Availability of Least-Cost Pathway Analysis for the Study of Inka Road System
Go Matsumoto (Department of Anthropology, Southern Illinois University at Carbondale)
[ABSTRACT] The inferred purposes of the Inca road system (e.g., relaying messages and transporting goods) indicate that minimum cost for traveling was
Zone 1 (Cañar-Azuay) Due to the unknown datum and the poor resolution of the scanned
METHOD
map, there is a high probability that the locations of the digitized polyline and point features
(roads and sites) are not very accurate. Given this fact, nonetheless, the northern half of the
one of the major concerns when designing the system. As previous studies indicate, however, other sociopolitical, cultural, and religious factors should
have operated intertwiningly. The least-cost pathway analysis based on anisotropic cost surface helps to inversely delineate non-economic factors for road
construction and route selection. This study will demonstrate the availability of this analytical technique, taking the Inka road system as an example.
The data source available for this study is quite limited. The famed Proyecto Qhapaq Ñan
Cusco has not yet published any comprehensive map of the Inka Road (Amado Gonzales 2005;
Ugarte Vega Centeno 2005). As of the moment, the only maps that can be scanned and
integrated into GIS overlay are those recorded by Hyslop (1984) from the late 70’s to the early
80’s. Out of the 12 zones that Hyslop recorded, the first three zones (Cañar-Azuay, Lambayeque-
shared path from Ingapirca to Achupallas to the north is relatively approximate to the
preserved/reconstructed road segments, while the southern half follows a completely different
route from the preserved/reconstructed (Figure 4). As clearly seen here, the calculated prefers
traveling on the valley floor to reduce the cost, while the preserved/reconstructed tends to
Figure 6: A Three-Dimensional View of the Calculated Paths for Zone 1
choose the shortest route even at the expense of additional energy expenditure.
Moche, and Cajamarca-Huamachuco) were selected due to the time constraints (Figure 2).
There is no description in either maps or texts about the map datum by which each of the
Paths
maps was projected. Taking into account that the original maps based on which Hyslop’s maps
INTRODUCTION
HYPOTHESIS
were created were all produced during the 60’s to the 70’s in Ecuador or Peru, I speculate that
the Provisional South American Datum 1956 (PSAD56) was used. Both preserved and
reconstructed segments of the road system were scanned, georeferenced, and digitized together
The least-cost pathway analysis, normally equipped in GIS packages, is an oft-employed
An escapeway from this cul-de-sac is to further refine the cost surface raster so as to more
analytical tool for simulating the “most economical” route for traveling between archaeological
accurately delineate the traveler’s concern for cost-efficiency. A recent shift in emphasis from
settlements and natural resources and. As Conolly and Lake (2006:252) note, this method has
isotropic to anisotropic costs enables us to represent different modes of travel across slope more
often been employed to predict the location of unpreserved ancient transport routes. As long as
precisely and faithfully to the reality. Anisotropic modeling takes into account the difference in
we keep using it solely for this purpose, however, we will never be able to link the results to the
cost depending upon the direction of travel (e.g., perpendicular to or parallel with the aspect?
true picture of the prehistoric road construction and route selection. This means that we cannot
upslope or downslope?), while isotropic one fails to do so and simply assumes a certain amount
assess the effectiveness of the method and may end up with wasting our limited resources just
of cost for each cell regardless of the travel direction. Consequently, the former may provide
for a guess. For example, it was because of its empirical modeling and testing with reference to
different routes for outward and homeward journeys, whereas the latter does not.
the actual distribution of archaeological settlements that the predictive modeling was widely
A recent field experiment directed by Yasuhisa Kondo (University of Tokyo) in Kozu Island,
accepted and eventually became a defining feature of archaeological application of GIS-aided
Japan revealed that the least-cost path calculated by means of an anisotropic, accumulated cost
analytical tools. In order for the least-cost pathway analysis to earn a reputation as a reliable
surface is by and large in close alignment with the route selected by travelers who favor the
method to solve authentic archaeological problems, the accuracy and precision of the modeling
”ease of traveling. In likewise, if the calculated path prioritizing the minimum cost for traveling
need to be refined through a comparative study of the calculated and the preserved routes (e.g.,
can adequately capture a possible, economic concern of those who built and traveled a road, in
Krist and Brown 1994). Furthermore, because the route selection may vary not only for different
theory, the gaps between the calculated path and the preserved road have to be explained from
purposes, but also in different times and areas, we should not assume an all-purpose model that
something other than an economic perspective. As a result, by focusing on those gaps, I suspect
is thoughtlessly applicable to any case. In this regard, the Inka Road System is an ideal subject of
that it would be possible to inversely shed light on the unknown, non-economic factors. This
this analysis, because it covers an immense area of diverse regional cultures and environmental
hypothesis is tested below by calculating some least-cost paths for the Inka Road System, which
characteristics and is, if partially, still preserved either on the ground or in the record. As the first
is thought to have been used for both economic (e.g., relaying messages and transporting goods)
step of my long-term research effort to elucidate interregional interactions in the Andes, this
and symbolic functions (e.g., representing ceque lines).
with archaeological sites and modern-day populations.
