Accurate flood prediction is an important tool for risk management and hydraulic works design on a watershed scale. The objective of this study was to calibrate and validate 24 linear and non-linear regression models, using only upstream data to estimate real-time downstream flooding. Four critical downstream estimation points in the Mataquito and Maule river basins located in central Chile were selected to estimate peak flows using data from one, two, or three upstream stations. More than one thousand paper-based storm hydrographs were manually analyzed for rainfall events that occurred between 1999 and 2006, in order to determine the best models for predicting downstream peak flow. The Peak Flow Index (I Q P) (defined as the quotient between upstream and downstream data) and the Transit Times (T T) between upstream and downstream points were also obtained and analyzed for each river basin. The Coefficients of Determination (R 2), the Standard Error of the Estimate (SEE), and the Bland-Altman test (ACBA) were used to calibrate and validate the best selected model at each basin. Despite the high variability observed in peak flow data, the developed models were able to accurately estimate downstream peak flows using only upstream flow data.
Floods are among the most powerful forces on the Earth [
Flood prediction has become an important social and economic component of risk management [
The study was implemented in two rivers located in the first-order administrative region of Maule in central Chile (34˚41'S and 36˚33'S latitudes) (see
Mataquito River is located in the northern part of the region, and drains an area of approximately 6190 km2. The river begins 12 km east of the city of Curicó at the confluence of two tributaries with headwaters in the Andes Mountains: the Teno River and the Lontué River, which drain the northern and southern parts of the basin respectively [
Instantaneous flow data from 13 stream gauging stations distributed up and downstream of both rivers was first used to make quantity and quality control (
In collaboration with DGA, the estimation points were mainly selected because the high recurrence of flooding. According to this criterion, the Mataquito in Licantén station (located near the town of Licantén) was selected for the Mataquito Basin. In fact, during May 2008, a flood of the Mataquito River resulted in the flooding of 70% of the town [
Once the estimation points were chosen, hydrographs were constructed from every identified flood event (storm event), and the estimation points were paired with one, two or three upstream stations. Only instantaneous flows data (m3∙sec−1) containing the date and time for the every upstream and downstream station were selected. Finally a total of 1000 flood events between 1999 and 2006 were chosen for further analysis.
In order to better understand the relationship between the peak flows recorded at the upstream and downstream stations for each basin, the Peak Flow Index (IQP) was created to describe the quotient between peak flow values at the downstream estimation point and the upstream stations. The index can be calculated as:
Map number | Station | River | Drainage area (km²) | Maximum instantaneous flow (m3/s) | Period | |
---|---|---|---|---|---|---|
Minimum | Maximum | |||||
1 | Mataquito in Licantén | Mataquito | 6190 | 16.49 | 4195.49 | 2000-2006 |
2 | Teno after Claro River | Teno | 1188 | 16.48 | 1155.88 | 2000-2006 |
3 | Palos before Colorado River | Palos | 514 | 10.50 | 518.57 | 2002-2006 |
4 | Colorado before Palos River | Colorado | 942 | 14.28 | 689.64 | 2000-2006 |
5 | Claro in Camarico | Claro | 684 | 1.10 | 1193.46 | 1999-2006 |
6 | Maule in Forel | Maule | 21,048 | 30.40 | 17212.94 | 1999-2006 |
7 | Claro in Rauquén | Claro | 3021 | 33.52 | 2210.01 | 1999-2006 |
8 | Lircay in Las Rastras bridge | Lircay | 375 | 1.26 | 1067.92 | 1999-2006 |
