the aggregation of the heterogeneity of the surface heat ¯uxes. The aggregation criteria proposed here are used to evaluate the aggregation error and to calculate the maximum values of the statistics of the distributing parameters with a given tolerance error. The intrapatch scale distribution of the surface ¯ux bulk transfer coecient can be aggregated within a tolerance error of 5 W=m 2 using the mean parameter method MPM) in all cases, but the evaporation eciency cannot be ag- gregated using MPM in some cases. However, aggregation is possible even when the moment method of second-order MM2) is applied. The interpatch scale heterogeneity at an area constituted of dierent land cover types can be aggregated with good accuracy using MM2. This result reveals that aggregation using the dominant land cover method DLM) and MPM could result in a sig- ni®cant error of more than 40 W=m 2 . Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Aggregation; Surface heat balance; Heterogeneity; Moment method; Tolerance error; Aggregation criteria 1. Introduction Surface hydrological processes are highly nonlinear because the land surface is the linking interface between two completely dierent systems, atmosphere and lithosphere. The land surface variables within an at- mospheric general circulation model AGCM) gird box vary widely because hydrological processes generate heterogeneities in themselves and the grid box usually consists of multiple land cover types. These character- istics make it dicult to aggregate hydrological pro- cesses such as surface heat ¯uxes and in®ltration. Subgrid scale variability in precipitation has a sig- ni®cant eect on the surface hydrology, with up to twice increase of surface runo and 15% increase in evapo- transpiration [14]. Variability in soil moisture aects the evaporation and the evaporation dierence between two dierent distributions with the same mean, and the variance of the soil moisture can reach 10% [25]. A numerical simulation indicated that soil moisture dis- tribution aects partitioning of areally averaged surface heat ¯uxes locally [7,8], and that, surprisingly hetero- geneous vegetation aects global-scale circulation pat- terns [4]. In addition to the surface heterogeneity, atmospheric conditions have signi®cant eects on the surface processes [31]. Many ®eld experiments have been conducted to ob- serve the distributing surface ¯uxes over heterogeneous terrain and to evaluate the in¯uence of the heterogeneity on estimating areally averaged ¯uxes, such as First In- ternational Satellite Land Surface Climatology Project ISLSCP) Field Experiment FIFE) [29] and the Lake Biwa Project [26]. These studies revealed that areally averaged ¯uxes cannot be estimated accurately without taking the heterogeneity into account, and therefore treatment of the heterogeneity is presently a crucial issue in hydrology [11]. Land surface heterogeneity has two classes of hier- archy. The upper class corresponds to a qualitative heterogeneity such as dierent land cover types in an area, and the lower class corresponds to the quantitative heterogeneity, such as the physical parameter distribu- tion in a single land cover type. The former is referred to as interpatch subarea/subgrid) scale heterogeneity and the latter is referred to as intrapatch intrasubarea/in- trasubgrid) scale heterogeneity [15]. Two implementa- tions are required to represent the hierarchy of the Advances in Water Resources 24 2001) 1159±1171 www.elsevier.com/locate/advwatres * Corresponding author. Tel.: +81-298-53-8601; fax: +81-298-55- 2552. E-mail address: tnakaega@mri-jma.go.jp T. Nakaegawa). 0309-1708/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII:S0309-170801)00041-0