In order to create the anisotropic cost surface, sets of free Digital Elevation Models (SRTM,
Degree Tiles) of relatively low resolution were utilized. For the large size of the study areas (e.g.,
Zone 1 of approximately 4,000 km² or 400,000 hectares), the 90-m resolution would be quite
adequate. Subsets including the study areas were clipped off of a combined image of the
Slope
(Min.)
Slope
(Max.)
Slope
(Mean.)
Cost
(Min.)
Cost
(Max.)
Cost
(sum)
Time
(hour)
1: Tomebamba
-> Achupallas
Calculated
97.77km
0.12
28.64
6.67
-26.40
23.86
-428.47
12.98
Preserved
87.29km
0.18
42.04
12.08
-26.28
41.86
887.11
22.44
2: Achupallas
-> Tomebamba
Calculated
97.77km
0.12
28.64
6.67
-26.40
23.86
-428.47
12.98
Preserved
87.29km
0.18
42.04
12.08
-26.28
41.86
887.11
22.44
3: Ricuarte
-> Deleg
Calculated
29.13km
0.55
24.71
4.27
-15.22
24.61
21.17
3.57
Preserved
12.60km
1.00
20.06
7.95
-14.04
11.17
-351.01
2.58
4: Deleg
-> Nazon
Calculated
15.38km
0.85
18.81
6.23
-18.81
11.94
-3.33
2.09
Preserved
8.59km
1.81
24.70
11.14
-24.70
17.99
26.88
1.87
5: Nazon
-> Ingapirca
Calculated
21.04km
1.38
14.60
5.98
-11.55
14.55
-233.90
2.55
Preserved
18.60km
0.74
22.85
9.29
-22.63
16.69
-217.82
4.11
adjacent DEM tiles. Patching data loss and defining UTM coordinates for the subsets were
conducted by means of 3DEM, free software available online. The procedures to create an
Length
Figure 5: Inter-Site Paths for Zone 1 (Reproduced from Hyslop 1984:20, fig. 2.1)
anisotropic cost surface are summarized below in the form of expressions used in ArcGIS Raster
Table 2: Comparison of total length, slope degree, travel cost and time for the ten paths in Zone 1
Calculator:
STEP 1:
[slope] = Slope([dem.tif])
…
STEP 2:
[bklink] = CostBackLink([origin], [slope])
…
STEP 3:
[flowdir] = FlowDirection([dem.tif])
…
STEP 4:
[diff] = Abs(Log2([flowdir]) + 1 - [bklink2])
…
study is aimed at examining the availability of the least-cost pathway analysis for the study of the
Inka Road System.
Create a raster dataset that identifies the rate of maximum change
in elevation value from each cell.
Creates a raster dataset that defines the next cell on the least
accumulative cost path to the destination. The direction of the
back link needs to be inverted by reclassifying the cell values: 1 ->
5; 2 -> 6; 3 -> 7; 4 -> 8; 5 -> 1; 6 -> 2; 7 -> 3; and 8 -> 4 ([bklink] ->
[bklink2]). See Figure 3-a.
Creates a raster dataset that defines the flow direction from each
cell to its steepest downslope neighbor (1 to 255). See Figure 3-b.
Create a raster dataset that not only distinguishes up and
downslope but also defines the types of movement across slope.
The resultant cell values are to be reclassified into eight categories,
integers from 0 to 7: 0, 1, 6, and 7 for downslope and 2, 3, 4, and 5
for upslope ([diff] -> [diff2]). See Figure 3-c.
In Table 2, the calculated paths are longer than the preserved/reconstructed road segments,
because the employed model always attempts to avoid steeper slopes that take more time to
travel. It is paradoxical, however, that this approach may lead to increase the travel time as in the
calculated paths 3 and 4. This is probably because the model cannot examine the topography as a
whole, but only considers “one step forward,” that is, the immediate eight cells that surround the
cell in question. Consequently, depending upon the topography and the locations of origin and
Figure 10: A Three-Dimensional View of the Calculated Paths for Zone 3
destination(s), an unrealistically long path may be provided. Nevertheless, the calculated paths 1
and 2 successfully reduced slope degree, travel cost, and travel time.