9 | Maule in Longitudinal | Lircay | 5800 | 6.51 | 2577.5 | 1999-2006 |
10 | Loncomilla in Las Brisas | Loncomilla | 10,046 | 9.20 | 7623.44 | 1999-2006 |
11 | Loncomilla in Bodega | Loncomilla | 7245 | 2.45 | 4227.8 | 1999-2006 |
12 | Ancoa in El Morro | Ancoa | 194 | 8.87 | 1080.89 | 1999-2006 |
13 | Achibueno in La Recova | Achibueno | 892 | 3.15 | 2436.1 | 1999-2006 |
Estimation point | Map number and gage stations | |
---|---|---|
Downstream | Upstream | |
EP1 | 1. Mataquito in Licantén | 2. Teno after Claro River |
3. Palos before Colorado River | ||
4. Colorado before Palos River | ||
EP2 | 6. Maule in Forel | 7. Claro in Rauquén |
10. Loncomilla in Las Brisas | ||
9. Maule in Longitudinal | ||
EP3 | 7. Claro in Rauquén | 5. Claro in Camarico |
8. Lircay in Las Rastras bridge | ||
EP4 | 10. Loncomilla in Las Brisas | 11. Loncomilla in Bodega |
13. Achibueno in La Recova | ||
12. Ancoa in El Morro |
where,
· IQP is the Peak Flow Index,
· QP(Downstream) is the peak flow recorded downstream,
· QPi(Upstream) is the upstream peak flow recorded at the upstream station
In the case of analyzing multiple upstream stations, the IQP was established by defining the denominator as the sum of the peak flow values for the “n” upstream stations. As the index is expressed as a quotient, it quantifies how many times the recorded peak flow increased from the upstream station to the downstream estimation point (
Both linear and non-linear models were used to determine if downstream peak flows were correlated with those values registered in upstream stations. The mathematical expression to define dependent and independent variables in each regression model was:
where,
· DSQP is the dependent variable considered as the peak flow for the downstream estimation point,
· USQP is the independent variable considered as the peak flow of the upstream station(s).
Within this context, 24 linear and non-linear mathematical models considering one, two, or three upstream stations were used to estimate peak flows at each estimation point (
For the model calibration stage, the quantity of data used to adjust every model varied according to the downstream estimation points and upstream stations being considered, due to the differences in the number of flood events for any given station. All storm events of the year were considered and no distinction was made regarding whether flood events occurred in the dry or rainy seasons. The Coefficient of Determination (R2) and the Standard Error of Estimate (SEE) were used to calibrate and validate the models and determine the best mathematical relation to estimate downstream peak flows (
where,
· y are the observed values,
· ŷ are the estimated values,
· n is the maximum value in the series,
· t corresponds to each storm-event considered in the analysis, where
Simple linear regression | Multiple linear regression | ||||
---|---|---|---|---|---|
Dependent variable | Independent variable | Dependent variable | Independent variable | ||
MAQP | = ƒ (COQP) | MAQP | = ƒ (COQP y PAQP) | ||
= ƒ (PAQP) | = ƒ (COQP y TEQP) | ||||
= ƒ (TEQP) | = ƒ (PAQP y TEQP) | ||||
MFQP | = ƒ (CRQP) | MFQP | = ƒ (CRQP y LBRQP) | ||
= ƒ (LBRQP) | = ƒ (CRQP y MLQP) | ||||
= ƒ (MLQP) | = ƒ (LBRQP y MLQP) | ||||
CRQP | = ƒ (CRQP) | CRQP | = ƒ (CCQP y LRQP) | ||
= ƒ (LBRQP) | |||||
LBRQP | = ƒ (LBOQP) | LBRQP | = ƒ (LBOQP y ACHQP) | ||
= ƒ (ACHQP) | = ƒ (LBOQP y ANCQP) | ||||
= ƒ (ANCQP) | = ƒ (ACHQP y ANCQP) | ||||
Note: MA is Mataquito in Licantén; MF is Maule en Forel; CR is Claro in Rauquén; LBR is Loncomilla in Las Brisas; ML is Maule in Longitudinal; CC is Claro in Camarico; LR is Lircay in Las Rastras bridge; LBO is Loncomilla in La Bodega; ACH is Achibueno in La Recova; ANC is Ancoa in El Morro; CO is Colorado before Palos; PA is Palos before Colorado; TE is Teno after Claro; QP is Peak flow.
