Runoff from the World’s Landmasses:
Amounts and Uncertainties at 2-degree Resolution
Trent Climate Note 89-3
J. Graham Cogley
August 1989; revised May 1995
Department of Geography
TRENT UNIVERSITY PETERBOROUGH ONTARIO CANADA
K9J 7B8
Runoff from the World’s Landmasses:
Amounts and Uncertainties at 2-degree Resolution
Trent Climate Note 89-3
J. Graham Cogley
August 1989; revised May 1995
Copyright (C) J.G.Cogley 1989, 1995
Department of Geography
TRENT UNIVERSITY PETERBOROUGH ONTARIO CANADA
K9J 7B8
Email: [email protected]
Telephone: 705-748-1440
Fax: 705-742-2131
Runoff from the World’s Landmasses:
Amounts and Uncertainties at 2-degree Resolution
Trent Climate Note 89-3
J. Graham Cogley
Abstract
A digital dataset has been prepared containing estimates of annual terrestrial runoff of water, runoff
of ice and the error in runoff of water. The spatial resolution is 2◦ × 2◦ . The quantities represented are
defined carefully in the context of the terrestrial water balance equation, and neglected terms are highlighted.
Limitations of the estimates are discussed; they include measurement errors, errors of interpolation (between
drainage basins with measurements), errors of extrapolation (from measured to unmeasured basins), and
mapmaking and mapreading errors. The principal source of information was the Atlas of World Water
Balance, from which estimates were made by eye. To investigate the errors in these visual estimates and
in the cartographic source, two map readers, working independently, read three different maps of the runoff
from northern North America (the Atlas of World Water Balance, and maps in Baumgardner and Reichel
(1975) and the Hydrological Atlas of Canada). Errors due to our mapreading method are about 6 percent,
ranging from about 4 percent in regions of smooth topography to about 10 percent in rugged regions (where
complicated contours make it difficult to estimate areal runoff). Neglecting regions where runoff is close to
or equal to zero, the total uncertainty in any one runoff estimate is given as the root sum square of standard
errors (presumed independent of each other) of measurement, interpolation and mapreading: 12 − 17 percent
in regions with data and 26 − 29 percent in regions of unmeasured runoff. However it is shown that these
estimates of uncertainty are themselves quite uncertain, suggesting that more work is warranted on this
problem.
1
Runoff from the World’s Landmasses:
Amounts and Uncertainties at 2-degree Resolution
Trent Climate Note 89-3
J. Graham Cogley
1. Introduction
Runoff is important in a variety of contexts other than the strictly hydrological, ranging from land
use management and aquatic toxicology at local and regional scales to the appraisal of water resources at
continental scales. This paper is about runoff on the largest scales: it describes a newly-digitized dataset
containing estimates of terrestrial runoff at a resolution of 2◦ × 2◦ , that is, for cells with areas of the order of
104 to 105 km2 . At this scale the drainage basins of big rivers are well resolved, but the main use envisaged
for the data is not in basin-scale hydrology but in global studies. Global hydrology is still a nascent science,
its growth stimulated by such problems as the prediction of climatic changes due to the burning of fossil
fuels, and the refinement of models of the Earth’s gravity field, and by such new instruments as the satellite
sensors which now yield almost routine estimates of the radiance of the Earth-atmosphere system. Much
progress has already been made on these problems, but much remains to be done, and there is no prospect
of runoff being inferred or measured directly from space in the near future. World society will soon face
a number of serious environmental decisions, and if these decisions are to be taken with confidence then
hydrological “ground truth” on the global scale is vital. Better and more easily usable runoff data are a
necessary part of this ground truth.
A recurring difficulty in studies of global change is the absence of information with which to answer
the question “Change from what?” . The solution to this difficulty is to compile systematic, organized and
preferably machine-readable records of the current or recent state of the Earth. Increasingly it is possible
to find such records in the scientific literature, but it is still common for the records to be unaccompanied
by reliable estimates of their uncertainty. The records of mean annual runoff which are described herein are
presented in parallel with estimates of their errors, but it will be shown that these errors are themselves
very poorly quantified. Much more precise error estimates — to say nothing of more precise data — will be
needed before the next question can be answered. The next question is “How much change?”, and it cannot
be answered with confidence when we cannot answer the first question given above.
Most of the primary information in this dataset comes from the Atlas of World Water Balance (Korzun
et al . 1977), which is documented fully by Korzun et al (1978). These two sources are referred to collectively
hereafter as map A. Map A is the most thorough and most reliable compilation of worldwide runoff data
available at present. However it refers to errors in the data only in passing, and to appraise these errors other
sources have been considered along with map A. In particular, map A has been compared with Baumgardner
and Reichel (1975 – map B hereafter) and the Hydrological Atlas of Canada (Min. Supply and Services 1978
– map C hereafter). None of this information appears to be available in digital form other than as presented
herein.
The next section of this paper defines the quantities with which we are dealing. Section 3 describes the
methods by which the runoff data were gathered and organized. Section 4 is about errors in the data and
about the steps taken to estimate and when possible to limit their magnitudes.