On the other hand, the Inka Road in some places seems to have been designed to link
adjacent sites with the shortest paths rather than to reduce the travel cost right in front. It is
likely that priority was placed on traveling through those sites in order. This could be for relaying
STEP 5:
[diff3] = Con([diff2] > 4, 7 - [diff2], [diff2])
STEP 6:
[travelcost] = - Cos([diff4])
LEAST-COST PATHWAY ANALYSIS
In the least-cost pathway analysis, the route is calculated based on the accumulated cost
surface that models the cost of traveling from a given origin to one or more destinations. As
…
…
Reclassifies the cell values of [diff2] into five different types of
movement. Then, the resultant cell values are again to be
reclassified: 0 -> 0; 1 -> 45; 2 -> 90 (or 75); 3 -> 135; 4 -> 180 ([diff3]
-> [diff4]). See Figure 3-d.
Creates a raster dataset that represents travel cost in terms of
movement type by calculating the cosine of the cells in [diff3]. The
resultant cell values will fall in the range from -1 (lowest cost) to 1
(highest cost).
STEP 7:
[aniso] = [travelcost] * [slope]
…
Create an anisotropic cost surface by multiplying travel cost by
slope degree.
STEP 8:
[time] = 1 / (100 * Exp( - 3.5 * Abs([aniso] *
3.1415926535 / 180 + 0.05)))
…
Calculates the time required to travel in each cell, based on
Tobler’s (1993) model.
messages by chaski or for expressing some symbolic importance just like the concept of “roads
through time” that represent symbolic links between religious features from different time
periods (Adler 1994; Fowler and Stein 1992). Regarding the transportation of goods by caravans
of camelids, however, the maximum slope of 42.04˚ may be the greatest obstacle. The road
system should have been critical in this area for the military campaigns to conquest the Cañari
Figure 8: The Least-Cost Paths for Zone 3 (Reproduced from Hyslop 1984:57, fig. 4.1)
prepared DEM. Since SRTM Degree Tiles (DEMs) involve elevation values not only for the terrain
accumulated cost surface and used to find the least-cost route over it. For the creation of an
Table 1: The procedures to create an anisotropic cost surface (originally coded by Yasuhisa Kondo,
University of Tokyo).
accumulated cost surface, most GIS packages require a minimum of three dataset: (1) traffic
surface, but also for the surface of the sea, the latter need to be excluded (or replaced with
unrealistically high cost values) in order to gain only an overland path. Otherwise, as shown in
origin; (2) destination(s); and (3) cost surface (or cost-of-traveling) raster.
Path 3 in Figure 7, water route may be given as the least-cost path. This is very noteworthy
because Path 3 indicates the possibility that the coastal population may have sailed in the sea for
some economic functions such as long-distance trade along the coast. Needless to say, the
surface conditions of the sea have to be carefully modeled by taking into account tide and
currents.
Most of the currently available models for the least-cost pathway analysis view terrain slope
a.[bklink2]
b.[flowdir]
c. [diff2] ([diff3])
d. Five types of movement
and aspect as a primary cost of traveling; therefore, the calculated path always circumnavigates
hills and mountain ridges. Unexceptional is the coastal area where the slope is trivial. The
Figure 3: The creation of an anisotropic cost surface.
preserved road segments on the coast, however, seems to persistently draw a straight line, even
in the rugged area (between Tambo Real and Canteras in Figure 7) allowing for additional energy
c. Cumulative Cost-surface
Figure 1: Finding the least-cost route by reiterating a basic spreading function over the cost-surface
(Reproduced from Conolly and Lake 2006:222, fig. 10.11).
The anisotropic cost surface is then processed to generate an accumulated cost surface by means
expenditure. For the travel on the coast, viewshed rather than ease of traveling may have been
of Cost Distance function of the ArcGIS Spatial Analyst extension. Based on the resultant,
prioritized. Thus, two notable gaps between the calculated paths and the preserved/
accumulated cost surface, the least-cost paths are calculated.
reconstructed road segments were observed: one between Tambo Real and Canteras and the
other between Desert Site to Chinquitoy Viejo.
