Overview of estimation methodology

The following flow diagram outlines the methods used within Qube for estimating the natural annual flow duration curve and mean flow from catchment characteristics.

Step 1: Estimate catchment characteristics and runoff derived annual mean flow.
Step 2: Use euclidean distance in catchment characteristics to identify a pool of nearest gauged catchments.
Step 3: Calculate natural annual flow statistics, adjusting for local data (and lakes in Scotland).
Step 4: Calculate natural monthly flow statistics.

Step 1: Estimate catchment characteristics and runoff derived annual mean flow

Using a catchment boundary for the site of interest, the relevant physical and catchment characteristics are extracted from digital grids to form the input into the models, or procedures, for estimating the flow statistics. This is illustrated in Figure 1.

graph TD A[Draw catchment boundary] --> B[Estimate catchment area] A --> C[Estimate catchment average annual runoff] A --> D[Estimate fractional extents of HOST classes] B --> E(Calculate annual runoff derived mean flow) C --> E D --> F(ROI: Use euclidean distance to identify nearest gauged catchments) C -.-> F

Figure 1 : Estimate catchment characteristics and runoff derived annual mean flow (note the dependence on runoff in the ROI is only for England, Wales and Scotland, whereas a zero weight is applied to runoff in Northern Ireland)

Mean flow is also then estimated using a 1km² resolution grid of average annual runoff for the selected period of record, as follows:

$Mean Flow (MF) = {RUNOFF \times AREA \times 10^6 \over 10^3 \times 3600 \times 24 \times 365}$

Step 2: Identify nearest gauged catchments

The model used to estimate the flow duration curve within Qube is a Region Of Influence (ROI) regionalisation approach as described in Holmes et. al. (2002a). The ROI approach to regionalisation is an approach that is used both within Qube and the Flood Estimation handbook (FEH) statistical methods (in which pools of self similar catchments are derived based upon catchment area, SAAR and BFIHOST).

Within England, Wales and Scotland the ROI approach includes a HOST based measure of hydrogeological similarity and a climatic similarity measure (catchment average annual runoff, RUNOFF). Within Northern Ireland, due to the relatively homogenous rainfall patterns the inclusion of a climatic similarity measure was not found to improve the predictive performance of the model.

The ROI approach develops an estimate of a flow statistic or hydrologic parameter at an ungauged ‘target’ catchment from observed values of that flow statistic or hydrologic parameter made at number of gauged catchments which are considered to be ‘similar’ to the ungauged catchment. Similarity is measured by catchment characteristics that can be obtained for any ungauged catchment in the UK.

The ROI based model seeks to reduce the variability of the dependent variable within the dataset by reducing it to a much smaller "region" of catchments that are ‘similar’ to the target catchment. In application to a catchment, the methods can be summarised as the following stages:

Catchment similarity is assessed by calculating a weighted Euclidean distance, with respect to the HOST variables and the logarithm (Base 10) of RUNOFF (LOGRUNOFF), between the target catchment and the catchments within the data pool.
A "region" of donor catchments is formed around the target catchment by ranking all of the catchments in the data pool by their weighted Euclidean distance and selecting the five catchments (three in IoM) that are “closest” to the target catchment.

The Euclidean distance measure is calculated as:

$de_{it} = \sum_{m=1}^{M}{W_m(X_{mi}-X_{mt})^2}$

where: $de_{it}$ is the weighted Euclidean distance from the target catchment, $t$ , to catchment, $i$ , from the similar data pool; $W_m$ is the weight applied to catchment characteristic, $m$ , and $X_{mi}$ is the value of catchment characteristic, $m$ , for catchment, $i$ .

The catchment characteristics, $X_m$ , used are the 30 fractional extents of the HOST classes within a catchment, which will vary between zero and unity, and LOGRUNOFF (base 10). The use of differing weights for individual HOST classes reflect the fact that relatively small proportions of certain HOST classes, strongly influence the variability of the flows within a catchment.