2. The Terrestrial Water Balance
The water balance of a vertical column extending from the top of the atmosphere to the bottom of the
groundwater reservoir is (Figure 1)
(Vin − Vout ) + (Rin − Rout ) + (Sin − Sout ) = ∆v + ∆r + ∆s,
(1)
where the uppercase quantities are horizontal fluxes with dimensions of mass per unit time, the lowercase
quantities are masses in storage, and ∆ represents a change of stored mass per unit of time. V , R and S
stand for the atmosphere, the surface (or near-surface layer) and the subsurface respectively. We assume
that fluxes through the top and bottom of the column are zero. Our concern is with the terrestrial water
balance, that is, with the water balance of land surfaces, so we ignore the atmospheric terms V and v in
(1). Exchanges between the atmospheric and surface parts of the column are precipitation, P , which is
2
nonnegative, and evaporation E, which may be either positive or negative. Over the oceans the terms in R,
r, S and s are interrelated in such a way that we can neglect the storage terms ∆r and ∆s for hydrological
purposes; we are interested primarily in how the currents R and S respond to imbalances between P and
E over seasonal and longer time scales. Over land our interests are similar, but the physical mechanisms at
work are quite different. Writing
R = Rout − Rin ,
S = Sout − Sin ,
zero flux
vapour flux
Vin
Δv
Vout
P
evaporation
*
***
* * *
* * * *
Rin
precipitation
E
Δr
Rout
runoff
mixed-layer current
Sout
groundwater flow
ice flow
deep-water current
I
groundwater recharge
glacier accumulation
and ablation
Sin
Δs
zero flux
Figure 1. An interpretation of the water balance, equation (1). Fluxes are defined to be
positive in the directions of the arrows.
for surface water runoff and groundwater runoff respectively, we can recast (1) for a column extending from
the land surface to the bottom of the groundwater reservoir as (Figure 2)
P − E − R − S = ∆r + ∆s = ∆w,
(2)
where ∆w is simply the change in total water storage.
The cross-sectional area of the column, and the time span over which we examine it, can make a big
difference in how we interpret (2). We may be able to assume ∆r = 0 and ∆s = 0 for periods longer than a
3
year, but we certainly cannot do so for periods shorter than a year. The greater the extent of the column,
the more likely is it that S will be negligible. To understand these generalizations we must be more explicit
about the contents of the reservoirs r and s and the exchange between them, I.
P
*
***
* * *
* * * *
E
Δr
R
I
Δs
S
zero flux
Figure 2. An interpretation of the terrestrial water balance, equation (2). Fluxes are
defined to be positive in the directions of the arrows.
The surface water reservoir r includes soil water and water in swamps, ponds, lakes and stream channels,
as well as water stored seasonally in the snowpack. The groundwater reservoir s includes not just groundwater
but also firn and glacier ice. I is groundwater recharge or discharge; in a glaciated column it can be thought
of as the rate of conversion of snow to firn or ice or of ice to meltwater. Whether the column is glaciated
or not, I can be either positive or negative. The depth at which I occurs need not be specified precisely:
for soil, we can take it to be one to a few metres; for ponds and the like, it occurs at the bed of the pond;
in glacier accumulation zones it occurs diffusely over a range of several to many metres, while in glacier
ablation zones it occurs at the surface in summer.
Our purpose here is to estimate climatological averages of R for the year over spatial extents great
enough that we can regard the simplified expression
P − E − R = ∆w
(3)
as a complete description of the water balance, with ∆w ' 0 for the year. In unglaciated regions, I is usually
positive (i.e., directed downwards) in places such as plateaux and hillslopes and negative in valley bottoms
and beneath streams, lakes and marshland. The spatial scale of this physical arrangement is such that in
most parts of the world I approaches zero as the radius of the column increases beyond several kilometres or
4
tens of kilometres. Consulting Figure 2, and noting that I − S = ∆s, we see that where we can assume I = 0
the only reason to expect S 6= 0 is that s, the quantity of stored groundwater, might differ from column to
column. If s were spatially variable, we should expect groundwater to flow down the resulting horizontal
pressure gradients. The timescale of this flow would be millennia or longer, since groundwater flows at rates
of dekametres/year and the distances to be traversed are tens of kilometres or greater. Large-scale flows
of this sort are certainly possible, and we have no way of estimating them directly. If they occur and are
significant they represent an influence exerted on the present hydroclimate by the past climatic states in
which the pressure gradients were set up. We assume that they are not significant.
Glaciers
Over ice sheets we cannot assume S = 0. On the contrary, we can assume that S = I in the accumulation
zones of ice sheets, at least for time spans not exceeding a few decades. In ablation zones we can also assume
S = I, but it is to be understood that S includes mass loss due to the calving of icebergs from floating
ice-sheet margins. We cannot apply (3) in either the accumulation zone or the ablation zone; rather we must
use
P − E − R − I = ∆r.
(4)
Almost all of Antarctica and about two thirds of Greenland are in the “dry snow zone”, where snow never
melts, or in the “percolation zone”, where there is no horizontal flux of liquid water because meltwater
is produced at the surface but refreezes at depth. We can be confident that in these zones R = 0 and
P − E − I ' 0 for periods of a year or longer. The periphery of Greenland and all of the smaller glaciers
elsewhere in the world have R > 0 and I < 0, so that we cannot close the water balance (4) without
independent estimates of all the terms, including the iceberg calving rate. The estimates of R given below
for glaciated regions are estimates of the rate of meltwater runoff.
