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%Submitted to Water Resources Research, December 1996
%\bigskip
\ni
{\bf Modeling landscape (processes)fluxes with spatially variable
level of detail}

\bigskip \ni
Lubos Mitas

\ni
National Center for Supercomputing Applications,
405 N. Mathews Ave,
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

\smallskip \ni
Helena Mitasova

\ni
Geographic Modeling and Systems Laboratory,
Department of Geography, 220 Davenport Hall,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801, USA

\bigskip
\bigskip
\ni
{\bf Abstract.}
\bs
\ni
{\bf 1. Introduction}
\medskip
\ni
Probelm formulation:
Collection and distribution of data moves from centralized
towards distributed. Digital geospatial data, gathered by different government
and private organizations are becoming readily available through Internet.
Due to this trend, it is possible to collect a diverse (heterogeneous)
data for a study area with different coverage, resolution, detail and 
accuracy (pozri abstrakty kapitol do BB2 na terminologiu a references). 
To fully exploit
the data available for given area for modeling (predictions and simulations)
algorithms should be able to handle and make effective use of such 
heterogeneous data sets, maximize the acuuracy and detail of predictions
while performing computations efficiently.

see also SDH conf in Canada, data fusion issues

Models, simulations which include only local spatial interactions
can be relatively simply segmented and areas with different accuracy
can be processesed independently with small overlaps. Processes
which involve fluxes, transport of mass are more difficult
to split because of long range spatial interactions where
a value/property in one location can have an impact on the simulation
results over along distance.
Typical processes with long range spatial relationship
which are difficult to partition are fluxes, such as water flow
sediment transport, chemicals transport, air flow, ...

The necessity for spatially heterogeneous or segmented modeling
of fluxes can arise due to various reasons:
\item{(a)} study area is represented by data with spatially variable resolution and accuracy
(study area has high resolution data however there is mass coming in
from an area for which we have only low resolution data);
\item{(b)} study area is large and reduction of resolution to reduce the 
computational demands is not acceptable (segmented processing is needed,
although this can be solved by computations by subwatersheds, however 
delineations of watersheds for large areas can be difficult too)
\item{(c)} study area is large, with spatially variable complexity for which 
simulation with spatially variable precission/detail is the most effective
(study area is large, but a certain part of it is homogeneous so it is 
unnecessary to run it all at high resolution,
only area of interest/highs patial variability will be simulated
at high resolution while the rest is low resolution.)

Various approaches are being developed. 

For characterization
at spatially variable resolution: variable block size kriging (REF).
reference to kriging with variable block size depending on data density
Csigo? (pozri co robil) 

For visualization: references to visualization with LODa (PERFORMA)

For simulations:
Finite element modeling approaches adapt the density of mesh to
the density of data/complexity of the process, however the mesh design
can be extremely complex and often requires manual adjustments

we present a general approach and algorithm for modeling fluxes
described by continuity equation at spatially variable resolution
and/or accuracy using hybrid data (heterogeneous accuracy and detail)
using the standard raster data format (avoid meshes) making it applicable/liunkavble
to stamdard GIS tools

, analyze the impact of difference in resolution an location on error
propagation and demonstrate the approach on a practical example, including 
visualization in 3D space.

\bigskip
\ni
{\bf 2. Methods}
\medskip
\ni
Our methodological framework for simulation of landscape processes 
is based on the principles defined in our previous work (mit 1997,98)
and briefly recalled here in relation to the multiscale 
landscape characterization and modeling

\item{a)} {\sl Representation of phenomena by 
multi-variate functions} (scalar or vector fields), 
as opposed to homogeneous hillslope segments or polygons
used in more traditional approaches. This concept assumes
that the values of modeled phenomena can be different in any point,
can change smoothly or abruptly from point to point.
This concept it also allows us
to support discrete representation of phenomena at multiple or
spatially variable scales and resolutions.

\item{b)} {\sl Description of landscape processes based on solving
 first principles equations} which determine the configuration or evolution
 of corresponding fields. This enables us to perform simulations
  formulated in terms of fundamental physical processes such as
  flux, diffusion, etc., and to use the known mathematical and
  physical apparatus for analyzing the solutions.

\item{c)} {\sl using a standard raster(grid) format} supported by
almost all GIS  - multiple raster files at different resolutions
 
Advanced GIS technology
is used to support the processing, analysis and visualization
of the data and simulation results [Mitasova et al., 1995a; Mitas et al., 1996;
Mitas et al., 1997; Brown et al., 1995].
The 3D visualization  design  includes techniques 
supporting multiscale visualization without
the need to create new data structures or meshes.

