The 3-D view of Budyko's framework and its 2-D projections
Figure shows our 3-D space (Φ-Ω-Ψ) within Budyko's
framework for the FAO agro-climatic stations (Fig. , left panel)
and catchments in the US and China (Fig. , right panel). Red dotes
represent observed data with E calculated with Turc's equation or the water
balance equation, respectively. In both cases, despite the scatter observed
in the plots, it can be seen that there is a surface that captures the data
sets within the proposed 3-D parameter space, which was obtained by
estimating E using Budyko's Eq. (). Grey lines on the faces of the
“cubes” represent the 2-D projections of the surface. Taking into
consideration how the three dimensionless variables were defined and that
they are not independent of each other, the equation of the surface
(Fig. ) can be easily obtained as Ψ=ΦΩ.
Figure also shows that red dots are limited to specific parts of
the surface. This means that although this surface comes from a valid
mathematical equation it does not necessarily mean that all parts of the
surface are physically feasible in nature. For example, there are no
environments with low Ω and low Φ at the same time or
environments with low Ψ and high Φ simultaneously. This leads us to
further explore the bi-dimensional projections of our 3-D space.
Bi-dimensional projections of the 3-D space for the FAO agro-climatic stations (a–c) and for
catchments in the US and China (d–f).
Figure shows the three bi-dimensional projections of the proposed
3-D space: Ψ vs. Φ, Ψ vs. Ω and Φ vs. Ω.
Blue dots denote actual data for the FAO agro-climatic stations
(Fig. a–c) and for the US–China catchments
(Fig. d–f). Thick black lines and dashed black lines represent
the Budyko curves (and their corresponding projections on the bi-dimensional
spaces of our 3-D approach) using Eqs. () and () with
n=2, respectively.
Figure a and d present the relationship Ψ vs. Φ
(traditional Budyko approach). On these panels, the Budyko curve has two
physically consistent limits denoting energy-limited and water-limited
evapotranspiration (Eq. ). Despite the observed scatter, which could
be explained by other factors affecting E such as soils, vegetation and
topography, amongst others, , data from agro-climatic stations
(Fig. a) and from catchments (Fig. d) follow the Budyko
curve. Figure b and e, illustrate the relationship Ψ vs.
Ω, which also has two physically consistent limits. The first limit is
a vertical line at Ω=1, because physically E can never exceed
Ep. Thus, Ω=1 corresponds to very humid environments
where precipitation is large, there is no water limitation and
E=Ep. The second physical limit in Fig. b and e is
a horizontal line at Ψ=1, where E=P, since on the long-term timescale,
E should not exceed P. In summary, in arid regions Ψ is large (with
a maximum value at Ψ=1) and Ω is small, whereas in humid regions
Ω is large (with a maximum value at Ω=1) and Ψ is low.
Figure c and f show the relationship between the remaining
dimensionless variables, Φ and Ω, which were related previously
using Eq. () and for which the physical inconsistency was found.
This relationship has one physical limit at Ω=1, for very humid
regions where E=Ep. It was demonstrated mathematically that
Eq. () requires that for Ω→1, Φ=0, as
revealed by both theoretical black curves in Fig. c and f.
However, actual data never reach zero at Ω=1, confirming what was
evidenced in Sect. 2.1.2. Given this physical inconsistency, found
particularly in wet environments, we propose a new way to address the Budyko
hypothesis in humid regions such as the Amazon River basin, as explained
next.
A scaling approach to Budyko's framework in humid environments
For many years power laws have been popular in the geosciences, mainly
because of their simplicity, unique mathematical properties and because of
the surprisingly physical mechanisms they represent . Many
hydrological, climatic, ecological processes, among others, exhibit emergent
patterns that manifest as power laws
, which
reveal certain types of universalities emerging from the complexity of nature
. By fitting empirical relationships using power laws, one
assumes that the system is essentially self-similar or fractal
, which suggests that the main characteristics of the
system exhibit an invariant organization that remains the same over a wide
range of scales.