Subglacial Meltwater
Large glaciers, and smaller glaciers in relatively warm regions, are expected to have
temperatures equal to the melting point over most of their beds, and therefore to yield subglacial meltwater.
Basal melting rates of up to a few mm a−1 can be estimated on theoretical grounds. We assume that if the
glacier is grounded its basal meltwater is a component of I; it constitutes a reduction of s, and it contributes
to R, but there is no way of identifying it explicitly with the methods used in this study. If the glacier is
afloat – that is, if it is an ice shelf – either net melting or net freezing is possible at the base. It is thought
(e.g. Jacobs et al . 1992) that there is probably net basal melting near the outer edges of most ice shelves
and net basal freezing in their interiors, but again we have no way of estimating this vertical flux explicitly,
in either sign or magnitude, and it is assumed to be zero. At a resolution of 2◦ × 2◦ only three ice shelves
– the Ross, Ronne-Filchner and Amery Ice Shelves in Antarctica – are resolvable, and all three are entirely
within the Antarctic dry snow and percolation zones and have R = 0.
Blowing Snow
For the purposes of this section we can think of the runoff R as
R = R0 + B
where R0 is runoff as understood conventionally and B is the horizontal flux of snow entrained from the
surface by the wind. Since B is confined to the bottom few metres of the atmosphere it is convenient to
think of it as a surface flux rather than an atmospheric flux. Considering the roughness of a typical dissected
and vegetated terrain, and the duration and windiness of a typical winter, there is no reason to expect the
flux of windborne snow to be distinguishable from zero at the spatial scales of this study. Antarctica and
Greenland, however, are very atypical, being smooth, cold and extremely windy. Blowing snow certainly
complicates the measurement of precipitation (Bromwich 1988), but it is an open question whether the
wind is responsible for significant long-distance horizontal transport of water substance in these extreme
environments. If so, one would expect the transport to be outwards, from the ice sheet interiors along
katabatic flow lines to sites of coastal deposition or export. Takahashi et al . (1988) deduced a loss of about
100 mm a−1 to the wind blowing over Mizuho station in Queen Maud Land, and constructed a model which
suggested potential divergence in the surrounding region of ± several hundred mm a−1 ; that is, the sign
of B was variable and was locally large, sufficient to explain tracts of bare ice as being due to complete
removal of the snow. However sites of divergence and of convergence were separated by distances of only
a few tens of kilometres. Likewise, but by an entirely independent line of reasoning, Pettre et al . (1986)
estimate the horizontal scale over which the wind redistributes snow in Adélie Land as less than 40 km. A
5
2◦ × 2◦ cell at these latitudes, about −70◦ , has dimensions of 220 km by about 75 km, so we conclude that
the blowing-snow flux B may be neglected because of its high frequency in the spatial domain. Even if it
were not negligible we would be unable to map its distribution, because information comparable to that just
cited is so scattered.
Large Lakes
Large lakes are a special case as far as equation (3) is concerned because their capacity w may be very
large. For 2◦ × 2◦ cells containing parts of large lakes we have recorded only the runoff from the land surfaces
of the cell. Cells contained within large lakes would be treated as oceanic – that is, their terrestrial runoff
would be recorded as zero – but in fact there are no such cells at 2◦ × 2◦ resolution.
Negative Runoff
In enclosed basins such as those of the Aral Sea, Lake Chad and Lake Eyre, the integral of R over
the basin must be zero. If R is positive around the edge of the basin it must be negative in the centre,
which must gain the water which runs off from the edge. The basins in question are all arid or semiarid.
Rivers like the Nile and the Indus, which flow to the sea across arid regions, present an analogous problem.
The water which they bring evaporates from their channels and flood plains, and should be accounted for
if the water balance of the lower reaches of their basins is to be correct. We ignore negative runoff because
it would be time-consuming to allow for it consistently and because the results would be very inaccurate.
Their coefficients of variation would be large because the amounts of negative runoff are absolutely small;
for example, using figures quoted by Korzun et al . (1978) for the discharge of the Nile at Wadi Halfa and
at its mouth, we can calculate a gain of −2 or −3 mm a−1 in each of the six 2◦ × 2◦ cells through which the
Nile flows below Wadi Halfa.
3. Methods
We prepared transparent acetate templates bearing graticules with spacings of 2◦ in latitude and longitude to match the contoured sheets of map A. With the template overlaid on the map sheet, we estimated
by eye the runoff in each 2◦ × 2◦ cell. Where contours were far apart we assumed that the runoff field was
smoothly-varying. Where contours were close together, mainly in rugged mountainous regions, we found it
difficult to integrate over the cell by eye and resorted to summing and averaging a substantial number of
point estimates, usually 25. When the independent work of two map readers is compared (v. infra), we find
that even with this precaution the estimates are least reliable where relief is large.
We tried to estimate the runoff for all the cells of a 2◦ × 2◦ “land mask”. The land mask was prepared
from the 1◦ ×1◦ resolution dataset GGHYDRO described by Cogley (1991). If a 2◦ ×2◦ cell contained at least
5 percent of dry land it was assigned a mask value of 1; otherwise it was assigned a mask value of 0. This
threshold eliminated many small islands, but several larger islands passed the test and were included in the
land mask. Map A does not give runoff estimates for all of these larger islands; for Hawaii we took estimates
from Armstrong (1973), while for Réunion, Mauritius, Kerguelen and Fiji we made our own estimates based
on precipitation and temperature.
