A novel variogram-based approach for determining the influence of catchment characteristics on river flow dynamics
Abstract:
IntroductionDue to between-catchment similarities, potentially resulting from natural self organisation or the co-evolution of climate, soils, vegetation and topography (Sivapalan, 2006), it should be possible to classify catchments into groups or regions sharing similar properties. This is based on the general assumption that some level of organisation and therefore predictability in catchment function (i.e. the translation of catchment input into river flow) exists (Bloschl et al., 2013). As catchment function is controlled, to a large degree, by catchment characteristics it should be possible to classify catchments, based on their characteristics, so that catchments with similar precipitation to flow relationships are in the same group or region. Being able to classify catchments using this method can provide an indication of the precipitation to flow relationship in un-gauged catchments.
As catchment charactersitics influence the precipitation to river flow relationship it stands to reason that they will also influence a river's response to temporal changes in precipitation. Published studies of hydrological change typically reveal significant heterogeneities in the response of catchments, even in regions with similar climate regimes. This is likely to be a product of variations in catchment properties, but standard change detection methodologies have a number of limitations which prevent the role of catchment function in hydrological change being fully elucidated. This is a gap in research, as there is a pressing need to understand how future changes in climate variability are likely to manifest themselves in catchments with different properties. Examples of limitations with current change detection methods include: calculating indicators, which only analyse changes in a pre-determined aspect of the flow regime. Thus, bias is introduced by selecting a specific aspect of the flow regime. Secondly, there are monotonic trend tests which are designed to detect incremental increases in the mean. Complex system interactions allow a change in one aspect (e.g. precipitation) to result in a non-linear change (e.g. river flow), which monotonic trend tests are not designed to capture. Finally, there are techniques such as wavelets. However, it is hard to relate the change in spectral shape to the hydrological regime (Smith et al, 1998).
This paper describes, and demonstrates the application of, a novel change detection method, based on characterising temporal dependence using a variogram-based technique, that aims to overcome these limitations and enables the impact that catchment characteristics have on moderating temporal changes in precipitation to be identified.
This paper is split into two sections. Section 1: investigation employs a catchment classification method to determine whether the variogram approach is sensitive to different types of catchments, with precipitation to flow relationships. Section 2 uses variograms to investigate temporal changes in river flow. The section examines the amount of temporal variability in each variogram parameter which can be explained by precipitation characteristics and identifies catchment characteristics which influence how river flow responds to temporal changes in precipitation.
Data
Catchment selection: 116 Near-natural UK 'benchmark' and 49 validation catchments with no more than 5% missing data between 1980 and 2012 were selected. These catchments only have modest net impacts from artificial influences (such as reservoirs or sewage discharge).
Catchment characteristics: Five elevation characteristics derived from the Integrated Hydrological Digital Terrain Model (Morris and Flavin, 1990). Land cover (grouped into four categories) derived from the Land Cover Map 2000. Nine physiographic characteristics from the Flood Estimation Handbook (FEH). Four different Hydrology Of Soils Types (HOST) soil types, based on the depth to gleyed layer (reduced from 29 HOST classes) and seven different hydrologically important rock types calculated from the 1:625000 scale digital hydrogeological map of the UK were identified.
Explanatory variables: Seventeen different precipitation characteristics (mean, standard deviation, percentiles (x5), max length above / below thresholds (x3), average length of time above / below thresholds (x3) and gradient of precipitation (x4)) were used. In addition the Soil Moisture Deficit (SMD), calculated from Met Office Rainfall and Evaporation Calculation System was included.
Background to variograms
The core of the new methodology is the (semi-)variogram. (Semi-)variograms are used to represent the temporal dependence structure in daily river flow rime series and hence capture differences in the precipitation to flow relationship between catchments and throughout time. Some brief background on variograms and how they are claculated is sketched here:
Firstly, daily river flows are standardised and deseasonalised to allow comparison between sites and through the year. Broadly the steps are: 1) in-fill the river flow data; 2) take logarithms, to create a near normal distribution 3) remove seasonality; 4) standardised the flow data for each catchment by deducting the mean and dividing by the standard deviation of the time series.
A semi-variogram was then calculated using the average squared difference between all pairs of values which are separated by the corresponding time lag. A variogram model was fitted to the semi-variogram in order to the variogram parameters: the nugget, which is the y intercept, represents a combination of measurement error and sub-daily variability; the sill is defined as the semi-variance where the gradient of the variogram is zero (the limit of temporal dependence). The range is the lag distance at which the variogram reaches the sill value. In addition to these variogram characteristics two more semi-variogram properties were calculated: the 3 and 20 Day Average Semi-Variance (DASV) (average of the first 3 and 20 points on the variogram respectively). These are added to capture short (3 days) and medium term (20 days) variability in the river flow time series.
