Introduction
Flood-envelope curves
For nearly a century, the flood-envelope curves (FECs), i.e., curves drawn
slightly above the largest measured flood discharges on a plot of discharge
vs. contributing area for a given hydroclimatic region (Enzel et al., 1993),
have been an important tool for predicting the magnitude of potential future
floods, especially in regions with limited stream-gauge data. FECs assume
that, within a given hydroclimatic region, maximum flood discharges for one
drainage basin are similar to those of other drainage basins of the same
area, despite differences in relief, soil characteristics, slope aspect, etc.
(Enzel et al., 1993). This assumption enables sparse and/or short-duration
flood records over a hydroclimatic region to be aggregated in order to
provide more precise constraints on the magnitude of the largest possible
(i.e., long-recurrence-interval) floods.
FECs reported in the literature have a broadly similar shape across regions
of widely differing climate and topography. For example, FECs for the
Colorado River Basin (Enzel et al., 1993), the central Appalachian Mountains
(Miller, 1990; Morrison and Smith, 2002), the 17 hydrologic regions within
the US defined by Crippen and Bue (1977), the US as a whole (Costa, 1987;
Herschy, 2002), and China (Herschy, 2002) are all concave-down when plotted
in log-log space, with maximum recorded flood discharges following a
power-law function of contributing area for small contributing areas and
increasing more slowly at larger contributing areas (i.e., the curve
“flattens”).
Traditional FECs also have the potential problem that the maximum flood
associated with smaller drainage basins may be biased upward (or the floods
of larger drainage basins biased downward) because there are typically many
more records of floods in smaller drainage basins relative to larger drainage
basins (because there are necessarily fewer large drainage basins in any
hydroclimatic region). That is, the largest flood on record for small
drainage basins within a hydroclimatic region likely corresponds to a flood
of a larger recurrence interval compared with the largest flood on record for
larger drainage basins. In this paper we present a method that includes
recurrence-interval information and avoids any sample-size bias that might
exist as a function of contributing area.
The use of FECs to quantify flood regimes is limited by the lack of
recurrence-interval information (Wolman and Costa, 1984; Castellarin et al.,
2005) and by the short length, incomplete nature, and sparseness of many
flood discharge records. Without recurrence-interval information, the data
provided by FECs are difficult to apply to some research and planning
questions related to floods. In the US for example, the 100- and 500-year
flood events are the standard event sizes that define flood risk for land
planning and engineering applications (FEMA, 2001).
Previously published studies have looked at new approaches to approve upon
the FEC method. Castellarin et al. (2005) took a probabilistic approach to
estimating the exceedance probability of the FEC for synthetic flood data.
The authors were able to relate the FECs of certain recurrence intervals to
the correlation between sites, the number of flood observations, and the
length of each observation. Later, Castellarin (2007) and Castellarin et
al. (2009) applied these methods to real flood record data and extreme
rainfall events for basins within northern–central Italy. Castellarin et
al. (2009) also created depth–duration envelope curves of precipitation to
relate extreme precipitation events to mean annual precipitation. This group
of studies was successful in incorporating recurrence-interval information
into the traditional FEC method. However, most of the models presented in
these studies were completed with synthetic data or created for design storm
processes and require additional analysis. Also, most of the precipitation
data used in these past studies were collected using rain gauges (point
sources), while only a small subset of data in Castellarin et al. (2009) was
sourced from radar-derived precipitation estimates. In contrast to these
studies we formulate a simplified method (i.e., the FMAC method) that is
readily applicable to any region of interest and can be directly compared to
already existing FECs. Also, we favor the use of spatially complete
radar-derived precipitation estimates in order to apply our methods to
ungauged basins.
To mitigate the uncertainty caused by short and incomplete flood discharge
records, this study uses a space-for-time substitution (e.g.,
regionalization) to lengthen the record for a given contributing area.
Previous studies have employed similar methods, including the index-flood
procedure first described by Dalyrymple (1960) and expanded upon by many
subsequent authors. The index-flood method uses data from multiple sites
within a region to construct more accurate flood-quantile estimates than
would be possible using a single site (Stedinger et al., 1993; Hosking and
Wallis, 2005). This method can also be used on precipitation data, where it
is referred to as the station-year method (Buishand, 1991). The index-flood
method is based on two major assumptions: (1) that observations from two or
more basins are independent; and (2) that observations follow the same
distribution (Wallis et al., 2007).
Here we use a regionalization method similar to the index-flood method in
order to calculate rainfall-intensity values associated with specific
recurrence intervals. The assumption of statistical independence of rainfall
(and associated flood) observations is one that we assume in this study but
understand may not be true for all samples in our natural dataset. This
assumption is difficult to definitively prove with natural data (Hosking and
Wallis, 2005). For example, a large rainfall event may affect two basins in a
similar way and therefore create correlated maximum rainfall-intensity
values. This spatial correlation is difficult to avoid and may cause biased
results. However, it has been shown that the index-flood method can be used
in the absence of fully statistically independent observations and still give
robust results (Hosking and Wallis, 1988; Hosking and Wallis, 2005). The
assumption that observations are sampled from the same distribution is also
somewhat difficult to prove with natural data, but by knowing the study areas
well a researcher can identify regions with similar rainfall and flood
mechanisms. Many examples of this type of area analysis can be found in the
literature, including Soong et al. (2004), who separated rural streams in
Illinois into hydrological regions based on basin morphology and soil
characteristics. Soong et al. (2004) used regionalization in their study to
increase the amount of flood data available for frequency analysis. Wallis et
al. (2007) employed a similar regionalization method to identify
hydroloclimatic regions in their study of precipitation frequency in
Washington. It should be noted that FECs in general use this type of
regionalization approach to analyze maximum flood data for hydroclimatic
regions with similar flood mechanisms. In this study we similarly attempt to
analyze regions based on their basic rainfall mechanisms, in this case by
separating the Upper and Lower Colorado River basins.
In this study, a new method for estimating flood discharges associated with
user-specified recurrence intervals is introduced that uses radar-derived
precipitation estimates (in this case rainfall only), combined with the
diffusion-wave flow-routing algorithm, to create frequency–magnitude–area
curves (FMACs) of flood discharge. Our method (i.e., the FMAC method) retains
the power of the FEC approach in that data from different drainage basins
within a hydroclimatic region are aggregated by contributing area, thereby
enabling large sample sizes to be obtained within each contributing-area
class in order to more accurately constrain the frequencies of past extreme
flood events and hence the probabilities of future extreme flood events
within each class. The method improves upon the FEC approach in that the
complete spatial coverage of radar-derived precipitation estimates provides
for large sample sizes of most classes of contributing area (larger
contributing areas have fewer samples). The radar-derived precipitation
estimates include only rainfall and therefore snow and other types of
precipitation are not included in the study. The precipitation estimates are
then used to predict flood discharges associated with specific recurrence
intervals by first accounting for water lost to infiltration and
evapotranspiration using runoff coefficients appropriate for different
contributing areas and antecedent-moisture conditions, and then routing the
available water using a flow-routing algorithm. Predicted flood discharges
are presented as FMACs on log-log plots, similar to traditional FECs, except
that the method predicts a family of curves, one for each user-defined
recurrence interval. These plots are then compared to FECs for the study
region (Enzel et al., 1993) and the US (Costa, 1987).
Study area
This study focuses on the Upper and Lower Colorado River basins (UCRB and
LCRB, respectively; Fig. 1) as example applications of the FMAC method.
Although the methods we develop are applied to the UCRB and LCRB in the
western US in this study, the methods are applicable to any region of
interest where radar-derived precipitation estimates are available (i.e., the
entire US and at least 22 countries around the world; Li, 2013; RadarEU,
2014). We focus on the UCRB and LCRB because they have been a focus of
flood-hazard assessment studies in the western US and hence the FECs
available for them are of especially high quality. In addition, the
distinctly different hydroclimatic regions of the UCRB and LCRB
(Sankarasubramanian and Vogel, 2003) make working in these regions an
excellent opportunity to test and develop the new methods of this study on
different precipitation patterns and storm types.
Map showing the locations of the Upper and Lower Colorado River
basins (UCRB and LCRB, respectively) outlined by the dotted line.