{\bf Notes}

Role of scale and interpolation
put into intro???
river slope is at scale 100m resolution, at higher resolutions - there are
more flat areas and depressions captured and even with accurate DEMs
water does not flow without explicit incorporation of channel.
Smmothing is equivalent to lowering resolution - show total stream length,
total first order streams/concentrated water flow at different resolutions 
and smoothing levels: high resolution, high tension has lots fo detail,
number of concentrated flow areas is high, however maijor streams are
difficult to extract due to flat areas and depressions and have to be
defined from stream map. At lower resolution/lowertension/smoothing number
of concentrated flow areas is substantially lower (TAB) - they are replaced 
by broad shallow 2D overland flow and main streams are simplified byt
water flow through. Compare water depth at outfoow at diff. resolutions
where the water goes through and then with and withouut pre-defined channel.
High resolution models have to be more complex and sophisticated,
capture more complex phenomena. At low resolution often semi-lumped 
(sibwatershed-based) models are
good enough because 2D flow is not represented correctly anyway

Whether water flows does not depend on resolution - it does not flow at 5 - 80m
resolution, smoothing makes the difference 
run SIMWE at 10 and 80m resolution and rfi=11 and 35
then run 80 to 10 (10 to 80 does not make sense)

Show movie terrain with streams - impact of tension and impact of scale

\bsn
{\bf 2.1 Continuity equations for fluxes of mass in landscape}

water and sediment flow
other fluxes (chemicals, air mass etc - see Lee, SWAT, etc.)
Fluxes in a 3D landscape can be described by bivariate (water,sediment,
nutrient flow oevr terrain) or trivariate (groundwater flow, ...)
differential equations (continuity, diffusion, etc...)
n this work we will focus on continuity equation 
in a form divq=sources-sinks

see/refer to CASC2d, Richards et al. TABS-MD *GIS and EM 93, 
Gao, Goodrich GISEM 93, etc maybe maidment

nested models Steyaert
dynamic, steady state: demonstrate both

\bsn
{\bf 2.2 Monte Carlo solution}

Equations can be solved by finite difference, finite element or stochastic
approach.
 Furthermore, solution through the Green's function given by equation (6) can be
reformulated for accommodation of {\it spatially variable accuracy}
 and {\it resolution} (MItas and Mitasova 1998-Vienna).  
The integral (6) can be multiplied by a {\it reweighting }
 function  $W({\bf r})$
$$
W({\bf r})\gamma({\bf r})= \int_0^{\infty}\int W({\bf r})G({\bf r},{\bf r'},p)
 {\cal S}({\bf r'})  d{\bf r'}dp =
\int_0^{\infty}\int G^*({\bf r},{\bf r'},p)
 {\cal S}({\bf r'})  d{\bf r'}dp
\eqno(8)
$$
 which is equal to the appropriate increase in accuracy ($W({\bf r})>1$)
 in the regions of interest while it is
  unity elsewhere. The function $W({\bf r})$ can change (abruptly or smoothly)
  between regions with unequal
 resolutions  and in fact, can be optimally adapted to the quality of
 input data (terrain, soils, etc) so that the accurate solution
 is calculated only in the regions with correspondingly accurate inputs.
 The reweighted Green's function $G^*({\bf r},{\bf r'},p)$, in effect,
  introduces higher density of sampling points in the region with
 large $W({\bf r})$. The statistical noise will be spatially
 variable as
    $\approx 1/[W({\bf r})\sqrt{M}]$, where $M$ is the average number
 of samples  resulting in the accuracy increase for the areas
 with $W({\bf r}) > 1$.

Large areas are often characterized by data with variable
resolution and the above {\it multi-scale} formulation
is able to make the best use of such heterogeneous data.
The system of equations (1-4) describes the water and sediment flow
at a spatial scale equal or larger than an average distance
between rills (i.e., grid cell size $\ge$ 1m) and therefore
the presented approach allows us to perform landscape scale
simulations at variable spatial resolutions from one to hundreds of meters,
depending on the complexity and importance of
studied subregions.

\bsn
{\bf 2.3 Implementation, GIS, visualization}

The proposed multiscale approach 
can be implemented within any  standard raster GIS  such as
ARCGRID, ARCvie spatial analyst, GRASS, IDRISI (REF) without 
the need for developing new data structures.

for each resolution a separate set of inputs is created with
overlapping region, 
simulation algorithm combines the data from different files 
by going through several passes of computation.

Algorithm produces a useful byproduct - identification of
borderline segments with inflow of matter and segments with outflow -
important for indetification of areas which require protection from
e.g. inflow of harmful chemicals or reas where outflow of matter
should be controlled.

To perform the modeling efficiently, customized interface
can be created for particular applications...

Visualization is supported by Nviz, using multiple masked surfaces



\bigskip
\ni
{\bf 3. Application and Results}
\medskip
\ni

To illustrate the concept we have used 
two examples :

Example 1: Modeling with spatially variable accuracy and homogeneous
resolution, used- to explain the principle
Scheyern

Example 2. Real world situation: spatially variable resolution due to the 
Entire waterhsed is represented by 400x400 DEM at 10m resolution.
Small subarea has available 1m contours interpolated to 2m DEM
(area is XxX at 10m resolution and 680x395 at 2m). For comparison/evaluation
we ran also simulations for the entire waterhsed at 2m resolution 
2000x2000 DEM.
spatially variable resolution/scale/detail of data: Hohenfels

Example 3. Radar data Ok for grass, bare soil area, but bad for forested
area - we will use different resolution and accuracy for each

\bigskip
\ni
{\bf 4. Discussion and Conclusion}
\medskip
\ni
further study : maximum acceptable difference in resolution, smooth change
of resolution

Presented examples still manageable at workstation level, IFSAR ft Hood
400 mil cells.

Theoretical implications in understanding the impact of changes at different locations..
can be used to better understand spatial aspects of human interaction
impact of location on error 

\bigskip
\ni
{\bf Acknowledgments}


\bsn
{\bf REFERENCES }

\bye