Test | Expression |
---|---|
Determination Coefficient (R2) | |
Standard Error of Estimate (SEE) |
Finally, the best three models as determined by the error measurements results (R2 and SEE) were validated by the Bland-Altman test (ACBA), which evaluates the degree to which the data obtained through direct observation differ from the theoretical response obtained from a model, and also determines if these differences are acceptable on a hydro-meteorological basis [
The model in which the observed and the estimated data have DP values closest to zero in absolute terms is considered the more accurate. In the case of an equal or minimal difference between the DP values, then the model with the smallest SD value and the narrowest limits of agreement is determined to be more accurate [
The fluctuation of streamflow and precipitation in the Aconcagua basin in Chile were studied by [
The Peak Flow Index (IQP) accurately determined the quotient increase for downstream peak flows in both the Mataquito and Maule Basins. For the downstream estimation point in the Mataquito Basin with one, two, or three upstream stations, the maximum IQP values varied from 3.1 to 17.4; for the estimation point in the Maule River the IQP values varied from 1.3 to 17.6 for one, two, or three upstream stations, which verified that IQP behaves similarly for the two basins (
On the other hand, IQP values corresponding to the sum of the peak flows for two or three upstream stations more closely approximated the observed value downstream than those values calculated with one upstream station, i.e. the IQP value was closer to 1 when two upstream stations were considered. Similarly, the IQP values obtained using three upstream stations suggest that the IQP increases are very similar to the increases in the observed data, as IQP values equal to 1 have less variability is observed in the data. Initially was thought that IQP values could be correlated with the peak flow values for the upstream stations or the downstream estimation points. However, the R2 results do not suggest a correlation. For some downstream estimation points, index values both increase and decrease with an increase in peak flows, while for others no definitive trend exists either way. However, in the case of the upstream stations a slight trend exists between the index and peak flow values, as the index values tend to slightly decrease with larger peak flows.
Peak flow Transit Time (TT) analyses that are reliable are imperative for flood prediction and mitigation. Some authors have stated that transit times may be mostly a function of distance between two stream gauging stations; however, other factors exist that might also influence the peak flow TT, e.g. channel morphology and vegetative cover, among others. In this context, [
In the absence of sufficient observations of flood extent, flooding risk areas are usually identified using numerical hydraulic models. This requires a dynamic approach to represent transient storage effects [
On the other hand, higher average peak flows downstream were related to shorter transit times, based on the relationship between average peak flows at the downstream estimation points and average transit time at the Mataquito and Maule rivers. Nevertheless, the linear correlation between peak flow transit time and peak flow values at each estimation point in both rivers, whether high or low, was very weak. Finally, as indicated previously, R2 and SEE were used as fitting measures to select the three best models for each estimation point; R2 values superior to 0.70 were found for the majority of the models. In the subsequent validation phase, the majority of the R2 values obtained indicated that, in general, the calibrated models accurately represented the variation in the data of the downstream estimation points. In only a few cases SEE and ACBA did not agree with the R2 results obtained. Finally, based on SEE and ACBA results, one model was chosen for based on one, two, or three upstream stations (