Precipitation and flooding in the LCRB are caused by convective-type storms,
including those generated by the North American Monsoon (NAM), and
frontal-type and tropical storms sourced from the Pacific Ocean and the Gulf
of California (House and Hirschboeck, 1997; Etheredge et al., 2004). In the
UCRB, the influence of the NAM and tropical storms is diminished and floods
are generally caused by Pacific frontal-type storms (Hidalgo and Dracup,
2003). In both regions, the El Niño–Southern Oscillation (ENSO) alters
the frequency and intensity of the NAM, tropical storms, and the Pacific
frontal systems, and can cause annual variations in precipitation and
flooding (House and Hirschboeck, 1997; Hidalgo and Dracup, 2003). Winter
storms in both regions are also intensified by the occurrence of atmospheric
rivers (Dettinger et al., 2011), which can cause total winter precipitation
to increase up to approximately 25 % (Rutz and Steenburgh, 2012). The
radar-derived precipitation estimates used in this study record this natural
variability in precipitation in the two regions.
The methods used in this study to calculate rainfall and flood discharges of
specified recurrence intervals from radar-derived precipitation estimates
require a few main assumptions. The first assumption is that of climate
stationarity; i.e., the parameters that define the distribution of floods do
not change through time (Milly et al., 2008). Climate is changing and these
changes pose a challenge to hazard predictions based on the frequencies of
past events. Nevertheless, stationarity is a necessary assumption for any
probabilistic analysis that uses past data to make future predictions. The
results of such analyses provide useful starting points for more
comprehensive analyses that include the effects of future climate changes.
The second assumption is that the sample time interval is long enough to
correctly represent the current hydroclimatic state (and its associated
precipitation patterns and flood magnitudes and risks) of the specified study
area. Our study uses data for the 1996 to 2004 water years and therefore may
be limited by inadequate sampling of some types of rare weather patterns and
climate fluctuations within that time interval. To address whether or not the
sample time interval used in this study includes major changes in circulation
and weather patterns, and therefore is a good representation of climate in
the CRB, we investigated the effect of the ENSO on rainfall intensity within
the UCRB and LCRB. ENSO is a well-known important influence on the
hydroclimatology of the western US (Hidalgo and Dracup, 2003; Cañon et
al., 2007). In general, winter precipitation in the southwestern US increases
during El Niño events and decreases during La Niña events (Hidalgo
and Dracup, 2003). The opposite effects are found in the northwestern
portions of the US (including the UCRB; Hidalgo and Dracup, 2003). The last
assumption of the method is that all basins of similar contributing area
respond similarly to input rainfall, i.e., that they have similar
flood-generating and flow-routing mechanisms. Specifically, the method
assumes that basins of similar contributing area have the same runoff
coefficient, flow-routing parameters, basin shape, and channel length, width,
and slope. This assumption is necessary in order to aggregate data into
discrete contributing-area classes so that the frequency of extreme events
can be estimated from relatively short-duration records. In this study,
high-recurrence-interval events (i.e., low-frequency events) can be
considered despite the relatively short length of
radar-derived-precipitation-estimate records because the number of samples in
the radar-derived record is extremely large, especially for small
contributing areas and short-duration floods. For example, for a 1 h
time-interval-of-measurement and a contributing area of 4096 km2 event
in the UCRB, there are approximately 40 (number of spatial-scale samples)
times 55 000 (number of temporal-scale samples in 9 years of data) samples
of rainfall-intensity values (and associated modeled discharges obtained via
flow routing). As contributing area and time intervals of measurement
increase there are successively fewer samples, within any particular
hydroclimatic region, thus increasing the uncertainty of the resulting
probability assessment for larger areas and longer time periods.
(a) Pseudocode describing the methods of the paper with schematic
diagrams shown below pseudocode in some cases. Equations within the text and
other figures are referenced in red text. (b) Pseudocode describing
the methods of the paper with schematic diagrams shown below pseudocode in some
cases. Equations within the text and other figures are referenced in red text.
Next-Generation-Radar (NEXRAD) data
The specific radar-derived precipitation estimates we use in this study come
from the Stage III Next-Generation-Radar (NEXRAD) gridded product, which is
provided for the entire US, Guam, and Puerto Rico. NEXRAD was introduced
in 1988 with the introduction of the Weather Surveillance Radar 1988 Doppler,
or WSR-88D, network (Fulton et al., 1998). The WSR-88D radars use the
Precipitation Processing System (PPS), a set of automated algorithms, to
produce precipitation intensity estimates from reflectivity data.
Reflectivity values are transformed to precipitation intensities through the
empirical Z–R power-law relationship
Z=αRβ,
where Z is precipitation rate (mm h-1), α and
β are derived empirically and can vary depending on
location, season, and other conditions (Smith and Krajewski, 1993), and
R is reflectivity (mm6 m-3; Smith and Krajewski, 1993;
Fulton et al., 1998; Johnson et al., 1999). Precipitation intensity data are
filtered and processed further to create the most complete and correct
product (Smith and Krajewski, 1993; Smith et al., 1996; Fulton et al., 1998;
Baeck and Smith, 1998). Further information and details about PPS processing
are thoroughly described by Fulton et al. (1998).
Stage III NEXRAD gridded products are Stage II precipitation products mapped
onto the Hydrologic Rainfall Analysis Project (HRAP) grid (Shedd and Fulton,
1993). Stage II data are hourly precipitation intensity products that
incorporate both radar reflectivity and rain-gauge data (Shedd and Fulton,
1993) in an attempt to make the most accurate precipitation estimates
possible. The HRAP grid is a polar coordinate grid that covers the
conterminous US, with an average grid size of 4 km by 4 km, although grid
size varies from approximately 3.7 km (north to south) to 4.4 km (east to
west) in the southern and northern US, respectively (Fulton et al., 1998).
Methods
NEXRAD data conversion and sampling
NEXRAD Stage III gridded products (hereafter NEXRAD products) for an area
covering the Colorado River Basin from 1996 to 2005 were downloaded from the
NOAA HDSG website
(http://dipper.nws.noaa.gov/hdsb/data/nexrad/cbrfc_stageiii.php) for
analysis. The data files were converted from archived XMRG files to ASCII
format (each data file representing the mean rainfall intensity within each
1 h interval) using the xmrgtoasc.c program provided on the NOAA HDSG
website. The ASCII data files were then input into a custom program written
in IDL for analysis.
Rainfall sampling over space
In this study we quantified hourly rainfall intensities (mm h-1) over
square idealized drainage basins (i.e., not real drainage basins, but square
drainage basins as shown schematically in Fig. 2a as brown squares) of a
range of areas from 16 to 11 664 km2 (approximately the contributing
area of the Bill Williams River, AZ, for readers familiar with the geography
of the western US) by successively spatially averaging rainfall-intensity
values at HRAP pixel-length scales of powers of 2 (e.g., 4, 16 pixel2)
and 3 (e.g., 9, 81 pixel2; Fig. 2, Step 1). Spatial averaging is done by
both powers of 2 and 3 simply to include more points on the FMACs than would
result from using powers of 2 or 3 alone. The number of samples within each
contributing area class limited the range of contributing areas used in this
study; i.e., at larger contributing areas there were too few samples to
successfully apply the frequency analysis.
UCRB and LCRB boundaries from GIS hydrologic unit layers created by the USGS
and provided online through the National Atlas site
(http://gdg.sc.egov.usda.gov) were projected to HRAP coordinates using the methods of Reed
and Maidment (2006). These boundaries were used to delineate the region from
which rainfall data were sampled from the NEXRAD products; i.e., when
averaging rainfall data by powers of 2 and 3, a candidate square drainage
basin was not included in the analysis if any portion of the square fell
outside of the boundaries of the UCRB or LCRB (Fig. 2a). Throughout the
analysis, the HRAP pixel size was approximated by a constant 4 km by 4 km
size despite the fact that HRAP pixel sizes vary slightly as a function of
latitude (Reed and Maidment, 2006). Our study's drainage basins span
latitudes between approximately 31 and 43∘ N, resulting in a maximum
error of 15 %. However, by keeping the pixel size constant, all pixels
could be treated as identical in size and shape, allowing us to sample the
NEXRAD products in an efficient and automated way over many spatial scales.
For larger contributing areas, necessarily fewer samples are available within
a given hydroclimatic region, thus increasing the uncertainty associated with
the analysis for those larger contributing-area classes. For the UCRB and
LCRB specifically, the uncertainty in the analysis becomes significant for
contributing-area classes equal to and larger than ∼ 103 to
104 km2, depending on the recurrence interval being analyzed. Of
course, if the hydroclimatic region is defined to be larger, more samples are
available for each contributing-area class and hence larger basins can be
analyzed with confidence.
