Part 3 – Regionalization of rainfall-runoff models – direct methods
8 Joining spatial proximity and physiographic similarity
In this chapter, we present and evaluate two approaches methods allowing a joint use of spatial proximity and physiographic similarity:
the intersection-based method, based on the assumption that good donor catchments are likely to be, at the same time, similar and geographically close to the ungauged catchment of interest. Thus, the best donors will belong to the intersection of the two ensembles ;
the union-based method, based on the assumption that the two approaches may identify good donors independently. Thus donors will be best identified by the union of the two ensembles.
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In the previous sections we covered two regionalization methods based on pure physiographic similarity. As already said, our attention has been focused on such methods first because of their expected robustness.
Another reason why an hydrologist should appreciate physical similarity more than spatial proximity is that it is less "black box": it does not really provide any outlook of the hydrological processes that dominate the catchments of a certain region, but at least it gives a possibility for a careful, indirect, rough interpretation.
However, spatial proximity should not totally be dismissed. On one side, there are situations (for instance very dense gauging networks) where its performances might be superior to those of approaches relying on physiographic measures. On the other, as Figure 18 shows, it is to some extent complementary to physiographic similarity.
In Figure 18 we can see a grey dashed line representing the performances of a pure spatial proximity regionalization on our dataset (using four donors), a black dashed line representing the performance of the backwards-sorting physiographic similarity covered in section 7.3, and a black solid line. Such black line represent an ideal (non-existent) method that would, for each ungauged catchment, be able to decide whether in that particular case spatial proximity would give a more accurate guess than physiographic similarity, or vice-versa. Its performances are clearly superior to the other two methods used alone. Of course, the reader should be aware that this example was constructed by "cheating" and is only used to show the complementarity of the two original regionalization approaches.
While we do not expect that a realistic method could come close to the performances of the
"ideal" case, we think that Figure 18 clearly shows the interest of investigating methods that combine some degree of physiographic similarity (as a way to ensure robustness and for its
"informative" value) with some degree of spatial proximity (whose only value is an eventual increase in performance). The next paragraphs will cover two simple propositions of how such a method could be constructed.
Figure 18: Performances of spatial proximity and physiographic similarity methods (dashed grey and dashed black lines) confronted with the performance of an ideal method perfectly combining the strenghts of the two approaches (solid black line)
8.2 An intersection-based method
8.2.1 description
The idea behind this method is that good donor catchments are likely to be, at the same time, similar and geographically close to the ungaged we are looking at.
To select donors that are close and similar, we proceeded as follows:
- The number of donors to be used was set. Let us say for the sake of this example that we wanted to use 10 catchments.
- Each time we considered a catchment as ungauged, the remaining ones were ranked in two lists of donors. The first was ranked according to geographical distance, the second was ranked for physiographic similarity (as in the backwards-sorting method shown in section 7.3
- We looked at the closest 10 catchments and at the most similar 10 catchments. If these two groups contained the same 10 stations, these would be the retained donors.
- In case we didn't have the same 10 catchments in the two groups, we would progressively increase the size of the two pools of candidates: for instance, the 11 closest one and the 11 most similar.
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- We would look at the intersection of the two pooling groups (i.e. catchments appearing both in the group of the closest and in the group of the most similar ones).
If 10 catchments were to be found in such an intersection, we would stop and retain these 10. If not, we would keep increasing the sizes of the two candidate groups until 10 candidates could be found.
8.2.2 Results
Figure 19 is a summary of the performance of the intersection regionalization method.
The optimal number of donors is six, with which a median C2M of 0.58 (equivalent to a NSE of 0.73) is obtained.
The performance gain, compared to pure physiographic similarity, is quite small (from 0.574 to 0.578). However, we actually notice a performance decrease in the "metrological desert"
robustness test. While this result should be expected as an effect of bringing spatial proximity into the regionalization method, it is quite strong (pure similarity already works better when the closest catchment is more than 20 km away) and makes the proposed "intersection"
method a poor choice.
Figure 19: Combining spatial proximity and physical similarity, results of the intersection regionalization method. Top left, median efficiency per number of donor catchments used. Top right, distribution of efficiencies compared to a random selection of donors (dashed line) and calibrated model (solid grey line). Bottom left, performance in a "metrological desert" situation.
8.3 A union-based method
8.3.1 Description
This approach is based on the idea that –for our dataset- pure spatial proximity and pure physiographic similarity will identify a certain number of "good" donor catchments when used alone.
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Pure spatial proximity works best (on our database) with four donor catchments, backwards-sorting physiographic similarity works best with 11 donors. We then simply pasted the two donor lists, obtaining a group of 15 donors. Notice that, when a catchment is both in the 4 closest and in the 11 most similars, it is counted twice.
8.3.2 Results
Figure 20 shows two charts about the performance of the union-based regionalization method: the distribution of the efficiencies obtained on the catchments we tested as ungauged, and the median performance in the "metrological desert" robustness test. In comparison to the previously treated proposals, we did not test different numbers of donors.
The median performance obtained is 0.58 in C2M, or an NSE of 0.74. This result is only marginally better than the intersection-based regionalization: however, the robustness of this approach seems to be much more satisfying. Pure physiographic similarity would only have a clear advantage on catchments who don't have any donor closer than 180 km, while when at least one donor closer than 100 km is available, the union-based method is superior.
Figure 20: Combining spatial proximity and physical similarity, results of the union regionalization method. Left, distribution of efficiencies compared to a random selection of donors (dashed line) and calibrated model (solid grey line). Right, performance in a
"metrological desert" situation.
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8.4 Comparison of the tested regionalization approaches
Figure 21 and Figure 22 allow a comparison of the performances of the tested direct regionalization approaches. All approaches perform very similarly for the better modelled catchments, with the most noticeable differences being concentrated between empirical frequencies 0.1 and 0.4.
Overall the "union" combination of spatial proximity and physical similarity performs best, even if it is not very far from the other three tested methods, and constitutes a marginal improvement over a backwards-sorting based similarity approach, despite a theoretically much bigger margin for improvement.
Figure 21: Distribution of the performances of the tested direct regionalizations, compared to two benchmarks: random donor selection (dotted grey line), calibrated model (solid grey line)
All methods appear to have similar robustness, with the possible exception of the
"intersection" one (which probably relies too much on spatial proximity). In all cases, a noticeable improvement over spatial proximity can be noticed.
Figure 22: Comparison of the performances of the tested direct regionalizations under the
"metrological desert" robustness test
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