Number of upstream stations | Estimation points (Downstream stations) | Peak flow (m3∙sec−1) | Upstream stations | Peak flow (m3∙sec−1) | IQP | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Average | Min | Max | Average | Min | Max | Average | Min | Max | |||
One upstream station | Mataquito in Licanten | 696.8 | 25.1 | 3603.7 | Colorado before Palos | 164.3 | 14.3 | 678.2 | 4.3 | 1.1 | 11.9 |
737.4 | 87.6 | 3603.7 | Palos before Colorado | 126.2 | 27.0 | 518.6 | 5.9 | 1.7 | 17.4 | ||
872.3 | 87.6 | 3603.7 | Teno after Claro | 306.4 | 34.5 | 1014.1 | 2.8 | 1.1 | 11.9 | ||
Maule in Forel | 2549.1 | 282.5 | 17212.9 | Claro in Rauquén | 621.9 | 61.1 | 2210.0 | 4.2 | 1.6 | 8.5 | |
3070.9 | 282.5 | 17212.9 | Loncomilla in Las Brisas | 1813.1 | 90.2 | 7623.4 | 1.8 | 1.1 | 4.0 | ||
2617.5 | 197.4 | 17212.9 | Maule in longitudinal | 548.3 | 165.6 | 2667.4 | 4.6 | 1.0 | 16.5 | ||
Claro in Rauquén | 621.9 | 61.1 | 2210.0 | Claro in Camarico | 183.9 | 17.5 | 1193.5 | 4.0 | 1.8 | 8.8 | |
611.9 | 61.9 | 2210.0 | Lircay in las Rastras bridge | 188.7 | 11.6 | 1067.9 | 3.9 | 1.4 | 8.2 | ||
Loncomilla en las Brisas | 1803.6 | 210.9 | 5140.2 | Loncomilla in la bodega | 1031.1 | 76.4 | 3205.0 | 1.9 | 1.4 | 2.8 | |
1543.8 | 190.6 | 6992.4 | Achibueno in la Recova | 408.5 | 27.6 | 2436.1 | 4.4 | 1.4 | 9.3 | ||
1627.6 | 144.6 | 6992.4 | Ancoa in El Morro | 192.9 | 12.5 | 940.5 | 8.7 | 2.2 | 17.6 | ||
Two upstream stations | Mataquito in Licanten | 746.9 | 87.6 | 3603.7 | Palos before Colorado | 127.8 | 27.0 | 518.6 | 2.4 | 0.7 | 6.3 |
Colorado before Palos | 189.4 | 32.1 | 678.2 | ||||||||
Mataquito in Licanten | 300.5 | 87.7 | 3603.7 | Colorado before Palos | 199.5 | 42. 5 | 678.2 | 1.6 | 0.5 | 3.8 | |
Teno after Claro | 300.5 | 76.6 | 1014.1 | ||||||||
Mataquito in Licanten | 9005.3 | 87.6 | 3603.7 | Teno after Claro | 148.3 | 41.3 | 518.6 | 1.8 | 0.6 | 3.4 | |
Palos before Colorado | 335.4 | 34.5 | 1014.1 | ||||||||
Maule in Forel | 2459.8 | 282.5 | 15752.2 | Claro in Rauquén | 574.9 | 65.1 | 2210.0 | 1.2 | 0.7 | 2.3 | |
Loncomilla in las Brisas | 1541.0 | 90.2 | 6692.4 | ||||||||
Maule in Forel | 3351.8 | 463.5 | 16665.6 | Claro in Rauquén | 706.1 | 75.7 | 1674.6 | 2.3 | 1.0 | 4.6 | |
Maule in longitudinal | 602.8 | 189.5 | 1964.3 | ||||||||
Maule in Forel | 2976.1 | 593.1 | 15752.2 | Loncomilla in las Brisas | 1728.0 | 334.9 | 6992.4 | 1.2 | 1.1 | 1.7 | |
Maule in longitudinal | 551.0 | 189.5 | 1128.0 | ||||||||
Claro in Raquen | 434.4 | 61.1 | 1376.8 | Claro in Camarico | 132.5 | 27.0 | 539.7 | 1.8 | 1.1 | 3.5 | |
Lircay in las Rastras bridge | 126.2 | 19.7 | 365.7 | ||||||||
Loncomilla in las Brisas | 1797.7 | 220.7 | 6992.4 | Loncomilla in la bodega | 405.5 | 54.4 | 1971.6 | 2.3 | 1.0 | 4.4 | |
Achibueno in la Recova | 454.1 | 58.2 | 1323.6 | ||||||||
Loncomilla in las Brisas | 1793.1 | 210.9 | 5140.2 | Loncomilla in la bodega | 461.7 | 76.4 | 2462.2 | 1.6 | 1.2 | 1.9 | |
Ancoa in El Morro | 707.5 | 12.5 | 3205.0 | ||||||||
Loncomilla in las Brisas | 1669.3 | 220.7 | 5140.2 | Achibueno in la Recova | 408.4 | 60.4 | 1323.6 | 2.2 | 1.1 | 5.7 | |