Rainfall sampling over time
In addition to computing rainfall intensities as a function of spatial scale,
we averaged rainfall intensities as a function of the time interval of
measurement ranging from 1 to 64 h in powers of 2 by averaging hourly
rainfall-intensity records over the entire 9-year study period (Fig. 2,
Step 1). This range in time intervals was chosen in order to capture rainfall
events that last on the order of ∼ 1 h (convective-type storms) to
days (frontal-type storms).
Rainfall data were sampled temporally by taking the maximum value of each
storm event. Storm events were identified as consecutive non-zero
rainfall-intensity values separated by instances of zero values in time for
each temporal scale. This allows for multiple maximum rainfall values in time
to be sampled within a year and throughout the entire 9-year study period.
This sampling method is similar to that used in the peak over threshold (POT)
method typically used on discharge data where a minimum threshold value is
set and maximum peaks above the threshold value are recorded as maximum
events. Here we set the minimum threshold value to zero, and hence the
maximum values of all individual storm events are considered in the analysis.
Rainfall recurrence-interval calculations
To determine the rainfall-intensity values with a user-specified recurrence
interval, maximum rainfall intensities of storm events sampled from the
NEXRAD data for each contributing-area and time-interval-of-measurement class
were first ranked from highest to lowest (Fig. 2a, Step 2). The relationship
between recurrence intervals and rank in the ordered list is given by the
probability-of-exceedance equation (i.e., the frequency–rank relationship):
RI=(n+1)m,
where RI is the recurrence interval (yr), defined as the inverse of
frequency (yr-1) or probability of exceedance, n is the total number
of samples in each contributing area and time-interval-of-measurement scaled
to units in years (resulting in units of yr), and m is the rank of the
magnitude ordered from largest to smallest (unitless). Here the recurrence
interval is prescribed (10, 50, 100, and 500 years); then, the rank
associated with this recurrence interval is computed using the
frequency–rank relationship (Eq. 2). The resulting rainfall intensities
associated with a user-specified recurrence interval and contributing-area
and time-interval-of-measurement class were then used to calculate the
Qp value.
At the end of the calculations described above we have datasets of
rainfall-intensity values for each combination of the eight
contributing-area classes, the seven time-interval-of-measurement classes,
and the four recurrence intervals. We then find the maximum values of
rainfall intensity associated with a given contributing-area class and
recurrence interval among all values of the time-interval-of-measurement
class (i.e., the values calculated for 1 to 64 h time intervals). This step
is necessary in order to find the maximum values for a given contributing
area class and recurrence interval independent of the
time-interval-of-measurement, i.e., independent of storm durations and
associated types of storms. The maximum values are used to be consistent
with the methods of the traditional FECs where the points represent the
largest possible storm for a given contributing area. These maximum values
are used to calculate Qp and Qfd (see next section).
Rainfall and runoff calculations
The first variable calculated from the maximum rainfall intensities found for
each contributing-area class and recurrence interval is the precipitation
(here rainfall only) discharge, Qp. The variable Qp is
defined as the average rainfall intensity over a basin and time interval of
measurement multiplied by the contributing area, resulting in units of
m3 s-1. This is a simple calculation resulting in a “discharge”
of rainfall to a basin. Qp is the input value for the flow-routing
algorithm that we employ to calculate the peak flood discharge (Fig. 2b,
Step 3).
The flow-routing algorithm we employ does not explicitly include infiltration
and other losses that can further reduce peak flood discharge relative to
input to the basin, Qp. In this study we modeled infiltration and
evaporation losses by simply removing a volume of water per unit time equal
to one minus the runoff coefficient, i.e., the ratio of runoff to rainfall
over a specified time interval, for three antecedent-moisture scenarios (wet,
med, and dry). We estimated runoff coefficients for each contributing-area
class and each of three antecedent-moisture scenarios using published values
for annual runoff coefficients for large basins within the UCRB and LCRB
(Rosenburg et al., 2013) and published values for event-based runoff
coefficients for small basins modeled with a range of antecedent-moisture
conditions by Vivoni et al. (2007) (Fig. 3). On average, estimated runoff
coefficients are higher for smaller and/or initially wetter basins. We found
the dependence of runoff coefficients on contributing area and antecedent
moisture to be similar despite the large difference in timescales between
event-based and annual values. Despite the difference in geographic region
between our study site and that of Vivoni et al. (2007) (they studied basins
in Oklahoma), the runoff coefficients they estimated are likely to be broadly
applicable to the LCRB and UCRB given that basin size and antecedent moisture
are the primary controls on these values (climate and soil types play a
lesser role except for extreme cases).
Logarithmic relationships between runoff coefficients and contributing
area using modeled data for wet (filled diamonds), medium (open squares), and
dry (filled circles) antecedent-moisture conditions (Vivoni et al., 2007) and
measured data for larger contributing areas (filled squares; Rosenburg et al.,
2013). The medium (open squares) and dry (filled circles) data separate into
two distinct groups relating to the precipitation event used to model them, with
the lower group and higher group relating to a 12 h, 1 mm h-1 event and
1 h, 40 mm h-1 event, respectively. All points were used in the
least-squares weighed-regression analysis.
We applied the estimated runoff coefficients for all three
antecedent-moisture scenarios by simply using them to remove a portion of the
Qp calculated for a specific time interval and basin area:
Qpm=C⋅Qp,
where C is the runoff coefficient calculated for the specific basin area
and antecedent-moisture scenario under evaluation. The newly formed
Qpm is now the Qp value for the wet, medium, or dry
antecedent-moisture scenario under analysis for each given
recurrence-interval and contributing-area class.
Flood discharge calculations
The second variable calculated in this study, and the end-result of our
methods, is the peak flood discharge, Qfd. The variable
Qfd is the peak flood discharge (m3 s-1)
calculated via the diffusion-wave flow-routing algorithm for a hypothetical
flood triggered by a rainfall discharge, Qpm, input uniformly
over the time interval of measurement to idealized square basins associated
with each contributing-area class (Fig. 2b, Step 4).
The flow-routing algorithm routes flow along the main-stem channel of
idealized square basins with sizes equal to the contributing area of each
contributing-area class. The choice of a square basin is consistent with the
square sample areas (see Sect. 3.1) and it allows for basin shape to remain
the same (and therefore comparable) over the range of contributing areas used
in this study. The main-stem channel, with a length of L (m), was defined
as the diagonal distance from one corner to the opposite corner across the
square basin (i.e., L is equal to the square root of 2 times the area of
the square basin). This main-stem channel was used in conjunction with a
normalized area function to represent the shape of the basin and the routing
of runoff through the drainage basin network. By including the normalized
area function, we can account for geomorphic dispersion (i.e., the
attenuation of the flood peak due to the fact that rainfall that falls on the
landscape will take different paths to the outlet and hence reach the outlet
at different times) in our analyses. The normalized area function, A(x)
(unitless), is defined as the portion of basin area, AL(x)
(m2), that contributes flow to the main-stem channel within a given
range of distances (x) from the outlet, normalized by the total basin area,
AT (m2; Mesa and Mifflin, 1986; Moussa, 2008). The normalized
area function is assumed to be triangular in shape, with a maximum value at
the midpoint of the main-stem channel from the outlet. Area functions, and
related width functions, from real basins used in other studies show this
triangular shape in general (Marani et al., 1994; Rinaldo et al., 1995;
Veneziano et al., 2000; Rodriguez-Iturbe and Rinaldo, 2001; Puente and Sivakumar, 2003; Saco and Kumar, 2008),
although not all basins show this shape. The triangular area function has
been shown to approximate the average area function of basins and that the
peak discharge and time to peak discharge is likely more important to the
shape of the flood wave (Henderson, 1963; Rodriguez-Iturbe and Valdes, 1979).
A one-dimensional channel with simplified width and along-channel slope
appropriate for channels in the CRB is used to approximate the geometry of
the main-stem channel of the idealized basin in the flow-routing algorithm.
In addition, values for channel slope, S (m m-1), and channel width,
w (m), are assigned based on the contributing area of the idealized basin
and the results of a least-squares regression to channel-slope and
channel-width data from the CRB. We assume here that the assigned channel
slopes and widths represent the average value for the entire idealized basin.
To find the best approximations for channel slope and width values, we
developed formulae that predict average channel slope and channel width as a
function of contributing area based on a least-squares fit of the logarithms
of slope, width, and contributing area based on approximately 100 sites in
the Colorado River Basin (CRB; Fig. 4). The data used in these least-squares
regressions included slope, width, and contributing area information from all
sites in the LCRB and southern UCRB presented in Moody et al. (2003) and
additional sites from USGS stream-gauge sites from across the CRB.