The way of creating cost surface varies depending on the type of cost that you assume for
RESULTS
the determination of accessibility. Most of the currently available models view the slope and
DISCUSSIONS
paths to different destinations, Paths 1 and 2, share a single path that is parallel to the
elapsed time or energetic expenditure (Bell and Lock 2000; Tobler 1993; van Leusen 2002). As
preserved/reconstructed road segments and is in complete alignment with the valley floors of
Setting the northernmost and southernmost sites in each zone as the traffic origins and the
rest of the sites located along the preserved/reconstructed segments of the Inka Road (both Inka
religious factors could have operated intertwiningly. Under the influence of postprocessualist
sites and modern-day populations) as destinations, both northward and southward routes were
thinking about space and landscape, Llobera (2000) integrates the cultural influence of
calculated. It was revealed that all but a few of the resulting routes share a single path and then
monuments into the energetic cost of traveling. Nonetheless, unlike the physical “frictions” that
branch off into a unique path to each destination (Figures 4, 7, and 8). Expectedly, in some areas,
can be measured and quantified objectively, it may be quite difficult to represent those,
both gaps and close alignments between the calculated paths and the preserved/reconstructed
oftentimes intangible, factors in the form of map with the same resolution and scale as those of
road segments came into focus. In order to clarify the reason for the gaps between the calculated
other map layers and to sophisticate the model to a satisfactory level for many archaeologists.
and preserved, additional inter-site routes were also calculated for Zones 1 and 3 (Figures 5 and
Here is a limitation of this analytical tool.
9).
Figure 2: Inka Road System (Taken from Hyslop 1984)
only for the study of Inka Road System, but also for the interregional interactions in the Andes. As
the next step of this long-term research, priority will be placed on: (1) reexamination and
Cajamarca and Condebamba Rivers (Figure 8). The unique paths branching off into the
As indicated above, there is a slight chance that new road segments may be found along the
calculated paths, especially along those in Zones 1 and 3. In order to reliably detect the presence
of those unseen road segments, improvement of data quality and ground-truth checking of the
already found segments by means of Global Positioning System (GPS) are essential. The study
areas are truly immense. Not all preserved roads could have been found by Hyslop. In fact, for
Zone 1, Fresco (2004) recently reported as “secondary roads” some new paths that had not been
recorded by Hyslop: three stretching east, west, and southwest from Tomebamba, one branching
off of the artery near Deleg and heading for the northeast, and another stretching northwest
from Ingapirca.
This study not only demonstrates that the least-cost pathway analysis is a useful analytical
tool, but also highlights some of the limitations and weaknesses of the current model and
calculation formula. First of all, the model cannot view the topography as a whole, but only
examines the travel cost in the immediate eight cells that surround the cell in question. As a
result, the resulting path may be unrealistically long, circumnavigating costly up and downslopes.
establishment of an additional model and formula to predict water routes.
Furthermore, as I noted elsewhere (Matsumoto 2005, in press), the improved model also
needs to involve postprocessualistic concerns for subjectivity, for example, sense of distance.
Commonly, adults walk longer and faster than children, and a caravan of men and animals travels
faster across a wider range of area when they do so without a heavy burden. Furthermore, the
perceived distance may not necessarily be commensurate with the amount of time that they
actually spend. It may also vary depending on certain factors such as the type of activity (e.g.,
messaging, pilgrimage, expedition, trade, farming, fishing, hunting, and so forth). This concept of
perceived distance would help us to quantify the perceived landscape. It will allow for a
refinement of the conventional ideal models such as region segmentation (e.g., central place
theory and Thiessen polygons) and optimum path analyses as well. By sorting the perceived
distance into different categories, one can generate a series of cost surfaces different in range
and apply them to create the sub-models that are more faithful to the past landscape. In order to
achieve this, the reference to the ethnohistorical and ethnographic records will be essential.
Secondly, the least-cost path may not be unique. Kondo (2007) demonstrates that different GIS
packages provide different least-cost paths. In this study, none of the calculated paths is in
perfect alignment with the preserved/reconstructed road segments. As Kondo (in press) argues,
a corridor as the same route. Thirdly, because the method always attempts to find the least-cost
path even in the area where the slope is trivial, the resulting route may make an unnecessary
detour. Thus, the current model would be more suitable for the study of movement in the
highlands and the interregional interactions between the highlands and the coast (Figure 11).
destinations are very approximate to the preserved road segment in the northern half of Paths 1
and 2, while those in the southern half show significant gaps. Two additional inter-site paths
were also calculated in order to clarify the reason for these gaps: Path 3 between Baños del Inka
and Namora and Path 4 between Ichocan and Cauday (Figure 9). Path 3 follows a totally different
CONCLUSION
path from the one reconstructed by Hyslop (1984) and is more approximate to the modern-day
road connecting Cajamarca and Namora. I suspect that the reconstructed segment may not have
REFERENCES CITED
refinement of the “one-step-forward” model; (2) improvement of data quality; and (3)
adopting the concept of “least-cost corridor,” we may have to consider the different paths within
Zone 3 (Cajamarca-Huamachuco) Like the previous zones described above, the calculated
aspect of terrain as a primary cost of traveling and attempt to quantify the cost in terms of
previous studies indicate (e.g., Kantner 1996), however, other sociopolitical, cultural, and
Figure 9: Inter-Site Paths for Zone 3 (Reproduced from Hyslop 1984:57, fig. 4.1)
used for cargo shipment but is yet to be found along the calculated path.