With the purpose of overcoming the physical inconsistency of Budyko's
framework for the relationship between Φ vs. Ω in humid
environments, we study the proposed 3-D Budyko space (and its bi-dimensional
projections) using data from 146 catchments in the Amazon River basin, with
values of Φ ranging from 0.43 to 1.55. Considering the behavior of the
data and bearing in mind all the physical limits of the Budyko hypothesis, we
suggest the following power law to represent the Budyko curve for the studied
catchments:
Ψ=kΦe.
Using a nonlinear least squares regression algorithm with confidence bounds
set at a 95 % confidence level, the coefficient and scaling exponent were
estimated as k=0.66 and e=0.83 (R2=0.93), respectively. Since our
interest is to capture the behavior of the data in humid environments in the
best way possible, we compared the performance of Eq. () with two
other approaches. First, we used Eq. () to model
the data in the Amazon River basin. However, instead of assuming n=2, we
used the same nonlinear least squares regression algorithm and obtained the
best value of n for this data set, which turned out to be n=1.58
(R2=0.85). Then, we followed the study by , who
justified the use of linear relationships to address the Budyko hypothesis.
For this approach we fitted the data to a linear relationship Ψ=aΦ+b,
with a=0.55 and b=0.11 (R2=0.91). However, it is worth remarking
that only assessed the inter-annual variability of the
water balance rather than long-term mean values.
Bi-dimensional projections of the 3-D space for the 146 Amazon River sub-catchments.
Our next step was to test the 3-D generalization of Budyko's framework in the
Amazon River basin, focusing on its three bi-dimensional projections
(Fig. ), with emphasis on the relationship between Φ and
Ω, which exhibited the physical inconsistency.
Figure a shows the traditional Budyko curve for the 146
sub-catchments within the Amazon River basin (blue dots) and the results of
the parameters obtained with Eqs. (), () and the linear
relationship. Figure b and c show the remaining bi-dimensional
projections of our 3-D space with actual data and the theoretical curves that
come from the previously mentioned equations. From Fig. a–c, and
taking into account the goodness of fit, it can be seen how both the power
law and the linear relationship are better suited to model the data in these
humid catchments. Also, both approaches theoretically overcome the physical
inconsistency found for the relationship between Φ and Ω
(Fig. c), since unlike the thick black line ,
the dashed black line and the thick red line (Power law)
do not approach zero as Ω increases. Nevertheless, the linear
relationship does not fully comply with the energy limit in the Budyko curve,
as can be seen in Fig. a. In fact, it can be seen that in order to
fulfill the energy limit, its intercept would have to be restricted to b=0.
For these reasons, we conclude that the power law is the best equation among
the three of them to capture the long-term mean coupled water–energy
balances in Amazonia.
One of the reasons that explains why both the power law and the linear
relationship work better than traditional Budyko-type equations, which was
first pointed out by , is that these relationships have
two parameters (k, e and a, b), while Eq. () has only one
(n). Particularly in this case, since we are dealing with humid
environments, it also has to do with the way these catchments partition water
and energy, in the sense that for most of these catchments in Amazonia
evapotranspiration is energy-limited rather than water-limited. This
observation locates the Amazonian catchments closer to the energy limit in
the Budyko curve (grey dashed line in Fig. a), along its “linear
1:1” part, and this makes data suitable for a power law.
With these results in mind, several questions arise: how do the parameters
change from catchment to catchment? Are the values of these parameters in the
long term the same as at the inter-annual scale? Can any of these parameters
be explained by landscape features or catchment properties? In order to
answer these questions our next goal is to study the inter-annual variability
of the coupled water and energy balances in Amazonia. By inter-annual
variability we mean the year-to-year variations within each one of the 146
sub-catchments, as explained in the following section.