Power-law relationships between channel slope and contributing
area (a) and channel width and contributing area (b) for the
Colorado River Basin.
The assigned channel slope and width values, together with the values of
Qpm modified for each antecedent-moisture scenario, were used to
calculate the depth-average velocities, V (m s-1), in hypothetical
one-dimensional main-stem channels of idealized square drainage basins
corresponding to each contributing-area and time-interval-of-measurement
class. In this study, flow velocity is not modeled over space and time, but
rather is set at a constant value appropriate for the peak discharge using an
iterative approach that solves for the peak depth-averaged flow velocity,
uses that velocity to compute the parameters of the diffusion-wave-routing
algorithm, routes the flow, and then computes an updated estimate of peak
depth-averaged velocity. To calculate the depth-averaged velocity, V, we
used Manning's equation, i.e.,
V=1nMR23S12,
where nM is Manning's n (assumed to be equal to 0.035), R is the
hydraulic radius (m) calculated with the assigned channel width, and
S (m m-1) is the assigned channel slope. In order to calculate
R, the water depth, h, of the peak discharge needed to be
determined. In this study h was iteratively solved for based on the
peak-flow conditions (i.e., the depth-averaged velocity, V, associated with
the peak-flood discharge, Qfd) with h set at 1 m for the first
calculation of the flow-routing algorithm. At the end of each calculation,
h is recalculated using Manning's equation. These iterations continue until
the water depth converges on a value (i.e., the change from the last
calculation of h to the next calculation of h is ≤ 0.1 m)
corresponding to a specific recurrence interval, contributing-area class, and
time-interval-of-measurement class.
The method we used to model flow through the main-stem channel is the
diffusion-wave flow-routing algorithm. This approach is based on the
linearized Saint-Venant equations for shallow-water flow in one dimension. To
find a simpler, linear solution to Saint-Venant equations, Brutsaert (1973)
removed the acceleration term from the equations, leaving the diffusion and
advection terms that often provide a reasonable approximation for watershed
runoff modeling (Brutsaert, 1973). Leaving the diffusion term in the
flow-routing algorithm includes hydrodynamic dispersion of the flood wave in
the calculation of the flood hydrograph. In the case where the initial
condition is given by a unit impulse function (Dirac function), the cell
response function of the channel, qd (units of s-1), is given
by
qd=x(2π)1/2btr3/2exp-(a-atr)22b2tr,
where x is the distance along the channel from the location where the
impulse is input to the channel, tr is the time since the impulse was
input into the channel, and the drift velocity a (m s-1) and
diffusion coefficient b2 (m2 s-1) are defined as
a=1+a0V,b2=V3gSFr21-a02Fr2,
where Fr is the Froude number, g is the acceleration due to
gravity (m s-2), and a0 is a constant equal to 2 / 3 when using
Manning's equation (Troch et al., 1994). The large floods modeled in this
study are assumed to have critical-flow conditions, and therefore the Froude
number is set to a constant value of 1.
The unit response discharge, qfd (m2 s-1), at the outlet
of a drainage basin can be computed from Eqs. (3) to (5) by integrating the
product of the cell response function qd(x, t) corresponding to a
delta-function input of the normalized area function, A(x), i.e., the
spatial distribution of rainfall input. The integral is given by
qfdtr=∫0tpQpwdt′∫L0qdx,tr-t′A(x)dx,
where tp is the time interval of measurement over which the unit
impulse input (i.e., Qp) is applied to the idealized square
drainage basin, and tr is the time after the input of the unit impulse
that is long enough to capture the waxing and waning portions and the flood
peak of the flood wave. The final peak discharge value, or
Qfd (m3 s-1), was calculated by multiplying the unit
discharge qfd (m2 s-1) by the channel width found
through the formula derived from CRB data in Fig. 4 and then selecting the
largest value from the resulting hydrograph.
Estimation of uncertainty
Confidence intervals (i.e., uncertainty estimates) were calculated to
quantify the uncertainty in calculated rainfall intensities and associated
Qp and Qfd values. In this study we estimated confidence
intervals using a non-parametric method similar to that used to calculate
quantiles for flow-duration curves (Parzen, 1979; Vogel and Fennesset, 1994).
Like quantile calculations, which identify a subset of the ranked data in the
vicinity of each data point to estimate expected values and associated
uncertainties, we estimated confidence intervals for our predictions based on
the difference in Qp values between each point and the next largest
value in the ranked list. This approach quantifies the variation in the
rainfall-intensity value for a given contributing area and recurrence
interval. In some cases the calculated uncertainties for rainfall intensities
and associated Qp and Qfd values are infinite due to the
values being past the frequency–magnitude distribution; i.e., there are not
enough samples for these values to be determined and there are no finite
numbers to sample. These values are not used in this study.
The resulting confidence intervals of rainfall intensity were used to
calculate confidence intervals for Qp and Qfd. Confidence
intervals for Qp values were equal to the confidence intervals for
rainfall intensity propagated through the calculation of Qp (i.e.,
multiplying by contributing area). Confidence intervals for
Qfd values were calculated to be the same proportion of the
Qfd value as that set by the rainfall-intensity value and its
confidence intervals. For example, if the upper confidence interval was
120% of a rainfall-intensity value, the upper confidence interval for the
Qfd value associated with the rainfall-intensity value is assumed
to be 120 % of the Qfd value. This approach to propagation of
uncertainty treats all other variables in the calculations as constants, and
additional uncertainty related to regression analyses of variables used in
the flow-routing algorithm such as slope, channel width, and runoff
coefficients was not included.
Testing the effects of climate variability
To quantify the robustness of our results with respect to climate
variability, we separated the NEXRAD data into El Niño and La Niña
months using the multivariate ENSO index (MEI). All months of data with
negative MEI values (La Niña conditions) were run together to calculate
the rainfall intensity and Qp values for contributing areas
of 16, 256, and 4096 km2, time intervals of 1 to 64 h, and for 10-,
50-, 100-, and 500-year recurrence intervals. This was repeated with all
months of data with positive MEI values (El Niño conditions). Figure 5
shows the distribution of negative and positive MEI values during the
1996 to 2004 water years used in this study.
Results
Channel characteristics and runoff coefficients
Least-squares regression of channel slopes and channel widths from the CRB
vs. contributing area was used to estimate channel slope, channel width, and
runoff coefficients for each idealized basin of a specific contributing-area
class. Channel slope decreases as a power-law function of contributing area
with an exponent of -0.30 (R2 = 0.39), whereas channel width
increases as a power-law function of contributing area with an exponent
of 0.28 (R2 = 0.65; Fig. 4). These results follow the expected
relationships among channel slopes, widths, and contributing area; i.e., as
contributing area increases, the channel slope decreases and the channel
width increases.
Multivariate ENSO index (MEI) of months included in Stage III NEXRAD
gridded products. Months are numbered from September 1996 to September 2005
with years shown in gray. Dashed black line MEI equal to zero. A positive MEI
indicates El Niño conditions, while a negative MEI indicates La Niña
conditions.
Runoff coefficients for wet, medium, and dry antecedent-moisture conditions
all decrease with increasing contributing area following a logarithmic
function, with the slope of the line decreasing from wet to dry conditions.
The fitness of the line to the data also decreases for the wet to dry
conditions, with the R2 values for wet, medium, and dry conditions equal
to 0.78, 0.45, and 0.04, respectively. Runoff coefficients decrease with
increasing contributing area due to the increased probability of water losses
as basin area increases. Also, as expected, runoff coefficients are highest
in basins with wet initial conditions that are primed to limit infiltration
and evapotranspiration.
Trends in rainfall intensity
Maximum rainfall intensities (i.e., the maximum among all
time-interval-of-measurement classes) for each contributing-area class and
recurrence interval decrease systematically as power-law functions of
increasing contributing area for all recurrence intervals with an average
exponent of -0.18 ± 0.06 (error is the standard deviation of all
calculated exponents found from a weighted least-squares regression; average
coefficient of determination R2 = 0.78). Note that
maximum-rainfall-intensity results are not presented because they are closely
related to the plots of Qp vs. contributing area in Fig. 6; i.e.,
Qp is simply the rainfall intensity multiplied by the contributing
area. The decrease in maximum rainfall intensity with contributing area can
be seen in Table 1, where maximum rainfall intensities over contributing
areas of 11 664 km2 are 45 to 8 % of maximum-rainfall-intensity
values for basin areas of 16 km2 in both the UCRB and LCRB (Table 1).