Map A gives no runoff information for Antarctica, although runoff is known to occur in some coastal
regions. Budd (1967) and Davis and Nichols (1968) describe meltwater runoff at Mawson and McMurdo
Sound respectively, while Mosley (1988) describes the Onyx River in the Dry Valleys region. The average
discharge of the Onyx River is equivalent over a 13-year record to about 3 mm a−1 of runoff, but it drains
to the enclosed Lake Vanda, and thus its net export of water is nil. We have assumed R = 0 throughout
Antarctica, except for those coastal cells where its value might be positive. Such cells were identified by
examining weather station records (Schwerdtfeger 1984) and extrapolating along the coastline from those
stations having at least one monthly mean temperature above −1◦ C. The runoff recorded for these cells, 1
mm a−1 , is entirely arbitrary.
For Greenland, map A shows (R + S), not R. To separate the flux of solid ice from the runoff would
require reliable information on the ice balance, I, which is not available. The ablation and accumulation
zones of the ice sheet can, however, be delineated reasonably confidently, as for example in Figure 101 of
Korzun et al . (1978). In the accumulation zone we take R = 0, which is a fairly accurate assumption since
only a narrow band around the edge of the accumulation zone exports any water. We also assume that S
can be approximated by I, which we take from Reeh’s map (1984) of accumulation.
In the ablation zone I is negative because of the melting of glacier ice. The only study which gives a useful
estimate of I for our purposes is that of Braithwaite and Olesen (1988), who present runoff estimates, based
6
on 8 − 10 years of measurements, for adjacent glaciated (R = 1100 mm a−1 ) and unglaciated (R = 690 mm
a−1 ) basins in the extreme south of Greenland. If the unglaciated basin is representative of the unglaciated
part (61 percent) of the glaciated basin, then Braithwaite and Olesen’s results imply I = −1060 mm a−1
averaged over the ablation zone. The magnitude of I will be less at higher latitudes, although even in
northernmost Greenland it will not be zero; we assume that I reaches about −400 mm a−1 in the northern
ablation zone of the Greenland Ice Sheet.
Over the Greenland Ice Sheet as a whole, our dataset yields R = 279 km3 a−1 and S ' I = 191 km3
−1
a , figures which are consistent with other published estimates. Clearly, however, they are only rough
approximations derived from sparse data, and should be used as such. In particular, climate models ought
not to be judged against such estimates.
The quantity appearing on map A for Spitsbergen, Franz Josef Land, northern Novaya Zemlya and
Severnaya Zemlya is also (R + S). We replaced the mapped quantity with information from Table 135 of
Korzun et al . (1978).
The runoff estimate for each cell was recorded in kg m−2 a−1 , which is equivalent to mm a−1 since the
density of the water is 1000 kg m−3 . The estimates are archived as integers in a file in which the numeral ‘0’
implies “no land” while the numeral ‘1’ implies “runoff equal to or indistinguishable from zero”. The file will
be supplied upon request in a run-length-encoded format. A similar format is used for the accompanying
error estimates and ice fluxes.
Ice Fluxes
So as to specify the horizontal fluxes completely, a separate file was prepared containing estimates of
the ice flux S, which in Antarctica was assumed to be equal to the accumulation as mapped by Giovinetti
and Bentley (1985) and in Greenland was estimated as described above. Thus information on S is presented
in a form analogous to that known in glaciology as the “balance flux” – that horizontal flux which would
occur if the column of ice had ∆s = 0.
Since we assume ∆s = 0 for every cell we are assuming in effect that the ice sheet as a whole is in a steady
state. Many workers have tried to estimate mass balances for greater or lesser portions of the Greenland
and Antarctic Ice Sheets, but none have been able to claim great accuracy. Among recent studies, Kostecka
and Whillans (1988) found 54 ± 72 mm a−1 in the accumulation zone of Greenland at 65◦ N and 0 ± 63 mm
a−1 at 69◦ -70◦ N. Allison (1979) reports 55 mm a−1 for the catchment of the Lambert Glacier in Antarctica,
disagreeing (Allison et al . 1985) with McIntyre’s claim (1985) that 36 mm a−1 is a more accurate estimate;
neither is able to compute formal estimates of uncertainty, but if Allison’s suggested uncertainty of about
30 percent is correct then these two estimates are indistinguishable. Hamley et al . (1985) suggested a mass
balance close to zero for the 90◦ -135◦ E sector of Antarctica. The least unreliable estimates of the mass
balance of parts of an ice sheet are those of Shabtaie et al . (1988), who studied the West Antarctic ice
streams feeding the Ross Ice Shelf. Of the six ice streams, five have ∆s < 0 and one has ∆s > 0; ∆s is
−24 ± 19 mm a−1 (of water) for the six considered together. For Ice Stream B, Whillans and Bindschadler
(1988) give a more detailed estimate of ∆s = −53 ± 38 mm a−1 , to be compared with Shabtaie et al .’s
−113 ± 22 mm a−1 .
Evidently it is premature to make much use of such rapidly-varying estimates. Repeated radar altimetry
from satellites is likely to reduce the uncertainty in them substantially, but at present there is no reason to
question the assumption that the Greenland and Antarctic Ice Sheets are each in a steady state as a whole.