Ancoa in El Morro | 636.8 | 12.5 | 3205.0 | ||||||||
Three upstream stations | Mataquito in Licanten | 926.6 | 87.6 | 3603.7 | Palos before Colorado | 151.2 | 45.9 | 518.6 | 1.2 | 0.4 | 3.1 |
Colorado before Palos | 222.9 | 45.3 | 678.2 | ||||||||
Teno after Claro | 342.8 | 34.5 | 1014.1 | ||||||||
Maule in Forel | 3062.2 | 593.1 | 15752.2 | Claro inRaquen | 725.6 | 106.5 | 2210.0 | 0.9 | 0.7 | 1.3 | |
Loncomilla in las Brisas | 1768.2 | 214.2 | 6992.4 | ||||||||
Maule in longitudinal | 570.3 | 189.5 | 2577.5 | ||||||||
Loncomilla in las Brisas | 1979.9 | 220.7 | 5140.2 | Loncomilla in la bodega | 442.8 | 91.3 | 1971.6 | 1.1 | 1.0 | 1.4 | |
Achibueno in la Recova | 491.3 | 60.4 | 1323.6 | ||||||||
Ancoa in El Morro | 842.0 | 12.5 | 3205.0 |
Estimation point | Upstream stations | Transit time (hours) | ||
---|---|---|---|---|
Min | Average | Max | ||
1. Mataquito in Licantén | Colorado before Palos | 9.92 | 20.02 | 33.55 |
Palos before Colorado | 2.93 | 20.45 | 42.93 | |
Teno after Claro | 12.5 | 21.37 | 45.50 | |
6. Maule in Forel | Claro in Rauquén | 1.35 | 9.30 | 29.02 |
Loncomilla in Las Brisas | 3.00 | 7.52 | 17.00 | |
Maule in Longitudinal | 3.98 | 11.62 | 25.32 | |
7. Claro in Rauquén | Claro in Camarico | 1.65 | 5.85 | 13.65 |
Lircay in Las Rastras bridge | 5.65 | 11.03 | 20.65 | |
10. Loncomilla in Las Brisas | Loncomilla in La Bodega | 0.70 | 3.86 | 19.70 |
Achibueno in La Recova | 1.98 | 11.27 | 25.28 | |
Ancoa in El Morro | 1.57 | 11.09 | 20.57 |
Estimation point | Model | R2 | SEE | ACBA (DP) |
---|---|---|---|---|
1. Mataquito in Licantén | 1) | 0.82 | 339.6 | −6.73 |
2) | 0.76 | 421.9 | −4.73 | |
3) | 0.84 | 337.8 | −174.3 | |
2. Maule in Forel | 1) | 0.91 | 729.7 | −12.22 |
2) | 0.96 | 222.3 | −56.59 | |
3) | 0.92 | 353.9 | −58.09 | |
3. Claro in Rauquén | 1) | 0.95 | 166.3 | 17.69 |
2) | 0.76 | 185.7 | 10.69 | |
4. Loncomilla in Las Brisas | 1) | 0.96 | 246.4 | −45.16 |
2) | 0.99 | 116.6 | 49.94 | |
3) | 0.80 | 206.1 | 59.80 |
Note: MA is Mataquito in Licantén; MF is Maule enForel; CR is Claro in Rauquén; LBR is Loncomilla in Las Brisas; ML is Maule in Longitudinal; CC is Claro in Camarico; LR is Lircay in Las Rastras bridge; LBO is Loncomilla in La Bodega; ACH is Achibueno in La Recova; ANC is Ancoa in El Morro; CO is Colorado before Palos; PA is Palos before Colorado; TE is Teno after Claro; QP is Peak flow.
The Peak Flow Index (IQP) accurately represented the relationship between upstream and downstream flows. The transit time (TT) was lower in Maule basin, despite its greater extent (which could imply more traveling time for the flooding wave). In general, for both Maule and Mataquito rivers various mathematical models were able to accurately estimate downstream peak flows using upstream data. However, there were points at which the linear models were more accurate than the more complex models using upstream information in the analyzed basins. Finally, it is important to point out that it is possible to use only upstream data to predict peak flows downstream. This is a good and relatively inexpensive approach for modeling flooding in real time, and it is useful when—as is often the case—extensive data needed for more complex models are unavailable. Furthermore, this simple streamflow-data approach can be used by national agencies like DGA to improve flood risk mitigation measures.
The authors would like to express their gratitude to the Dirección General de Aguas de Chile (DGA) and a directorate of the Ministry of Public Works for providing the paper-based dataset used in this study.