Assessing inter-annual variability of the coupled water and
energy balances in the Amazon River basin
With 27 years of information for each of the 146 sub-catchments in Amazonia,
the inter-annual variability of the water balance is studied. Once more the
three approaches that were tested before are compared. For this purpose,
Eqs. (), () and the linear relationship are fitted to the
data, and thus the parameters k, e, a, b and n are obtained for
each catchment. The first question that we will try to answer is whether the
values of these parameters in the long term are the same as at the
inter-annual scale in order to search for signs of space–time symmetry of
the coupled water and energy balances
. Later on, signs of
catchment co-evolution will be explored by linking the parameters with
characteristic landscape features within the Amazon River basin.
(a) Between-catchment variability of the long-term mean Budyko curve and (b) between-year
variability of the Budyko curve. Colors denote different sub-catchments within Amazonia. The insets show the results
of fitting Eqs. () and () and the linear relationship, with their respective parameters.
Space–time symmetry
In hydrology, the term space–time symmetry has been adopted when an equation
or model can be used to depict between-catchment variability of long-term
mean annual water balances and the corresponding between-year variability
within individual catchments, thus implying ergodicity of the hydrological
system at the catchment scale
. Figure
presents a comparison of the long-term mean Budyko curve (Fig. a)
and the inter-annual Budyko curve for the 146 catchments in Amazonia
(Fig. b), as well as the results of the implementation of
Eqs. (), () and the linear relationship. Each triangle
represents one catchment (Fig. a) and each dot represents 1 year
of the 27 years of data of each catchment (Fig. b). It should be
pointed out that Figs. a and a are essentially the
same; however, in Fig. a catchments can be distinguished from each
other by different colors. In the previous section we described how we used
a nonlinear least squares regression algorithm to obtain the parameters for
the three equations that are shown in Fig. a. Now, for the case of
Fig. b, the procedure is the same, although all catchments with
their 27 years of data were taken together as if they were a single data set.
With confidence bounds set a 95 % confidence level, the coefficient and
scaling exponent of the power law, the slope and intercept of the linear
relationship and the parameter from Eq. () were estimated as
k=0.67, e=0.87, a=0.59, b=0.08 and n=1.64, respectively. These
results appear to reveal that there is indeed space–time symmetry within
these 146 catchments in the Amazon River basin, especially for the power law
equation, given that the values of the parameters k and e do not seem to
change much.
Moreover, for the case of the power law and the linear relationship, the
scatter present in the inter-annual variability of the water balance does not
seem to affect the goodness of fit, and actually both coefficients of
determination increase (R2=0.94 and R2=0.92, respectively);
nevertheless, bearing in mind the physical limits of the Budyko framework
that have been discussed throughout this paper, it can be seen once more how
the linear relationship does not fully satisfy the energy limit at the
inter-annual scale (Fig. b). The increase in the goodness of fit
for both equations could be possibly attributed to an increase in data points
and also because in Amazonia, the scatter in the inter-annual variability is
evidenced somehow parallel to the energy limit, that is, in the direction of
the power law and linear relationships (Fig. b). In contrast, the
scatter present in the year-to-year variations does affect the performance of
Eq. (), as reflected in a decrease of R2. In fact, in Amazonia,
the inter-annual variability of the water balance diverges from the
traditional Budyko curve as Φ increases (Fig. b). This is
because while in arid environments moisture available for E comes mostly
from P and E/P→1 as Ep/P increases, in humid
environments there can be other sources of moisture besides P, such as
water stored in soils and vegetation, which in the Amazon River basin can be
significant. This is also the reason why at the inter-annual timescale (and
at shorter timescales) values of E>P are feasible in humid environments
(Fig. b). In addition, the complementary relationship between
actual and potential evapotranspiration becomes relevant, because in
environments like the Amazon River basin, where E is significant,
Ep can decrease and thus, even if P diminishes, in these
catchments the aridity index does not increase as much.
(a) Between-catchment and (b) between-year complementary relationships for the 146
sub-catchments of the Amazon River basin.
Relationship between (a) ∂E/∂Ep and (b) ∂E/∂P
with Ep/P.
Relationship between k (Eq. ), a and Ω.