Regional imbalances of the order of tens of mm a−1 are probable, but we are unable to map them.
Methods of Map A
Except as noted above for some islands and for Antarctica and Greenland, we did not modify the
estimates of map A in any way. Therefore the descriptions of how the map was prepared (Korzun et
al . 1978) may be consulted to appraise the mapmaking procedures. Map A represents a large amount of
careful work, done mainly by hand and relying little on computers. It is based on source maps in national
and regional atlases, with corrections made by reference to runoff data from long-established measurement
stations and adjustments made where adjoining source maps disagreed. These corrections and adjustments
appear to have been both thoughtful and widespread. Over about 30 percent of the landmasses, where
either source maps or data were unavailable, the mapmakers extrapolated from neighbouring regions, using
empirical relationships between runoff and potential evaporation (or between runoff and precipitation in
some humid equatorial regions). In northern Canada, the Arctic islands of the U.S.S.R., and parts of the
Amazon basin, runoff was estimated as precipitation minus evaporation.
7
The discharge records on which map A is based are of variable epoch and duration. The mapmakers
used the longest available discharge records as checks on the runoff derived from their source maps. They
also appear to have used long records to check the representativeness of shorter records. However map A
does not represent a well-defined interval of time.
4. Errors and Quality Control
Measurement Errors
Estimates of runoff usually derive from records of water level or “stage” at a selected stream cross-section
and a knowledge of the area tributary to that section. The volume of water flowing through the cross-section
is inferred from a “rating curve”, which is generated by measuring the discharge directly at as many different
water levels as possible. Most discharge measurements consist of simultaneous measurements of velocity and
cross-sectional area. Dickinson (1967) has investigated in some detail the errors to be expected in estimates
of runoff. The weakest points in the procedure are probably the direct measurement of discharge and the
use of the rating curve.
It is often difficult and even hazardous to sample adequately the variation of velocity and depth across
a stream cross-section. Dickinson deduces measurement errors of only 5 percent in the best circumstances;
however he found errors of 10 − 12 percent on the average over a large number of discharge measurements in
Colorado. These figures are deviations of measurements from estimates made from rating curves; Dickinson
argues plausibly that the deviations arise mostly from measurement error.
Another way to appraise measurement error is to make replicate measurements. Table I summarizes
discharge estimates made simultaneously by six groups of students on a 30-m reach of a small stream near
Peterborough, Ontario. The students had never measured discharge before, and velocities were slow enough
to raise questions about inconsistent performance of the velocity meters. On the other hand the flow was
steady and working conditions were good. The result of this experiment was a standard error for any one
discharge measurement of about 12 percent.
Table I
Replicate Measurements of Discharge
Dam Creek, 1100h-1130h, 16 October 1986
Distance downstream
(m)
Discharge, Q
3
(m s
−1
Statistics
)
5
0.162
n=5
10
0.143
Q̄ = 0.172
15
0.174
s(Q) = 0.021
20
0.199
σ(Q) = 0.0094
25
(0.072)*
100 c(Q) = 12.2
30
0.181
* Outlier : discard
s is the standard deviation (the estimated standard error of any single measurement), σ is the standard error
of the mean Q̄, and c = s/Q̄.
At a stable cross-section, unaffected by continuing scour and fill, the rating curve commonly explains a
large part of the variance in the relationship between discharge and stage. The hydraulic principles of flow
in channels lead us to expect such a relationship, and a rating curve which is statistically powerful will give
a more accurate estimate of discharge than will any single measurement. The trouble is that while many
streams have stable cross-sections, many do not. It is common, for example, to find sections which are stable
most of the time except when their rating curves are altered by floods, and also sections whose rating curves
vary periodically – usually with the time of year: rating curves are especially uncertain when ice covers are
forming or breaking up. Much may therefore depend on how much attention can be given to keeping the
rating curve accurate and relevant. Dickinson (1967) shows that when a rating curve is used to estimate a
8
series of discharges the errors in the members of the series are perfectly correlated; he also shows that the
error is proportional to the discharge. This means that the error in the sum of the series is the same as
the error in any one member. Thus, assuming that the measurement cross-section is closely monitored, an
estimate of monthly or annual discharge is subject to an uncertainty of the order of the 1σ uncertainty on
the rating curve, which for Dickinson’s dataset is about 2 percent. The more statistically powerful the rating
curve the less the error, and the more unstable or the less well-attended the cross-section the greater the
error. We suspect that many hydrographic agencies are unable to monitor their measurement cross-sections
as closely as is needed, and that in practice the error in typical estimates of annual discharge is significantly
greater than 2 percent. We adopt 10 percent, the typical discharge measurement error, as a working figure
for errors in annual discharge. It is very probable that the actual error is less.
Errors of Interpolation
The uncertainty in the area tributary to a discharge measurement cross-section is usually insignificant.
However to convert a number of point estimates of discharge to a comparable or greater number of areal
estimates of runoff it is necessary to interpolate, and the interpolation will introduce additional uncertainty.
Extrapolation to places where there are no discharge measurements will be yet more uncertain. Korzun et al .