Section 1: Identifying which catchment characteristics influence the precipitation to flow relationship.
The aim of this analysis was to classify the UK benchmark catchments based on their variogram properties and relate the clusters to catchment characteristics. This is split into two sections:
A)Hierarchical clustering based on Ward's minimum variance method was applied to a Euclidean squared distance matrix, calculated using the whole of the semi-variogram to maximise the information going into the clustering algorithm. Dendrograms and agglomeration plots were used to decide the number of clusters, four was deemed to be the optimum number.
B) Discriminant analysis was used to investigate how many of the 116 benchmark and 49 validation catchments could be clustered without the flow data. Discriminant analysis identifies whether the mean of a catchment characteristic differs between clusters, if it does then the catchment characteristic is included to help distinguish between the clusters. Discriminant models were created for the benchmark catchments and blind validation was carried out using the validation catchments.
The difference in the temporal dependence structure between the clusters (and the location of the clusters) is illustrated in Figure 1 with increases in range, and decreases in the sill and nugget from clusters 1 to 4. An increasing range indicates a longer dependency in the daily mean river flow, while a decreasing sill is caused by less temporally autocorrelated variability throughout the 30 year record. The key catchment characteristics in distinguishing between the clusters were found to be: elevation (particularly values describing the highest pasts of the catchment); average drainage path slope; base flow index; soil type; the amount of arable land and aquifer type. Cluster 4 was found to be dominated by catchments which overlay highly productive fractured aquifers causing a vastly different precipitation to flow relationship than catchments in the other three clusters. Moreover it was found that over 70% of un-gauged catchments could be correctly clustered using 5 catchment characteristics (slope, 3 different soil characteristics and the percentage of arable land).
Section 2: do the catchment characteristics influence how a river responds to changes in precipitation?
This section builds on section 1 (which established that catchment characteristics control the shape of the variogram), and aims to examine the over-arching question of how catchments modulate changes in precipitation. Variograms were created using 5 year overlapping moving windows to investigate how the temporal dependence structure changes through time. Two methods are used to investigate temporal changes in the temporal dependence structure:
A)Analysing the influence of precipitation: Multiple Linear Regression (MLR) was undertaken in order to give an indication of how much temporal variability in the variogram parameters could be explained by a combination of different precipitation characteristics (also calculated over 5 year moving windows). As precipitation characteristics are correlated with each other, a procedure which penalises extra model parameters is required. Information Theory (IT) base on Akaike's Information Criterion (AIC) was used to analyse how much information is added by each characteristic. For each catchment the model with the lowest AIC score is used to obtain the R2 value.
B) The influence of catchment characteristics:The amount of temporal change in the variograms will, at least partly, be caused by how much the catchment characteristics are moderating the changes. The influence of the catchment characteristics in moderating temporal changes in river flow will be calculated by looking at the correlation between the amount of the variability explained by the precipitation characteristics and the catchment characteristics.
The correlation between each variogram parameter and the precipitation characteristics are shown in Table 1, identifying that different precipitation characteristics are driving changes in each of the variogram parameters.
Table .1 Percentage of catchments which have a significant (at the 95% CI) correlation between the precipitation and variogram characteristics and the average R2 value for the catchments with the percentage of catchments with significant correlations in brackets.
Precipitation characteristic |
Sill |
Range |
3 DASV |
20 DASV |
Mean |
42 (0.22) |
28 (-0.51) |
32 (0.46) |
59 (0.66) |
Standard deviation |
51 (0.45) |
33 (-0.28) |
41 (0.53) |
71 (0.64) |
Average length of wet period (above 1mm) |
51 (-0.23) |
51 (-0.41) |
41 (-0.20) |
49 (0.12) |
Average length of dry period (below 1mm) |
41 (0.14) |
45 (0.51) |
37 (-0.10) |
52 (-0.29) |
25th percentile |
33 (0.29) |
23 (0.15) |
28 (-0.10) |
32 (0.20) |
Median |
33 (-0.15) |
32 (-0.52) |
26 (0.42) |
49 (0.50) |
75th percentile |
29 (0.01) |
28 (-0.48) |
29 (0.30) |
54 (0.48) |
90th percentile |
40 (0.27) |
30 (-0.38) |
44 (0.40) |
49 (0.59) |
95th percentile |
41 (0.34) |
26 (-0.42) |
34 (0.51) |
54 (0.60) |
DJF gradient |
29 (-0.37) |
57 (-0.44) |
44 (-0.51) |
24 (0.02) |
MAM gradient |
30 (0.04) |
23 (-0.18) |
36 (0.29) |
28 (0.36) |
JJA gradient |
26 (0.27) |
45 (0.39) |
37 (0.42) |
11 (-0.25) |
SON gradient |
29 (-0.42) |
60 (-0.47) |
30 (-0.33) |
17 (0.24) |
Soil moisture deficit |
27 (0.17) |
40 (0.57) |
68 (-0.59) |
50 (-0.53) |
MLR was carried out using the combinations of the precipitation characteristics (Table 1) with the lowest AIC score (calculated from the IT analysis). The precipitation characteristics explain a large amount of the variability in the variogram parameters (on average: 60, 78, 79 and 86% for the sill, range, 3 DASV and 20 DASV respectively) although the amount explained did vary between catchments. The correlation between the amount explained and the catchments characteristics (for the catchment characteristics with significant correlations) are shown in Table 2.