The complementary relationship from the perspective of the
scaling approach
use the mathematical derivatives of Budyko-type equations
to determine whether in a catchment changes in E are mostly dominated by
changes in P or Ep. For our power law relationship
(Eq. ) the derivatives are as follows:
∂E∂Ep=k(1-e)EpPe∂P∂Ep+eEpP(e-1),∂E∂P=k(1-e)EpPe+eEpP(e-1)∂Ep∂P.
For the analytical derivation of their Budyko–type equation,
considered P and Ep to be completely
independent of each other, and thus the terms ∂P/∂Ep and ∂Ep/∂P were neglected. On
the other hand, while in Fu's mathematical derivation there is
no such supposition, studies that have used this formulation, like the one
carried out by , do not consider these derivatives when
interpreting the complementary relationship based on the Budyko hypothesis.
As mentioned previously, assuming that P and Ep are
independent variables is not valid since they are correlated through E via
the complementary relationship of evapotranspiration, as shown in
Fig. for the 146 sub-catchments of the Amazon River basin.
Figure a shows between-catchment variability among E and
Ep vs. P, while Fig. b shows their between-year
variability. Triangles and diamonds are used to denote the relationship
between long-term mean E and Ep with respect to P where
each marker represents one catchment, whereas circles and squares are used to
depict the relationship between inter-annual E and Ep vs.
P, where each marker represents 1 year. Once more, colors are used to
separate catchments. Figure reflects another sign of space–time
symmetry within the Amazon River basin, since both inter-annual variability
and the long-term mean relationship between E and Ep with P
exhibit the same pattern: E increases as P increases, while
Ep seems to decrease. To quantify the symmetry between both
cases, a linear relationship was fitted to the data. For the case of E, the
slope and intercept from Fig. a were estimated as 0.04 and 1094,
respectively, while from Fig. b they were calculated as 0.03 and
1119. For the case of Ep, both slopes were estimated as
-0.05, while the intercepts were 1840 and 1825, respectively. However, in
both cases the coefficients of determination (R2) were very small, and
thus these results are not statistically significant. This means that an
equation for the relationship between P and Ep could not be
obtained empirically. This issue requires further studies, and we believe an
effort should be made towards the development of either an empirical or an
analytical formulation for the relationship between P and Ep
in humid environments, as they are evidently not independent
(Fig. ). Nevertheless, since this formulation is still not
available and in order to compare our analytical derivations with those of
, in the present study, the terms ∂P/∂Ep and ∂Ep/∂P are not considered.
Figure a and b show the theoretical relationships between
∂E/∂Ep and ∂E/∂P with
Ep/P using the differential forms of Eqs. () and
() for different values of the parameters e and n. For the case
of the power law the value of k was fixed at k=0.6. This figure shows
that for small values of Ep/P, the value of ∂E/∂Ep is larger compared to the value of ∂E/∂P, which means that in humid catchments changes in E are mostly
governed by changes in Ep rather than in P. Also, it can be
seen in Fig. a how Budyko-type equations suggest that for very
humid environments (Ep/P→0) changes in E are equal
to changes in Ep (∂E/∂Ep=1),
which is not necessarily true . Results remain the same if
instead of Eq. () the equation proposed by is used to
depict Fig. . However, our scaling approach allows E to change
more than Ep, which is consistent with the asymmetrical nature
of the complementary relationship.
Between-catchment variability of the parameters at the
inter-annual scale
To explore how the parameters change from catchment to catchment,
Eqs. (), () and the linear relationship were fitted to
represent the observed inter-annual variability, and thus individual values
of k, e, a, b and n were obtained for each catchment. Regarding the
goodness of fit (with 95 % confidence bounds in all cases) we found that
for the power law the estimated range of the parameters were as follows:
0.52<k<0.82 and 0.85<e<1.08 with 0.88<R2<0.99. For the linear
relationship, 0.51<a<0.86, -0.05<b<0.12 with 0.87<R2<0.99 and for
the Budyko type equation 1.07<n<3.54, 0.30<R2<0.88. Once more the data
set is best modelled by either the power law equation or the linear
relationship, which in both cases exhibit higher R2 than
Eq. (). Also, results found for the power law and for the linear
relationship are very similar, as can be seen in the values of both
parameters k and a. This happens mainly because (i) the intercepts of the
linear relationship (b) are close to zero, and (ii) the scaling exponents
of the power law (e) are close to 1. Both characteristics make both
equations resemble each other, and assuming e=1 and b=0, the power law
and the linear relationship become the same equation:
Ψ=kΦ1=aΦ+0thereforea=k.