The largest decrease in maximum-rainfall-intensity values between the
smallest and largest contributing areas were found for the largest recurrence
interval (e.g., 500-year) for both the UCRB and LCRB. The decrease in maximum
rainfall intensity with increasing contributing area suggests that there is a
spatial limitation to storms of a given rainfall intensity.
Frequency–magnitude–area (FMA) curves of Qp
vs. contributing area for recurrence intervals (RIs) of 10, 50, 100, and
500 years for the Upper Colorado River Basin (UCRB; a) and the Lower
Colorado River Basin (LCRB; b).
Maximum rainfall intensity and Qp for the Upper Colorado
River Basin (UCRB) and Lower Colorado River Basin (LCRB). Note that data are
all sampled from time intervals of measurement ≤ 2 h.
RI
Area
Intensity (mm h-1)
Qp (m3 s-1)
(km2)
UCRB
LCRB
UCRB
LCRB
10
16
28.0 ± 0.0
36.6 ± 0.0
125 ± 0
162 ± 0
10
64
25.4 ± 0.1
32.5 ± 0.0
451 ± 1
578 ± 0
10
144
25.1 ± 1.1
29.5 ± 0.4
1004 ± 44
1182 ± 16
10
256
23.7 ± 0.2
27.3 ± 0.0
1682 ± 13
1944 ± 1
10
1024
19.8 ± 1.5
19.7 ± 0.4
5644 ± 427
5610 ± 114
10
1296
20.7 ± 2.4
21.7 ± 3.5
7439 ± 873
7820 ± 1268
10
4096
15.5 ± 3.0
15.9 ± 0.8
17 682 ± 3462
18 134 ± 890
10
11 664
12.6 ± 1.7
11.0 ± 2.6
40 914 ± 5571
35 521 ± 8586
50
16
55.9 ± 0.7
56.2 ± 0.1
248 ± 3
250 ± 0
50
64
55.1 ± 1.2
47.7 ± 0.0
980 ± 22
847 ± 1
50
144
55.3 ± 3.5
43.3 ± 0.9
2211 ± 142
1734 ± 38
50
256
54.9 ± 1.4
40.9 ± 0.5
3901 ± 101
2908 ± 32
50
1024
50.8 ± 5.5
33.6 ± 1.4
14449 ± 1569
9560 ± 393
50
1296
50.8 ± 25.0
32.5 ± 3.9
18 287 ± 9011
11 704 ± 1410
50
4096
27.6 ± 22.2
30.0 ± 5.2
31 382 ± 25 313
34 126 ± 5969
50
11664
21.1*
15.4 ± 8.3
68 434*
49 764 ± 26 874
100
16
92.3 ± 0.3
68.6 ± 0.0
410 ± 1
305 ± 0
100
64
91.9 ± 2.5
54.5 ± 0.2
1635 ± 44
970 ± 3
100
144
90.1 ± 3.0
51.9 ± 1.0
3606 ± 118
2075 ± 41
100
256
88.7 ± 4.3
48.4 ± 0.4
6305 ± 307
3440 ± 27
100
1024
63.8 ± 11.0
42.5 ± 2.2
18 155 ± 3139
12 085 ± 630
100
1296
78.5 ± 50.1
43.2 ± 7.8
28 257 ± 18 022
15 544 ± 2820
100
4096
40.8*
32.0 ± 10.4
46 422*
36 425 ± 11 803
100
11 664
21.1*
20.1*
68 434*
65 011*
500
16
254.0 ± 0.8
81.9 ± 0.5
1129 ± 3
364 ± 2
500
64
229.0 ± 3.1
68.6 ± 1.5
4071 ± 55
1219 ± 26
500
144
219.1 ± 11.9
68.6 ± 4.7
8762 ± 476
2743 ± 187
500
256
219.4 ± 7.3
68.6 ± 3.4
15 600 ± 517
4877 ± 242
500
1024
166.0 ± 44.1
68.6 ± 3.1
47 229 ± 12 554
19 507 ± 884
500
1296
174.6 ± 85.3
65.6 ± 31.3
62 862 ± 30 696
23 624 ± 11 279
500
4096
81.6*
53.6*
92 844*
60 930*
500
11 664
21.1*
20.1*
68 434*
65 011*
* Values with infinite confidence intervals; not used in
this study.
Differences among maximum rainfall intensities for the four recurrence
intervals as a function of contributing area are larger in the UCRB than in
the LCRB (Table 1). This larger “spread” in the maximum rainfall
intensities in the UCRB relative to the LCRB is also propagated throughout
the maximum rainfall and flood discharge calculations. For both the UCRB and
LCRB, the difference between the 50- and 100-year recurrence-interval values
was the smallest (Table 1). These trends show that maximum rainfall
intensities vary much more as a function of recurrence interval in the UCRB
compared with the LCRB.
Maximum rainfall intensities associated with a 10-year recurrence interval
are similar in the LCRB and UCRB, while intensities were higher in the UCRB
than the LCRB for recurrence intervals of 50, 100, and 500 years (Table 1).
The results of the comparison between the two basins suggest that common
(i.e., low-recurrence-interval) rainfall events will have similar maximum
rainfall intensities in the UCRB and LCRB, but that rare (i.e.,
high-recurrence-interval) rainfall events will have higher maximum rainfall
intensities in the UCRB than in the LCRB for the same recurrence interval.
Maximum precipitation intensities associated with the four defined recurrence
intervals are similar to previously published values. In general the values
we calculate for the LCRB and the UCRB for the 10-, 50-, and 100-year
recurrence intervals are on the order of 10 s of mm h-1. This is
similar to the spread in values reported on precipitation intensity maps for
the same duration and recurrence interval in Hershfield (1961). However, the
values reported by Hershfield (1961) are slightly higher (by less than
20 mm h-1) in the LCRB for the three recurrence intervals and in the
UCRB for the 10-year recurrence interval than values calculated in this
study. The values calculated here are also broadly consistent with presented
precipitation frequency estimates for points within the LCRB and UCRB
provided by the NOAA Atlas 14 Point Precipitation Frequency Estimates website
(http://hdsc.nws.noaa.gov/hdsc/pfds/pfds_map_cont.html). Due to the
difference in how precipitation intensities are measured and how the
frequencies are calculated, the values are expected to be slightly different
but within the same order of magnitude.
Trends in Qp
Maximum precipitation (here only rainfall) discharges (Qp
hereafter) increase with contributing area as power-law functions with an
average exponent of 0.82 ± 0.06 (error is the standard deviation of all
calculated exponents) based on weighed least-squares regressions on the data
(R2 = 0.98) for all recurrence intervals and for both the UCRB and
LCRB (Fig. 6). These Qp values for a given contributing-area class
and recurrence interval are the largest values taken from the multiple values
calculated for each of the seven time intervals of measurement as explained
in Sect. 3.3. By taking the maximum values, the resulting Qp FMACs
approximate the upper envelope of values of a given recurrence interval. In
this study the FMAC follows a power-law function that shows that Qp
increases predictably across the range in contributing areas. As with the
maximum rainfall-intensity results, differences between Qp values
of different recurrence intervals for a given contributing area were larger
for the UCRB than the LCRB (Fig. 6).
Maximum Qfd for the Upper Colorado River Basin (UCRB) and
Lower Colorado River Basin (LCRB). Note that data are all sampled from time
intervals of measurement ≤ 2 h.