Step 3: Calculate natural annual flow statistics

The method for calculating the natural annual flow statistics is illustrated in Figure 2 and is described in the following sections.

graph TD E(Annual runoff derived mean flow) --> H F(ROI nearest gauged catchments) --> G[Calculate standardised annual FDC] G --Lake adjustment in Scotland--> H[Calculate ROI Annual FDC] E --if local data is available--> X[Calculate LDG adjusted annual mean flow] H --if local data is available--> Y[Calculate LDG adjusted annual FDC]

Figure 2 : Overall estimation procedure for natural annual flow statistics

Standardised annual flow duration curve

An assumption central to the estimation procedures in Qube is that when low flows are expressed as a percentage of the long term mean flow (standardised), the dependencies on the climatic variability across the country and the effect of catchment area are minimised. As a result, the estimation of standardised flow duration curves is largely dependent on the hydro-geological and soils characteristics of the catchment. For the UK the Hydrology of Soil Types (HOST) classification of soils (Boorman et. al, 1995) is used to represent these characteristics in Qube.

A standardised annual flow duration curve is estimated for the ungauged site by taking a weighted combination of the standardised flow duration curves for the “region” of donor gauged catchments formed about the ungauged catchment from Step 2.

The process for estimating the standardised annual flow duration curve for the target site from those within the "region" involves weighting each of the donor catchment flow duration curves by the inverse absolute Euclidean distance of that donor to the target catchment. Thus, greater weight is given to catchments that are more similar in HOST and RUNOFF characteristics to the target catchment. This is expressed mathematically as:

$QP_{EST_t}(x) \% MF= \sum_{i=1}^{n} \Bigg[ { \Big( {{1 \over {de_{i}}^{0.5}}\Big)} \over \sum_{j=1}^{n} \Big( {1 \over {de_{j}}^{0.5} } \Big) } \Bigg] \times QP_{OBS}(x) \%MF_i$

where: $QP_{EST_t}(x) \% MF$ is the estimate of the flow for the target catchment, $t$ , at exceedance percentile $P(x)$ ; $QP_{OBS}(x)\%MF_i$ is the observed value of QP(x) for the $i$ th source catchment in the region of donor catchments and dei is the weighted Euclidean distance of the $i$ th catchment from the target catchment, $t$ , in HOST and LOGRUNOFF space.

Lake adjustment

The storage associated with natural lakes within a catchment tends to maintain (increase) base flow and hence low flows and attenuate high flows. Qube incorporates a procedure for estimating the influence of natural lakes and lochs in Scotland on the estimation of flow duration statistics. See Lake Adjustment for further details.

Improving the annual FDC and mean flow using local data

The UK regional models within Qube do not explicitly take into account local hydrometric data in the context of the estimation of long term mean flow and only partially within the estimation of the flow duration curve. Therefore the estimation of natural annual FDC and mean flow in catchments with local gauged data can be improved through the explicit incorporation of that gauged data.

Where upstream and/or downstream local data gauges (LDG) are available, the LDG adjusted annual mean flow and flow duration curve are calculated using the methods described in Local Data Algorithm.

Step 4: Calculate natural monthly flow statistics

The driver for developing methods for estimating monthly statistics within Qube's predecessors was the estimation of artificial influences which are incorporated as a monthly influence profile (Young et al. 2003). This allows both the seasonal variations in flows and influences to be taken into account. The method for calculating the natural monthly flow statistics is illustrated in Figure 3 and is described in the following sections, based upon Holmes et al., (2002c).

graph TD Z(ROI nearest gauged catchments) --> F[Calculate monthly % runoff volumes] Z --> G[Calculate standardised monthly FDCs] E("(LDG adjusted) annual mean flow") --> H[Calculate monthly mean flow] F --> H H --> I[Calculate monthly FDCs] G --> I Y("(LDG adjusted) annual FDC") --> K["Calibrate monthly FDCs to the (LDG adjusted) annual FDC"] I --> K

Figure 3 : Overall estimation procedure for natural monthly flow statistics

Monthly Mean Flow

The monthly flow duration curve is estimated as a percentage of the monthly mean flow. As discussed previously, the magnitude of the variation in UK flow regimes is dominated by catchment hydrogeology. Expressing these flow statistics in dimensionless form removes the overlying effects of the scale of catchment hydrological processes and enables statistical relationships between these low flow statistics and catchment characteristics to be developed. Estimates of the monthly mean flows are required so these estimated low flow statistics can be expressed in m³/s.