Spatial distribution of the long-term mean annual P, E and Ep across the sub-catchments of
the Amazon River basin.
Moreover, in the context of our 3-D approach, since Ψ=E/P and
Φ=Ep/P, then mathematically from Eq. (),
a=k=Ω=E/Ep. This was tested for the data and is shown in
Fig. , where k, a and Ω were plotted against each
other. These results indicate that the slope (a) of the linear relationship
and the coefficient (k) of the power law are linked through the way that
each catchment partitions its energy via evapotranspiration. Nonetheless,
taking into account the goodness of fit (Fig. ) k is actually
closer related to Ω (R2=0.95) than a (R2=0.69). This
outcome suggests that our scaling approach (Eq. ) for the Budyko
Curve implicitly incorporates the complementary relationship of
evapotranspiration (in terms of Ω), and thus k becomes a sign of
energy limitations in a catchment. This is a consequence of the dependent
nature of the studied variables within our 3-D space but also of the
physically mutual interdependence between E, Ep and P. In
particular, in humid environments, where normally there is little water
stress, changes in E are mostly dominated by changes in Ep
rather than changes in P, as evidenced in Fig. and as pointed
out by , and . The
latter studied 547 catchments in the continental US and noticed that
catchments in subtropical and humid regions exhibited larger slopes and
smaller intercepts. They also reformulated the linear relationship to
E=aEp+bP in order to explain physically both parameters, and
found that the slope a and intercept b reflect the variability of E
with respect to Ep and P, respectively. Accordingly, in this
study k represents the variability of E due to Ep, which is
in agreement with results for the long-term mean annual E, Ep
and P for each catchment, shown in Fig. . As for the scaling
exponent, it could also be thought of as a measure of the dependence of E
on Ep. The more humid the catchment is, the more likely is
e=1, and thus the more dependent is E on Ep rather than on
P. Figure a shows the distribution of long-term mean annual P
in the Amazon River basin, with values ranging from 1179 to up to
3735 mmyr-1. A consistent spatial pattern can be found, with the
highest precipitation occurring in the Colombian Amazon (north-west region)
and the lowest precipitation taking place near Peru (western region), Bolivia
(south-western region) and some parts of Brazil (south-eastern region). This
is consistent with macroclimatic factors such as the migration of the
Inter-tropical Convergence Zone (ITZC), and the South Atlantic Convergence
Zone (SACZ), land surface–atmosphere interactions as
well as interactions between the Amazon and the Andes and the Atlantic Ocean
. Yet, the most
distinctive spatial pattern can be identified in the regional distribution of
Ep (Fig. c), with highest values (up to
1885 mmyr-1) over the eastern region of the Amazon River basin,
decreasing systematically westwards. Regarding E (Fig. b)
values range from 869 up to 1313mmyr-1 and although some
regionalization can be observed it is not as clear as with P or
Ep. However, it should be noticed that the highest values of
E are found not where P is greater, but where Ep is
greater, even if values of P are not the highest in that region. In general
it can be seen that in the Amazon River basin E follows Ep
more than it follows P.
Parameter k vs. landscape features.
Searching for signs of catchment co-evolution: power
law parameters vs. landscape features
As stated by , catchment co-evolution studies the process
of spatial and temporal interactions between water, energy, landscape
properties such as bedrock, soils, channel networks, sediments and
anthropogenic influences that lead to changes of catchment characteristics
and responses. In particular, landscape organization determines how
a catchment filters climate into hydrological response in time. For this
reason catchment co-evolution is not studied in the time domain, but from
a spatial perspective .