RI
Area
Wet Qfd (m3 s-1)
Med Qfd (m3 s-1)
Dry Qfd (m3 s-1)
(km2)
UCRB
LCRB
UCRB
LCRB
UCRB
LCRB
10
16
65 ± 0
86 ± 0
36 ± 0
47 ± 0
20 ± 0
26 ± 0
10
64
246 ± 1
263 ± 0
137 ± 0
151 ± 0
75 ± 0
89 ± 0
10
144
465 ± 20
489 ± 7
268 ± 12
290 ± 4
156 ± 7
175 ± 2
10
256
657 ± 5
748 ± 0
388 ± 3
449 ± 0
244 ± 2
283 ± 0
10
1024
2363 ± 179
2194 ± 44
1423 ± 108
1326 ± 27
892 ± 68
820 ± 17
10
1296
2244 ± 263
2384 ± 387
1459 ± 171
1543 ± 250
1010 ± 118
1066 ± 173
10
4096
5594 ± 1095
5304 ± 260
3665 ± 718
3375 ± 166
2507 ± 491
2315 ± 114
10
11664
14603 ± 1966
11048 ± 2670
9010 ± 1213
6978 ± 1687
6105 ± 822
4942 ± 1195
50
16
131 ± 2
131 ± 0
73 ± 1
73 ± 0
41 ± 1
41 ± 0
50
64
553 ± 12
387 ± 0
307 ± 7
222 ± 0
172 ± 4
130 ± 0
50
144
1145 ± 73
720 ± 16
636 ± 41
424 ± 9
355 ± 23
259 ± 6
50
256
1772 ± 46
1119 ± 12
1043 ± 27
676 ± 7
639 ± 16
421 ± 5
50
1024
6127 ± 665
3062 ± 126
3665 ± 398
1928 ± 79
2291 ± 249
1308 ± 54
50
1296
7076 ± 3487
3562 ± 429
4265 ± 2102
2300 ± 277
2682 ± 1321
1571 ± 189
50
4096
15 716 ± 12 650
8487 ± 1485
9451 ± 7607
5850 ± 1023
6076 ± 4890
4343 ± 760
50
11 664
44 482*
15 700 ± 8478
28 783*
10 176 ± 5495
19 770*
7138 ± 3855
100
16
216 ± 1
160 ± 0
120 ± 0
89 ± 0
67 ± 0
50 ± 0
100
64
924 ± 25
442 ± 1
514 ± 14
255 ± 1
286 ± 8
150 ± 0
100
144
1807 ± 60
860 ± 17
1041 ± 35
508 ± 10
610 ± 20
309 ± 6
100
256
2888 ± 140
1324 ± 10
1706 ± 83
798 ± 6
1037 ± 50
499 ± 4
100
1024
10 586 ± 1830
3812 ± 199
6366 ± 1101
2438 ± 127
3979 ± 688
1662 ± 87
100
1296
9564 ± 6100
4713 ± 855
5752 ± 3668
3058 ± 555
3619 ± 2308
2104 ± 382
100
4096
29 415*
10 319 ± 3344
19 095*
6654 ± 2156
13 116*
4698 ± 1522
100
11 664
59 600*
18 607*
38 667*
12 904*
26 747*
9609*
500
16
594 ± 2
192 ± 1
330 ± 1
107 ± 1
184 ± 1
59 ± 0
500
64
1855 ± 25
556 ± 12
1068 ± 14
320 ± 7
628 ± 8
188 ± 4
500
144
3631 ± 197
1138 ± 77
2141 ± 116
670 ± 46
1306 ± 71
408 ± 28
500
256
6012 ± 200
1879 ± 93
3618 ± 120
1130 ± 56
2266 ± 75
709 ± 35
500
1024
19049 ± 5059
6139 ± 278
11478 ± 3048
3945 ± 179
7186 ± 1909
2660 ± 120
500
1296
19 075 ± 9314
7153 ± 3415
12 370 ± 6041
4656 ± 2223
8499 ± 4150
3198 ± 1527
500
4096
43 688*
14 892*
28 354*
10 460*
19 481*
7800*
500
11 664
65 705*
23 062*
42 738*
16 198*
29 364*
12 080*
* Values with infinite confidence intervals; not used in
this study.
Maximum rainfall intensity and Qp values for 10-, 50-,
100-, and 500-year recurrence intervals during negative (neg) and
positive (pos) multivariate ENSO index (MEI) conditions within the Lower
Colorado River Basin (LCRB) and Upper Colorado River Basin (UCRB). Note that
data are all sampled from time intervals of measurement ≤ 2 h.
Basin
MEI
Area
Intensity (mm h-1)
Qp (m3 s-1)
(km2)
10 yr
50 yr
100 yr
500 yr
10 yr
50 yr
100 yr
500 yr
LCRB
neg
16
39
56
69
77
175
250
305
343
neg
256
31
46
53
69
2206
3251
3741
4877
neg
4096
21
32
43
54
23 856
36 425
48 363
60 930
pos
16
40
64
74
130
179
284
330
576
pos
256
27
38
47
52
1943
2690
3369
3721
pos
4096
13
20*
20*
20*
15 229
22 689*
22 689*
22 689*
UCRB
neg
16
41
98
162
254
186
435
721
1129
neg
256
33
101
155
254
2366
7172
11 012
18 055
neg
4096
22
34
41
82
25 556
39 013
46 422
92 844
pos
16
26
51
56
74
115
225
248
330
pos
256
18
40
51
56
1255
2810
3601
4018
pos
4096
10
26
27*
27*
10 822
30 034
31 044*
31 044*
* Values with infinite confidence intervals; not used in
this study.
Qfd frequency–magnitude–area curves of 10, 50, 100, and
500-year recurrence intervals (RIs) and for wet, medium, and dry conditions
for the Upper Colorado River Basin (UCRB) and the Lower Colorado River
Basin (LCRB). Published FECs (black lines) for the Lower Colorado River Basin
(solid black line) from Enzel et al. (1993) and the US (dashed black line)
from Costa (1987) are also shown.
In general, confidence intervals for Qp values
increase with increasing contributing-area class (Table 1 and Fig. 6). The
large values of the highest contributing-area classes and highest recurrence
intervals show the spatial limitation of the method, meaning that at these
contributing-area classes and recurrence intervals the values are sampled
from the largest ranked value and have infinite confidence intervals. These
values include the 50-, 100-, and 500-year recurrence intervals for the UCRB
and the 100- and 500-year recurrence intervals for the LCRB at the 11 664 km2
contributing-area class. These values also include the 100- and
500-year recurrence intervals for the UCRB and the 500-year recurrence
intervals for the LCRB at the 4096 km2 contributing-area class. Values
with infinite confidence intervals are not included in Fig. 6 due to their
high uncertainties.
Trends in Qfd
Maximum Qfd values (hereafter Qfd), i.e., the largest
values taken for the multiple values calculated for each time interval of
measurement for a given contributing-area class and recurrence interval, were
used to plot FMACs for wet, medium, and dry conditions for both the UCRB and
LCRB (Fig. 7). In general, FMACs for Qfd values follow the
power-law relationship shown in the Qp FMACs until contributing
areas of ∼ 1000 km2, where the curves begin to very slightly
flatten or decrease. As with the Qp values, Qfd values representing
some of the higher recurrence intervals converge to the same value (i.e., the
value corresponding to the highest rainfall intensity for the
contributing-area class) at contributing areas of
≈ 10 000 km2 and the confidence intervals become infinite
(Table 2). This convergence of Qfd values at the largest
contributing areas is due to the reduction in the range of values and the
number of samples from which to calculate the associated values for each
recurrence interval.
In general, the UCRB Qfd FMACs (Fig. 7a, c, and e) are slightly
higher in magnitude and span a larger range of magnitudes than the FMACs for
the LCRB. For both basins, FMACs for the wet, medium, and dry conditions
resulted in the highest, middle, and lowest magnitudes, respectively. This
trend is expected due to the lowering of runoff coefficients and available
water as conditions become drier.
FMACs of Qfd for the LCRB plot below published FECs for the LCRB
and US (Fig. 7b, d, and f) at low contributing areas, but meet and/or exceed
the LCRB FEC for contributing areas above ≈ 1000 and
≈ 100 km2 for dry and wet antecedent-moisture conditions,
respectively. The FMACs for the LCRB do not exceed the US FEC. All of the
FMACs of Qfd for the UCRB exceed the LCRB FEC for wet conditions,
with the FMACs of lower recurrence intervals exceeding the curve at higher
contributing areas than the FMACs of higher recurrence intervals (Fig. 7a).
The 500-year FMAC for wet conditions approximates the US FEC for contributing
areas between ≈ 100 and 1000 km2. These results suggest that
under certain antecedent-moisture conditions, and in basins of certain
contributing areas, the LCRB produces floods that exceed the maximum recorded
floods in the LCRB, and the UCRB produces floods of magnitudes on par with
the maximum recorded floods in the US.
The effects of ENSO on rainfall
Definitive differences in maximum rainfall intensities and
Qp values were found between months with positive vs. months with
negative MEI values (Table 3). For very small contributing areas
(16 km2) in the LCRB, maximum rainfall intensities and
Qp values are similar during negative and positive MEI conditions.