(1978, p.143) state that for regions where runoff can be estimated from data rather than by extrapolation “the
error of total runoff determined from the map does not exceed 10 percent compared with runoff computed
from actual observations”. Something may have been lost in the translation, but this appears to be an
estimate of the error of interpolation. If so, and if we assume as seems reasonable that the errors of discharge
estimation and of interpolation are uncorrelated, we can add their squares and take the root of their sum;
the result, about 14 percent, is the coefficient of variation of those estimates of runoff on map A which are
not obtained by extrapolation. If the figure quoted by the mapmakers is a range rather than a standard error
then we are again being overcautious, and 14 percent is too pessimistic. Later we shall refine this estimate,
reducing it somewhat and allowing for the influence of topographic roughness on the mapmaking process.
It is much easier to acknowledge the uncertainty of extrapolation than to put a figure on it. In the next
subsection we describe a procedure for quality control and suggest that it may yield at least an informal
estimate of these “worst case” errors.
Quality Control
Errors in mapped estimates of runoff arise when the mapmaker assesses, either subjectively or objectively, the variability of runoff within the area tributary to a measurement station, or the difference in runoff
between a measured catchment and neighbouring unmeasured catchments. To compile all the measurements
and interpolate objectively between them would be a substantial task – one which was not done for map A,
but which would offer a means of appraising the errors of interpolation at the same time as the quantities
being interpolated. However we can quantify the errors in subjective estimates of runoff, at least in a rough
way, by comparing the work of different mapmakers. In an attempt to make useful estimates of the uncertainties in our global runoff dataset, we have compared runoff maps of northern North America from three
different sources: map A, which is the origin of the data in the global dataset, map B (Baumgardner and
Reichel 1975), and map C, the Hydrological Atlas of Canada (Min. Supply Services 1978).
Map B is based on data compiled by Marcinek (1964) and UNESCO (1969). However the mapmakers
appear, from a description which is obscure both in translation and in the original German, not to have
mapped these data. Rather they have plotted runoff as the residual of the water balance equation. Only their
precipitation and evaporation maps represent independent information and their runoff maps simply show
the difference between the other two. Thus map B is probably less reliable than map A, and in particular
we expect it to yield underestimates of runoff because the corrections applied for the known tendency of rain
gauges to catch too little precipitation are decidedly too small.
Map C is one of the sources from which information for map A was taken, so that A and C are not
independent. However, A was based on a preliminary (1969) version of map C, and inspection shows that
the two are not particularly similar: C is a more generalized map, and in A some trouble has been taken to
refine details, especially in remote mountainous regions. Map C was compiled using “actual and synthesized
data” from a number of sources, including consultants’ reports. The aim was to represent the annual runoff
“at its place of origin . . . where it first collects in stream channels as streamflow”.
9
Map-reading Error
There is extra uncertainty in the dataset due to our having converted the map data
to numerical form by eye. We try to quantify these errors by comparing the estimates of two map readers
who worked independently to produce duplicate digital versions of the three maps. The readers are referred
to as reader 1 and reader 2. Reader 1 produced the entire global dataset; reader 2 worked only on northern
North America. The extent of this test domain is shown in Figure 3.
70
70
8
-24
-16
40
40
Figure 3. Test domain for quality control experiments. The dashed lines enclose the
restricted domain of map C, the Hydrological Atlas of Canada.
Denote a runoff estimate from map i by reader j as rij (λ, θ), where λ is longitude and θ is latitude.
Usually we shall not mention explicitly the dependence on position. Thus rB1 denotes the runoff estimated
from map B by reader 1 at some unspecified longitude and latitude. The average of the estimates from map
i of the two readers is
ri = (ri1 + ri2 )/2.
(5)
We assume that ri1 and ri2 are samples from the work of an infinite population of independent map readers.
The standard error of the estimate of ri is the standard deviation of the sample divided by the square root
of 2, the number of elements in the sample:
√
σi = si / 2 = |ri1 − ri2 |/2 = |δi |/2.
(6)
Thus the standard error is simply one half the absolute value of the difference δi between the two readings.
If the average difference is zero then it is reasonable to interpret σi as the standard error not just of ri but
of its component ri1 ; we shall find that this is indeed so. It is more convenient to work with the coefficient
of variation
ci = σi /ri ,
(7)
which is the absolute value of the difference divided by the sum.
With such a small sample it is wise to be cautious in interpreting this fractional uncertainty: the
standard error of the estimate of ci is
γi
=
s
= ci
[σ(σi
r
)]2
∂ci
∂σi
2
1
+ c2i
2
+ [σ(ri
)]2
∂ci
∂ri
2
(8)
√
where σ(σi ) ≡ σi / 2 is the standard error of the standard error and σ(ri ) ≡ σi is the standard error of the
mean.
We interpret the mean
1 XX
Ci =
ci (λ, θ),
(9)
Ni
λ
θ
10
where Ni is the number of coefficients of variation obtained from map i, as a measure of “map-reading error”
– the random error to be expected when map data are converted by eye to digital form. The mean
Γi =
1 XX
γi (λ, θ),
Ni
λ
(10)
θ
is interpreted as a measure of our uncertainty about Ci . That is, we think of the map-reading error as
probably lying between (Ci − Γi ) and (Ci + Γi ).
Figure 4 shows the frequency distribution of δA /2 rA , which, except that we retain the sign of the
difference (cf. eq. 6), is the same as the coefficient of variation betweeen map readers working on map A.
Note that it is zero on average. The distributions for maps B and C are very similar in shape and are not
shown.