Considering the distribution of the total volume of annual runoff between the months of the year, the distribution is clearly a function of the magnitude and seasonal distribution of rainfall, the strong seasonality of evaporation demand and the presence of soil moisture deficits that suppress the generation of runoff. Furthermore the distribution is strongly influenced by catchment hydrogeology. For example, the lowest flows in catchments with groundwater will typically occur in the autumn when groundwater levels are at their lowest, whilst the lowest flows in low storage impermeable catchments will typically occur in the summer months when evaporation demand is highest. The distribution of annual runoff within dry impermeable catchments will tend to be more skewed towards the winter months than in wet impermeable catchments. This is a function of the enhanced role that soil moisture deficits will play in suppressing summer runoff within drier catchments.

The stages required for the derivation of the monthly mean flows are illustrated in Figure 3. The percentage of the annual runoff volume that occurs within each month, termed the Monthly Runoff Volume ( $MRV$ ), is estimated using the “region” of donor gauged catchments formed about the ungauged catchment from Step 2. The monthly runoff volumes for the ungauged catchment are then estimated as a weighted average of those for the gauged catchments forming the "region":

$MRV\%_{tj}= \sum_{i=1}^{n} \Bigg[ { \Big( {{1 \over {d_{St}}}\Big)} \over \sum_{k=1}^{n} \Big( {1 \over d_{St} } \Big) } \Bigg] \times MRV\%OBS_{ij}$

where; $MRV_{tj}$ = the estimate of $MRV$ for month $j$ for target catchment $t$ ; $MRV\%OBS_{ij}$ = the observed $MRV$ for month $j$ for the $i$ th catchment in the region of $n$ catchments closest to the target catchment; $d_{St}$ = the modulus of the difference in LOGRUNOFF between the target catchment $t$ and the $i$ th catchment.

The $MRV$ is then rescaled to Monthly Mean Flow ( $MMF$ ) using the (LDG adjusted) annual mean flow from Step 3. Remembering that the $MRV$ for a month is the percentage of the total annual runoff volume occurring within that month, the $MMF$ for the month ( $m$ ) is estimated using:

$MMF_m = MRV\%_m \times MF_{Ann} \Bigg({360 \over {100 \times DAYS_m}}\Bigg)$

where; $DAYS_m$ is standardised to 30 days in each month.

Standardised monthly flow duration curves

The method for the estimation of the standardised monthly flow duration curves is based on the region of influence approach using “nearest neighbour” gauged catchments from Step 2.

In the UK, periods of high flows are typically experienced in the winter months and periods of low flows in the summer months, when the evaporation demand is highest. The hydrogeology of a catchment, as with the annual FDC, also has a strong influence of the variability of flows within a month.

The "region" of catchments identified using the ROI approach for estimating the annual flow duration curve within an ungauged catchment are also used to estimate the monthly flow duration curves for the ungauged catchment. The monthly flow duration curves for the ungauged site are estimated by taking a weighted average of the observed monthly flow duration curves (standardised by the relevant monthly mean flows). In the weighting procedure the observed monthly FDC points for the individual catchments in regions are weighted by absolute distance in weighted HOST and LOGRUNOFF space. This procedure is exactly the same as the direct estimation of the annual flow duration curve.

Calibrated monthly flow duration curves

The natural (LDG adjusted) annual flow duration curve can be reconstructed from the 12 standardised monthly flow duration curves and the monthly mean flow estimates. It is important that the reconstructed annual flow duration curve is identical to the natural (LDG adjusted) annual flow duration curve for consistency.

The area under a flow duration curve is the equal to the total volume of water over the period in question. The sum of the areas under the 12 estimated monthly flow duration curves (once they have been re-scaled by the appropriate estimate of monthly mean flow) should equal the area under the estimated annual flow duration curve. To ensure that this is the case a fitting routine is used.

The monthly flow duration curves (with units of m³/s) are each disaggregated into the 101 flows at equally distributed percentiles [0.1 ,1 … 99, 99.9]. The resultant 1212 flow values are placed within a pool whilst retaining a flag against each flow value to identify the month from which the flows are derived. A composite flow duration curve is then derived from the pooled values by ranking the 1212 flows in order of decreasing size. The annual probability of exceedence, $P(x)$ , is then calculated for each of the 1212 flows using:

$P(x) = x {100 \over 1212 + 1}$

where $x$ is the rank

The 1212 points are then compared with the natural (LDG adjusted) annual flow duration curve. The values within the composite curve are adjusted to ensure that the two curves coincide and the monthly curves are then reconstructed from the adjusted composite curve values. To achieve this the annual flow duration curve (defined by 101 points) has to be interpolated using a log normal distribution to each of the probabilities calculated for the composite curve.