Larger contributing areas (256 and 4096 km2) show higher maximum
rainfall intensities during negative MEI conditions regardless of recurrence
interval. Values of Qp show the same trend as the maximum rainfall
intensity in the LCRB. In the UCRB, maximum rainfall intensities and
Qp values during negative MEI conditions are higher than those
during positive MEI conditions regardless of recurrence interval.
Discussion
Use and accuracy of NEXRAD products
NEXRAD products are widely used as precipitation inputs in rainfall–runoff
modeling studies due to the spatially complete nature of the data necessary
for hydrologic and atmospheric models (Ogden and Julien, 1994; Giannoni et
al., 2003; Kang and Merwade, 2011). In contrast to past studies similar in
scope to this study (Castellarin et al., 2005, 2009; Castellarin, 2007), we
did not use rain-gauge data and only used NEXRAD products to determine the
FMACs for precipitation and flood discharges. We favor NEXRAD products due to
the spatial completeness of the data.
Intuitively, NEXRAD products that are spatially complete and average
precipitation over a 4 km by 4 km area would not be expected to match
rain-gauge data within that area precisely (due to the multi-scale
variability of rainfall), although some studies have tried to address this
discrepancy (Sivapalan and Bloschl, 1998; Johnson et al., 1999). Xie et
al. (2006) studied a semi-arid region in central New Mexico and found that
hourly NEXRAD products overestimated the mean precipitation relative to
rain-gauge data in both monsoon and non-monsoon seasons by upwards of 33 and
55 %, respectively. Overestimation of precipitation has also been noted due
to the range and the tilt angle at which radar reflectivity data are
collected (Smith et al., 1996). Underestimation of precipitation by NEXRAD
products relative to rain-gauge data has also been observed (Smith et al.,
1996; Johnson et al., 1999), however.
Under- and over-estimation of precipitation by NEXRAD products in relation to
rain-gauge data is partly due to the difference in sampling between areal
NEXRAD products and point data from rain gauges and partly due to sampling
errors inherent to both methods. For example, NEXRAD products include
problems such as the use of incorrect Z–R relationships for
high-intensity storms and different types of precipitation, such as snow and
hail (Baeck and Smith, 1998). Also, because of its low reflectivity, snow in
the NEXRAD products is measured as if it were light rain (David Kitzmiller,
personal communication, 10 January 2012). This means the NEXRAD products
likely underestimate snowfall and therefore snowfall is not fully accounted
for in this study. Due to snowfall not being included in this study,
associated snowpack and snowmelt effects were also not accounted for. Rain
gauges can also suffer from a number of measurement errors that usually
result in an underestimation of rainfall (Burton Jr. and Pitt, 2001). In
addition, gridded rainfall data derived from rain gauges are not spatially
complete and therefore must be interpolated between point measurements to
form a spatially complete model of rainfall. It is impossible to discern
which product is more correct due to the differences in measurement
techniques and errors, but by taking both products and combining them into
one, the Stage III NEXRAD precipitation products generate the best
precipitation estimate possible for this study. Moreover, it should be noted
that 100-year flood magnitude predictions based on regression equations have
very large relative error bars (ranging between 37 and 120 % in the
western US; Parrett and Johnson, 2003) and that measurements of past extreme
floods can have significant errors ranging from 25 to 130 %, depending on
the method used (Baker, 1987). As such, even a ∼ 50 % bias in
NEXRAD-product-derived precipitation estimates is on par or smaller than the
uncertainty associated with an analysis of extreme flood events.
As stated previously, the NEXRAD precipitation estimates used here do not
include snowfall and other non-rainfall precipitation types. In this study we
also do not include snowpack information in our flood discharge calculations.
The omission of snowpack is a reasonable assumption for our low-elevation,
warm regions within most of the UCRB and LCRB. However, we acknowledge that
some of our higher elevation areas at higher latitudes may underestimate the
maximum flood discharge by only including rainfall-derived runoff. If the
methodology in this paper were applied to a snowmelt-dominated region,
snowpack would need to be added to accurately estimate the maximum flood
discharge.
Comparison of FMACs to published FECs
FMACs of Qfd exhibit a similar shape and similar overall range in
magnitudes to previously published FECs, derived from stream-gauge and
paleoflood records, for the LCRB and the US (Fig. 7). In general, the FMACs
exceed or match published FECs at larger contributing areas, and are lower
than or on par with published FECs at the smallest contributing areas
(Fig. 7).
All FMACs except the 500-year recurrence-interval curve for the UCRB under
wet conditions are positioned well below the US FEC presented by Costa (1987;
Fig. 7a). The similarity between the 500-year recurrence interval
Qfd FMAC for the UCRB under wet conditions and the US FEC suggests
that the US FEC includes floods of larger recurrence-intervals, which are
similar in magnitude to the 500-year recurrence-interval floods within the
UCRB. The approximation of the US FEC by the 500-year UCRB FMAC is a
significant finding due to the fact that the US FEC includes storms from
other regions of the US with extreme climatic forcings (i.e., hurricanes,
extreme convection storms).
The Qfd FMACs for the LCRB can be directly compared to the FEC for
the LCRB presented by Enzel et al. (1993). At contributing areas smaller than
approximately 100 km2, Qfd FMACs for wet conditions and all
recurrence intervals are positioned below the LCRB FEC, but at larger
contributing areas Qfd FMACs exceed or approximate the LCRB FEC.
Qfd FMACs calculated for medium and dry antecedent conditions show
the same trend, but exceed the LCRB FEC at larger contributing areas
(≥ 1000 km2). This comparison suggests that although the FMACs
overlap the overall range of flood magnitudes of the LCRB FEC, the two
methods are not capturing the same trend for extreme flood discharges and the
LCRB is capable of producing floods larger than those on record.
The difference in the slope of the FMACs, and specifically the exceedance of
the published LCRB FEC, suggests that the two methods are not capturing the
same information. This difference may be due to the difference in how the
data are sourced for each method. FECs are created as regional estimates of
maximum flood discharges and are based on stream-gauging station and
paleoflood data. The FECs are then used to provide flood information for the
region, including ungauged and unstudied drainage basins. FECs are limited
to the number of stream gauges employed by public and private parties and do
not include all basins within a region. In general, FECs may underestimate
maximum floods in larger basins, relative to smaller basins, because there
are a larger number of smaller basins to sample than larger basins. This
sample-size problem introduces bias in the record where flood estimates for
smaller contributing areas may be more correct than estimates for larger
basins. In this study, the regional precipitation information given by the
NEXRAD network is used to form the FMAC, therefore taking advantage of the
entire region and using precipitation data to calculate flood discharges,
rather than directly measuring flood discharges. This sampling scheme allows
for much larger sample sizes for the range of contributing areas, therefore
minimizing the sample bias of the traditional FEC.
This study aimed to introduce the new method of the FMAC and therefore
improve upon the traditional methods of the FEC. By calculating FMACs we
provide frequency and magnitude information of possible flood events for a
given region, in contrast to the FECs that only provide an estimate of the
largest flood on record. This information is vital for planning and
infrastructure decisions and the accurate representation of precipitation and
flooding in design-storm and watershed modeling. In addition, the fact that
the FMACs match the FECs for large (500-year) recurrence intervals and do not
exhibit the same trends suggests that the FMACs are capturing different
samples than the FECs. This indicates that by using the NEXRAD products, the
FMACs may provide a more inclusive flood dataset for a region (especially
ungauged areas) than the traditional stream-gauge records.
Conceptual diagram of the characteristic concave-down shape of the FEC
(observed) shown in comparison to a power-law function between Qp and
contributing area. The “gap” between the observed curve and the predicted
power law is caused by precipitation limitations and mechanisms occurring during
the routing of water over the landscape.
Precipitation controls on the form of the FEC
Qp FMACs were shown to have a strong (average
R2 = 0.93) power-law relationship between Qp
and contributing area for all recurrence intervals. Figure 8 shows a
conceptualized FEC where the concave-down shape is created when the observed
envelope curve diverges from the constant positive power-law relationship
between Qp and contributing area. This diversion creates a
“gap” between the two curves and indicates that flood discharge is not a
simple power-law function of contributing area. Three mechanisms have been
proposed to explain the “gap” and characteristic concave-down shape of
FECs: (1) integrated precipitation (i.e., total precipitation over an area)
is more limited over larger contributing areas compared to smaller
contributing areas (Costa, 1987), (2) a relative decrease in maximum flood
discharges in larger contributing areas due to geomorphic dispersion
(Rodriguez-Iturbe and Valdes, 1979; Rinaldo et al., 1991; Saco and Kumar,
2004), and (3) a relative decrease in maximum flood discharges in larger
basins due to hydrodynamic dispersion (Rinaldo et al., 1991). The first
explanation, proposed by Costa (1987), suggests that there is a limitation
to the size of a storm and the amount of water that a storm can precipitate.