%
50
40
30
20
10
0
-80
-60
-40
-20
0
20
40
60
80
signed cv(r1 vs r2) (%)
Figure 4. Variation in performance of two map readers working on map A. The abscissa
is the signed coefficient of variation (i.e. the difference between readings divided by their
sum) of spatially-coinciding runoff estimates by the two map readers. The number of
readings is 650.
Table II summarizes the results for the three maps. The simplest way to summarize Table II is to say
that the exercise of analogue-to-digital conversion has introduced additional errors of the order of 2 − 10
percent beyond those on map A. With the exception of a topographic effect to be discussed below, there is
no reason to expect these errors to be correlated with the measurement and interpolation errors discussed
in the last section. This being so, adding CA in quadrature to the earlier errors increases the coefficient of
variation of a single runoff estimate from 14 percent to about 16 percent. Recall, however, that this applies
only to estimates in areas where extrapolation was not necessary.
Table II
Random Error in Map-to-Digital Conversion
Map
i
Ci − Γi
(%)
Ci
(%)
Ci + Γi
(%)
Ni
A
B
C
1.5
2.4
2.3
6.0
9.7
8.7
10.5
17.0
15.1
650
647
371
11
The larger errors on maps B and C may be attributed to their more generalized contours, which make
digital estimates quite arbitrary over wide areas.
Mapmakers’ Errors
For the analysis of differences in runoff by different mapmakers we consider the
maps in pairs, comparing the work of each map reader on map A with the work of the same reader on maps
B and C. By analogy with (5) to (10) we obtain from the raw quantities
pB1 = (rA1 + rB1 )/2,
σB1 = |rA1 − rB1 |/2,
(11)
(12)
the summary statistics
CB1 =
1 XX
cB1 (λ, θ)
NB1
(13)
1 XX
γB1 (λ, θ),
NB1
(14)
λ
and
ΓB1 =
θ
λ
θ
with three pairs of analogous expressions obtained by substituting subscripts C for B and 2 for 1.
Table III shows the results of the comparison of maps. At first sight the worst-case errors (Ci1 + Γi1 )
are rather depressing: 30 percent is bad enough, but if the coefficient of variation were really as large as 50
percent we would have difficulty in saying anything useful about the world water balance. But the picture
is not as black as this. Note that CCj is less than CBj for both j = 1 and j = 2; this suggests systematic
differences between maps A and C on one hand and map B on the other. Figures 5a and 5b show that B is
indeed biased with respect to A while C is not. We interpret these observations as follows. Map A depends
indirectly on map C; thus CCj ' 15 − 17 percent is a measure of the disagreement which may arise between
hydrographers working from similar raw material to estimate runoff. Map B is largely independent of both
A and C, and CBj ' 23 − 27 percent is a measure of the disagreement between two widely-used sources of
global runoff information, one of which – map B – systematically underestimates runoff.
Table III
Random Error in Runoff Estimates by Different Mapmakers
Reader 1
Map Pair
A:i
Ci1 − Γi1
(%)
Ci1
(%)
Ci1 + Γi1
(%)
Ni1
A:B
A:C
4.4
3.6
26.9
16.7
49.4
29.8
647
354
Map Pair
A:i
Ci2 − Γi2
(%)
Ci2
(%)
Ci2 + Γi2
(%)
Ni2
A:B
A:C
4.5
3.5
23.3
15.5
42.4
27.5
685
371
Reader 2
The underestimate is believed to arise because the runoff on map B is a residual of the water balance
equation and the precipitation data which were used have not been corrected sufficiently for undercatch. The
makers of map A relied on the water balance equation only in remote areas, and at least within the southern
part of our test domain the runoff on map A is uncontaminated by errors in precipitation. It is beyond our
scope here to advance arguments to support the assertion that map B underestimates precipitation, but this
seems the least unlikely explanation of its bias with respect to maps A and C.
12
%
20
a
10
-80
-60
-40
-20
0
20
40
60
80
60
80
signed cv(A vs B) (%)
%
20
b
10
-80
-60
-40
-20
0
20
40
signed cv(A vs C) (%)
Figure 5. Variation in performance of reader 1 working on a) maps A and B, b) maps
A and C. The variation is a measure of differences between maps, not of the performance
of the map reader. The abscissa is the signed coefficient of variation (i.e. the difference
between readings divided by their sum) of spatially-coinciding runoff estimates from the
two maps.
The worst-case errors of 30 − 50 percent in Table III are exactly as plausible as the best-case errors of
3 − 5 percent. That we cannot make a firm estimate of the uncertainty in the data, because the uncertainty
in the uncertainty is of the same order of magnitude, simply emphasizes how much work remains to be done.
If the interpretation offered above is correct, then CC1 = 16.7 percent is an informal estimate of the
coefficient of variation of our estimates of runoff. It is fortuitously close to the 16.3 percent derived more
formally above (the root sum square of measurement, interpolation and mapreading errors of 10, 10 and 6
percent respectively), but the coincidence encourages us to suggest that 15 − 20 percent is a good working
estimate of the error of our runoff estimates where they are based on discharge neasurements.
13
Coefficient of variation (%)
Errors of Extrapolation
In extrapolating to remote unmeasured regions the makers of map A used procedures not unlike those
used in the preparation of map B, namely the application of conservation principles in conjunction with
measurements or estimates of components of the water balance other than runoff. It is reasonable, therefore,
to suggest that CBj ' 20 − 30 percent is not very far from an estimate of the total error, including errors of
extrapolation where applicable. However we cannot exclude the possibility that the actual errors might be
even worse.