Table 2. Correlation between the percentage of explained variance in the variogram parameters after regressing against precipitation and the catchment characteristics for values which are significant at the 95% CI.
Precipitation characteristic |
Sill |
Range |
3 DASV |
20 DASV |
Area |
 |
|
 |
 |
Arable |
-0.3 |
-0.46 |
|
|
Grassland |
0.21 |
0.31 |
|
|
Median elevation |
0.36 |
 |
||
ax elevation |
0.27 |
0.42 |
|
 |
Base flow index (HOST) |
-0.40 |
-0.36 |
|
 |
PROPWET |
0.24 |
0.50 |
 |
|
Average drainage path slope |
0.21 |
0.43 |
 |
-0.22 |
Highly productive fractured rock |
-0.55 |
-0.43 |
|
 |
Low productivity fractured rock |
0.34 |
0.37 |
 |
 |
Free draining soil |
-0.3 |
|
 |
|
Peat soil |
0.21 |
0.38 |
|
 |
Table 2 identifies that the short and medium term variability (3 and 20 DASV) are mainly driven by precipitation and the influence of the catchment characteristics is limited. However, the amount of variability in the sill and the range (total amount of variability and the length of temporal dependence) is highly dependent on the catchment characteristics.
Discussion
This study has developed a novel suite of approaches to catchment classification and change detection based on the concept of the variogram. Variograms are a very powerful tool for use in hydrology. A variogram is able to capture the temporal dependence structure of daily river flow which is driven by catchment characteristics. The catchment characteristics which drive (or are highly correlated with characteristics which drive) the precipitation-to-flow relationship influence either the pathway from precipitation to discharge and/or the amount of storage in a catchment. As the catchment characteristics drive the shape of the variogram, over 70% of the benchmark and validation catchments could be correctly clustered without the use of river flow data. The method therefore has potential application for prediction of temporal dependence structure (and thus, broadly, catchment function) in un-gauged catchments. The variogram parameters represent different aspects of the river flow regime hence, have different relationships with the precipitation characteristics. Combinations of precipitation characteristics explain 60, 78, 79 and 86% of the variability for the sill, range, 3 DASV and 20 DASV respectively. The unexplained proportion is caused by either: other precipitation characteristics not included, land management change or the catchment characteristics moderating the change in precipitation. Changes in catchments which have fast precipitation to flow relationships generally can be explained by precipitation whereas catchments with high infiltration and storage are not as well explained by precipitation. This could be because the catchments with more infiltration and storage have more memory and hence mitigate short term precipitation anomalies but exacerbate anomalies which occur over longer time periods.
As demonstrated in this paper a (semi-)variogram has a wide variety of implications. Firstly, clustering catchments into groups with similar precipitation to flow relationships provides knowledge about which catchment characteristics are driving the precipitation to flow relationship. Furthermore it enables un-gauged catchments to be clustered, giving information about their precipitation to flow relationship. Secondly, creating moving window variograms is a new method for detecting change in hydrological regimes. Finally, variograms can be used to investigate how catchment characteristics moderate how a river responds to changes in precipitation; providing understanding into how future climate variability may impact different catchments.
References
Bloschl G, Sivapalan M, Wagener T, Viglione A, Savenije H. 2013. Runoff Prediction Ungauged Basins, Synthesis across Processes, Places and Scales Cambridge University press.
Morris DG, Flavin RW. 1990. A Digital Terrain Model for Hydrology. Proc 4th International
Symposium on Spatial Data Handling. 1: 250-262.
Sivapalan M. 2006. Pattern, Process and Function: Elements of a Unified Theory of Hydrology
at the Catchment Scale. DOI: 10.1002/0470848944.hsa012.
Smith, L., Turcotte, D., Isacks, B., 1998. Stream flow characterization and feature detection using a discrete wavelet transform. Hydrol. Process. 12, 233-249.