To determine the possible links between the power law parameters (k and
e) and the chosen landscape features (topography, groundwater levels and
vegetation, described in Sect. 2.2), the Spearman's rank correlation
coefficient (ρ) was used. The Spearman's ρ estimates the
statistical dependence between two variables and how well their relationship
can be described using a monotonic function, thus it is a measure of
nonlinear co-dependence. If there are no repeated data values, a perfect
correlation is obtained when ρ=±1. The statistical significance of
the results was tested by p values with significance levels set at 5 %
as presented in Fig. for k. As remarked by
, high correlations indicate that there might be
a relationship between climate and landscape; however, the lack of
correlations does not necessarily indicate independence.
In this study, the strongest connection appears to be with vegetation and
mean elevation above sea level. Specifically k (ρ=0.57,
Fig. c) and e (ρ=0.47) seem to increase with average
maximum green fraction. This result is not surprising, considering that the
Amazon River basin is predominantly covered by tropical rainforest, given the
relationship that we found previously between k and Ω and the role
of vegetation in evapotranspiration. In addition, both k (ρ=-0.58,
Fig. a) and e (ρ=-0.28) seem to decrease with mean
elevation above sea level. The influence of elevation on evapotranspiration
(mainly on Ep) has been studied previously by
for the Colombian Andes, where a decreasing exponential
relationship between reference evapotranspiration and elevation for the Cauca
and Magdalena river basins was found. Moreover, the inverse relationship
between elevation and Ep could be due to the way
Ep was estimated, given that equation
is temperature-based and temperature decreases as elevation increases.
Results also show that k (ρ=-0.42, Fig. b) and e
(ρ=-0.30) appear to decrease with water table depth, which is also
connected to elevation above sea level. Water table depths are larger in the
mountains and shallower at lower elevations. Correlations between the other
parameters (a, b and n) were also carried out for comparison purposes.
The slope of the linear relationship (a) exhibited similar results to k,
while the intercept (b) exhibited similar results to e, but in general
k and e showed stronger connections to landscape features than a and
b. As for the parameter n, none of the landscape features shows any
statistically significant ρ. Thus, even though we are aware that our
power law is an empirical relationship, landscape features and catchment
characteristics are somehow embedded in the values of the associated
parameters.
An alternative equation for evapotranspiration
Our results show that among the three tested equations the best equation to
represent data from the Amazon River basin is the power law (Eq. ),
not only from the perspective of Budyko's framework in humid environments and
its physical limits for the 3-D space, but also from the perspective of the
complementary relationship, space–time symmetry and catchment co-evolution.
At this point it is worth mentioning that the emerging power law was tested
for the entire set of 527 catchments in the US and China, with k=0.63 and
e=0.46 (R2=0.61). It was also tested just for the humid catchments
(Φ<1) from the MOPEX data set with k=0.74 and e=0.96 (R2=0.60).
In addition, the mean aridity index and evaporation ratio for these data sets
are, respectively, ΦUS-China=1.28, ΦMopex=0.74,
ΩUS-China=0.63 and ΩMopex=0.76, while for
the catchments within the Amazon River basin, ΦAmazon=0.86 and
ΩAmazon=0.68. Recalling that for Amazonia k=0.66,
e=0.83, this results seemingly show that Ω≃k. They also appear
to reveal that k decreases with Φ while e increases. This suggests
that our scaling approach could possibly be used as an alternative to the
traditional Budyko-type equations in other catchments with different climatic
conditions, although it might work better for humid environments.
Nevertheless, it should be remembered that for the US and China E was
calculated from the water balance equation and therefore estimates of E and
P are not mathematically independent, whereas for the Amazon River basin
E was estimated using an independent data set based on remote sensors and
meteorological observations. For this reason, the universality of the
coefficients (k) and scaling exponents (e) of the power law should be
explored in the future using diverse data sets. In particular, calculations
should be carried out using different and more reliable estimates of
Ep considering that there are still difficulties in
appropriately estimating the potential of evaporation in very humid
environments, especially when there are tropical rainforests and mountains
involved such as in the Amazon River basin.