These regions occupy about 34 percent of the world’s land surface: 9 percent in the dry snow zones
of Antarctica and Greenland, where total error in estimates of runoff is zero because the runoff itself can
confidently be assumed to be zero; 2 percent in the hyperarid desert interiors where the error is also zero;
and 23 percent elsewhere. We assign a “measurement error” of 25 percent to this 23 percent, acknowledging
that it is only a nominal figure.
20
a
16
12
8
4
0
0
32
64
128
256
512
1024
2048
4096
2048
4096
Elevation range (m)
Number of readings
200
b
160
120
80
40
0
0
32
64
128
256
512
1024
Elevation range (m)
Figure 6. a) Average coefficient of variation between map readers working on map A,
plotted as a function of the elevation range of the cell being read. The thin lines are error
bounds on the estimates of CA (shown as dots). The thick line is the function adopted
to represent map-reading error. b) Number of readings in each elevation-range category.
The boundaries of the categories are made logarithmic in an attempt to achieve more
uniformly robust estimates of CA (η).
14
Topographic Influences
In an effort to refine error estimates for particular cells we examined the dependence of the coefficient
of variation for map A, cA , on topography within the test domain. Two measures of topography were
considered: the average elevation of each 2◦ × 2◦ cell, and the range of elevation within the cell. The terrain
heights were taken from the 1◦ × 1◦ dataset GGTOPO used by Cogley (1984). The standard errors of these
heights are unquantified, and so the runoff cells were categorized only roughly by elevation and elevation
range. Similar results were obtained for the two topographic measures, but we regard the elevation range
as the more useful of the two because the influence being investigated is that of terrain roughness on mapreading (and probably also map-making) errors. It is not particularly difficult to read contours over high
plateaux, where runoff tends to vary slowly in space, but rugged mountain ranges tend to have complex and
hard-to-read runoff fields even when their average elevation is not very high.
Writing η for the elevation range, and recalling eqs. (7) to (10), we now have cA ≡ cA (λ, θ, η) and
γA ≡ γA (λ, θ, η). The means CA and ΓA are also functions of η. They are shown in Figure 6, from which we
see that CA is probably 1 − 7 percent in very smooth terrain and probably 2 − 18 percent in the roughest
terrain; the most likely values are about 4 percent in smooth terrain and about 10 percent in rough terrain.
The envelope C ± Γ is wide enough to permit the assumption C(η) = const, but we choose to regard the
influence of topography as being real. Korzun et al . (1978) do not describe their errors of interpolation in
detail, but they appear to have taken about as much trouble in dealing with the influence of topographic
roughness as we did. Therefore we regard this influence as affecting the work of the mapmakers as well as
our own map readings; we assume that the error of interpolation and the map-reading error are equal to
each other and are given by the following prescription:
Range of Elevation
Error of Interpolation
Map-reading Error
(m)
(%)
(%)
< 100
4
4
100 to 500
6
6
> 500
10
10
Low Runoff
The fractional errors deduced above are useful only for moderately large values of runoff. In places
where the runoff is less than about 50 mm a−1 the errors are greater. In all such regions it is true either
that the runoff is flashy and it is difficult to obtain reliable measurements in either the long or the short
term, or that the region is remote and poorly-monitored. To assign more reasonable coefficients of variation
where the runoff is low, the absolute error is prescribed to be not less than 5 mm a−1 . Thus the coefficient
of variation reaches 100 percent for R = 5 mm a−1 and 250 percent for R = 2 mm a−1 .
In the hyperarid interiors of deserts, however, it is reasonable to assume not just that the runoff is zero
but also that its error is zero. To define “hyperarid” we have turned once more to GGHYDRO (Cogley
1991); cells with less than 2 mm a−1 of runoff are assigned a coefficient of variation of zero if they have
no perennial or intermittent streams or lakes, and of 500 percent otherwise. (Recall that negative runoff is
possible in such cells.)
The accumulation zones of Greenland and Antarctica have zero runoff and are assigned errors of zero.
Summary
Errors in the recording of discharge, and therefore of runoff, can be as low as 2 percent but are probably
higher. We adopt 10 percent as an estimate of measurement error, believing it to be on the cautious (i.e.,
pessimistic) side. The compilers of our source map give 10 percent as their estimate of the coefficient of
variation, or possibly the error range, of their interpolated information; going by experience in the reading
of the maps, we judge that the error of interpolation is probably greatest in rugged terrain, and therefore
we set this error equal to the map-reading error. Map-reading error, from a detailed analysis, is probably 6
percent, but this estimate is quite uncertain, and it varies from 4 to 10 percent depending on topographic
roughness. Low runoff is assigned greater uncertainty by specifying a minimum absolute error of 5 mm a−1 ,
although zero runoff is assigned zero uncertainty. Low or zero runoff aside, the total uncertainty in any one
runoff estimate in our dataset is the root sum square of measurement, interpolation and map-reading errors:
12 − 17 percent in regions with data and 26 − 29 percent in regions of unmeasured runoff.
Acknowledgements
I thank Stacey Bonyai for her able assistance and NSERC (Canada) for financial support.
15
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