were calculated by means of the accumulated cost surface that was generated from a specially
accumulated by a spreading function (Figure 1). The resultant cost raster is then called
b. Cost-surface
Figure 7: The Least-Cost Paths for Zone 2 (Reproduced from Hyslop 1984:38, fig. 3.1)
and the Puruhua to the north. It is undeniable that there may have been another road that was
Zone 2 (Lambayeque-Moche) The northward and northward least-cost paths in Zone 2
moving over the cost-of-traveling raster map to each destination, the cost value in each cell is
a. Origin and Destination
Figure 4: The Least-Cost Paths for Zone 1 (Reproduced from Hyslop 1984:20, fig. 2.1)
Through a series of calculations of the most economical routes between given two loci, the
exited and that another path similar to the modern-day road linked Cajamarca and the preserved
least-cost pathway analysis proved to be a useful analytical tool, although it still leaves some
segment near Namora. Likewise, Path 4 does not cross the Crisnejas River as the preserved road
weaknesses to be improved. Once these weaknesses are overcome and its availability is further
does, but rather takes a route similar to the modern-day road.
validated with reference to the preserved road segments, this method will be very powerful not
Figure 11: Inter-Artery Paths Between Zones 2 and 3
Adler, Michael
1994 Population aggregation and the Anasazi social landscape: A view from the Four Corners. In The
Ancient Southwestern Community, edited by W. H. Wills and R. D. Leonard, pp. 85-102. University of
New Mexico Press, Albuquerque.
Amado Gonzales, D. D.
2005 Sistema vial andino en el valle de Cusco. Qhapaq-Ñan del Tawantinsuyu 1:7-23.
Bell, T. and G. Lock
2000 Topographic and Cultural Influences on Walking the Ridgeway in Later Prehistoric Times. In Beyond
the Map: Archaeology and Spatial Technologies, edited by G. Lock, pp. 85-100. IOS Press, Amsterdam.
Conolly, J. and M. Lake
2006 Geographic Information Systems in Archaeology. Manuals in Archaeology. Cambridge University
Press, Cambridge.
Fowler, Andrew P. And John R. Stein
1992 The Anasazi Great House in space, time, and paradigm. In Anasazi Regional Organization and the
Chaco System, edited by D. E. Doyel, pp. 101-122. Maxwell Museum of Anthropology Anthropological
Papers No. 5. University of New Mexico, Albuquerque.
Fresco, A.
2004 Ingañán: La red vial del imperio inca en los Andes ecuatoriales. Banco Central del Equador, Quito.
Hyslop, J.
1984 The Inka Road System. Studies in Archaeology. Academic Press, Orlando.
Kantner, J.
1996 An Evaluation of Chaco Anasazi Roadways. Paper presented at the 61st Annual Meeting of the Society
for American Archaeology, New Orleans, Louisiana.
Kondo, Y.
2007 Rethinking GIS-based Travel Cost Modeling. Paper presented at the the 24th Semiannual Meeting of
Japan Society for Archaeological Information, Keio University, Tokyo, Japan.
in press From Pathways to Corridor: Rethinking GIS-based Travel Cost Modeling. Archaeological
Information 14(1).
Krist, F. J. and D. G. Brown
1994 GIS Modeling of Paleo-Indian period caribou migrations and viewsheds in northern Lower Michigan.
Photogrammetric Engineering and Remote Sensing 60:1129-1137.
Llobera, M.
2000 Understanding Movement: A Pilot Model Towards the Sociology of Movement. In Beyond the Map:
Archaeology and Spatial Technologies, edited by G. Lock, pp. 65-84. IOS Press, Amsterdam.
Tobler, W.
1993 Three presentations on geographical analysis and modeling. Technical Report 93-1, National Center
for Geographic Information and Analysis, University of California.
Ugarte Vega Centeno, D.
2005 Prologo. Qhapaq-Ñan del Tawantinsuyu 1:5.
van Leusen, P. M.
2002 Pattern to Process: Methodological Investigations into the Formation and Interpretation of Spatial
Patterns in Archaeological Landscapes, Unpublished Ph.D. Dissertation, University of Groningen.