The effect of precipitation limitations may be evidenced by the decreasing
maximum rainfall intensities with increasing contributing area. However, the
strong power-law relationship between Qp and contributing area
for all recurrence intervals indicates that Qp is, in general,
increasing predictably over the range of contributing areas used in this
study. Even if precipitation limitations affect the shape of the curve, this
single hypothesis does not account for all of the concave-down shape of each
FEC suggesting that other mechanisms are important to creating the
characteristic shape. However, it is important to note that the importance
of each mechanism may be different for different locations.
Climate variability in the NEXRAD data
The results from comparing negative and positive MEI conditions in the UCRB
and LCRB are generally consistent with ideas about ENSO and how it affects
precipitation in the western US. In the LCRB, during negative MEI
conditions, small, frequent storms have similar or slightly higher maximum
rainfall intensities and Qp values than during positive MEI
conditions. This similarity between the two conditions may be explained by
the balancing of increased winter moisture during El Niño in the
southwestern US (Hidalgo and Dracup, 2003) and increased summer moisture
through the strengthening of the NAM system and the convective storms it
produces during La Niña conditions (Castro et al., 2001; Grantz et al.,
2007). In general, the strengthening of the NAM may explain the higher
maximum rainfall intensities and Qp values during negative MEI
conditions in the LCRB. Strengthening of the NAM may be due in part to the
large temperature difference between the cool sea surface of the eastern
Pacific Ocean and the hot land surface of the southwestern US and
northwestern Mexico during La Niña conditions. The large temperature
gradient increases winds inland, bringing the moisture associated with the
NAM (Grantz et al., 2007). In the UCRB it is during negative MEI conditions,
where the highest maximum rainfall intensities and Qp values
for all recurrence intervals occur. This suggests that the UCRB is affected
by ENSO much like the northwestern US, where wetter winters are affiliated
with La Niña and not El Niño conditions (Cayan et al., 1999; Hidalgo
and Dracup, 2003). It is important to note that this comparison is of
intensity rates and not total precipitated moisture so the MEI condition
resulting in wetter conditions is not known.
In addition to the ENSO analysis, by investigating previous studies we see
that, along with natural yearly precipitation variability, the 1996 to
2004 water years included many atmospheric river events (Dettinger, 2004;
Dettinger et al., 2011). It is important that these events were included due
to their ability to greatly increase winter precipitation in the UCRB and
LCRB (Rutz and Steenburgh, 2012). Atmospheric river events (sometimes known
as Pineapple Express events) can also be tied to major Pacific climate modes
such as the ENSO (Dettinger, 2004; Dettinger et al., 2011), the Pacific
Decadal Oscillation (PDO; Dettinger, 2004), and the North Pacific Gyre
Oscillation (NPGO; Reheis et al., 2012) in southern California.
Unfortunately, correlations between atmospheric river events are unknown
and/or less clear for the interior western US. However, all three of these
Pacific climate modes shifted during the 9-year study period in
∼ 1998 to 1999 (Reheis et al., 2012), indicating that both positive and
negative conditions of the ENSO, PDO, and NPGO exist in the NEXRAD products
used in this study.
The presence of distinct trends in maximum rainfall-intensity and
Qp values calculated for negative and positive MEI conditions, as
well as the information in the literature on atmospheric river events,
indicates the NEXRAD products used in this study incorporate
circulation-scale weather patterns. In addition, the patterns in maximum
rainfall intensities and Qp values during different MEI conditions
agree with common understanding of the effects of ENSO on the western US and
provide evidence that the data and methods used in this paper to analyze
precipitation are reliable. This analysis shows that the NEXRAD products
worked well in this location and that using radar-derived precipitation
products may be useful for identifying precipitation and climatic trends in
other locations where the FMAC method can be applied.
Conclusions
In this study we present the new FMAC method of calculating precipitation and
flood discharges of a range of recurrence intervals using radar-derived
precipitation estimates combined with a flow-routing algorithm. This method
improves on the traditional FEC by assigning recurrence-interval information
to each value and/or curve. Also, instead of relying on stream-gauge records
of discharge, this method uses up-to-date and spatially complete
radar-derived precipitation estimates (in this case NEXRAD products) to
calculate flood discharges using flow-routing algorithms. This study presents
an alternative data source and method for flood-frequency analysis by
calculating extreme (high recurrence interval) event magnitudes from a large
sample set of magnitudes made possible by sampling the radar-derived
precipitation estimates.
The FMACs for Qp and Qfd for the UCRB were similar to
those produced for the LCRB. In general, all recurrence-interval curves
followed the same general trend, indicating that the mechanisms of
precipitation and flood discharge are similar for the two basins. However,
there were some differences between the two basins. Overall, there were
larger differences between curves of different recurrence intervals for the
UCRB than the LCRB, suggesting a larger range in maximum rainfall
intensities, and therefore Qp and Qfd, in the UCRB
relative to the LCRB. For both the UCRB and LCRB the 50- and 100-year
recurrence-interval curves for all precipitation and discharge FMACs were the
most similar. This similarity may mean that although historical discharge
records are short, having a 50-year record may not underestimate the 100-year
flood as much as one might expect. Also, for Qp and Qfd,
low recurrence-interval values were slightly higher in the LCRB than in the
UCRB. This relationship was opposite for high recurrence-interval values.
This likely points to a general hydroclimatic difference between the two
basins, with the LCRB receiving high-intensity storms annually due to the NAM
and the UCRB receiving more intense and rarer winter frontal storms.
Power-law relationships between maximum rainfall intensity, Qp, and
contributing area were also found in this study. Maximum rainfall intensities
decreased as a power-law function of contributing area with an average
exponent of -0.18 ± 0.06 for all recurrence intervals.
Qp values for all recurrence intervals increased as a power-law
function of contributing area with an exponent of approximately
0.82 ± 0.06 on average. Based on the constant power-law relationship
between Qp and contributing area, the “gap” or characteristic
concave-down shape of published FECs is likely not caused by precipitation
limitations.
In general, the FMACs of Qfd calculated in this study are lower
than, and exceed, the published FECs for the LCRB at lower and higher
contributing areas. All FMACs of Qfd were positioned well below the
US FECs except the UCRB 500-year FMAC, which approximated the US FECs during
wet antecedent-moisture conditions. All FMACs of Qfd for all
moisture conditions in the LCRB closely approximated the same magnitudes as
the published LCRB FEC, but exceeded it for larger contributing areas. The
higher estimates of flood discharges at larger contributing areas may be the
result of the difference of sampling methods and are likely not erroneous and
may be proved true by future events.
Lastly, the approximately 9 years of NEXRAD products were found to be a good
representation of climate in the CRB. This conclusion was made based on
differences in precipitation between positive and negative ENSO conditions in
both the UCRB and LCRB and additional data found in the literature. In
general, the UCRB was found to have a hydroclimatic regime much like that of
the northwestern US, where El Niño conditions result in lower maximum
rainfall intensities and amounts and La Niña conditions result in higher
maximum rainfall intensities. The LCRB showed a more complex trend with
similar maximum rainfall intensities for both El Niño and La Niña
conditions.
Here this method is applied to the UCRB and LCRB in the southwestern US, but
could be applied to other regions of the US and the world with variable
climate and storm types where radar-derived precipitation estimates are
available. In this study we used set values for contributing area, drainage
basin shape, time intervals of measurement, and recurrence intervals that can
be changed based on the focus of future studies. However, it is also
important to note that a number of assumptions were made in this study that
simplified our analysis; most importantly, (1) space-for-time substitution,
or regionalization, was used to increase the number of samples and assumed
that observations were independent and sampled from the same distribution;
(2) it was assumed that the time period length and the spatial and temporal
sampling scales were sufficient to create a representative sample from the
observations; and (3) it was assumed that similar flood-generating and
flow-routing mechanisms (and related variables such as runoff coefficients)
were present in each basin regardless of size or location. These assumptions
allowed us to form and apply the methods described here to our study area,
but may not apply to all areas. Other variables such as snowpack, elevation,
land use, and climate change that were not included in this study should be
explored in conjunction with this methodology to better understand controls
on precipitation and flooding. The absence of these elements from the method
here may limit the application of this method to other locations.