Method for forecasting the flood risk in a tropical country

The definition of flood risk is a concept adapted to multiple and different empirically estimated variables. The definition of flood risk (FR) has evolved from the beginning of the 21st century to the present. The FR was a diffuse variable inferred indirectly through hazard and vulnerability maps. From the second decade of the 21st century, the flood risk was weighted by combining the hazard and vulnerability variables, which were represented on the map (Farias et al. 2020a).

The flood hazard is determined by the variables that cause flooding. Two main causes or mechanisms are attributed to flooding events, atmospheric and geotechnical (Sene 2008). The atmospheric mechanisms comprise frontal depressions, thunderstorms, monsoon, tropical cyclones (hurricanes and storms), snowmelts, and ice jams. The geotechnical mechanisms include tsunamis and debris flow.

The variables associated with these mechanisms and linked to the flood hazard are depth, velocity, extent and debris. The vulnerability to flooding is characterized by the location and number of people likewise, the properties under risk, whose variables constitute the basis for the community-level plans, and organizational structures (Farias et al. 2020a).

Until the present, the focus for forecasting has been the flood event. The flood forecasting models are based on observations of river levels and rainfall higher in a catchment (for river flooding), or of tidal levels, wave heights, wind speed and other parameters (for coastal flooding) (Schumann 2011).

Process-based methods for modeling river levels, flows and velocities, on the floodplain can include (Sene 2008): one-dimensional models for the main river channel, with projection of levels onto the floodplain, or separate pathways for main channel and floodplain flows; one-dimensional models including floodplain pathways represented via spill units, compartments and/or cells; two-dimensional models of the floodplain using digital terrain models based on mass conservation, or including momentum effects; fully two- or three-dimensional models of the floodplain incorporating features on the floodplain such as buildings, embankments, and gulleys among others, and possibly urban drainage networks.

In recent studies, Farias et al. (2020b) developed a method for estimating flood risk, supported by the physical process of rainfall–runoff determined by applying the US Soil Conservation Service (US-SCS) to obtain the effective precipitation (EP). The US-SCS-EP is linked to the exceedance probability used as an input factor to the equation for the water-control design due to the occurrence of extreme hydrological events (Chow et al. 1988; Guevara & Cartaya 2004; Farias et al. 2020a). The US-SCS-EP is estimated based on water balance in the soil surface adjusted by a correction factor considering the previously stored water in the soil matrix. The US-SCS-EP method tends to produce homogeneous values due to the exclusion of the effects of variation in terrain elevation, slope and relief.

In this study, a novel method of forecasting flood risk is proposed, called CIHAM-UC-FFR. The method is based on the rainfall–runoff process. The CIHAM-UC-FFR method consists of three stages: (1) calibration and validation for the effective precipitation model, named the CIHAM-UC-EP model, (2) calibration of forecasting models for components of the CIHAM-UC-EP model, (3) proposed model for forecasting of gridded flood risk, named CIHAM-UC-FR. The CIHAM-UC-EP model has a mathematical structure derived from a conceptual model obtained by applying the principle of mass conservation combined with the principle of Fick's law. The CIHAM-UC-FR model is a stochastic equation based on the exceedance probability of the forecast effective precipitation.

The main difference of the proposed method when comparing with the historical flood risk definition is the relationship with a variable, such as the effective precipitation, that represents the flow on the watershed surface, considering its use and coverage such as urban, agricultural, rangeland, vegetation, water bodies, and degraded soil, among others. In the preceding definitions, the flood risk is indirectly associated with the hydrological variables through the flooding hazard. Moreover, the hydrological variables are connected, in a restricted way, to the flooding caused by a river in its adjacent areas. In the proposed method, the river is part of the elements that cause flooding, however it can be generated by surface flow on the ground in paved areas (urban) or on impervious soil.

In CIHAM-UC-EP, effective precipitation is estimated by combining the principle of conservation of masses together with a first-order law established by Fick to explain the temporal dynamics of the rate of substance consumption associated with the dissipation gradient of the concentration of substances as their molecules move in a predominant direction within an aqueous medium. In the proposed equation, this principle is applied to explain the dynamics of the rate of attenuation of water depth following a specific flow path by estimating an attenuation coefficient that represents the spatio-temporal dynamics. The attenuation coefficient implies loss by infiltration, bed roughness, and evaporation. Furthermore, the proposed equation incorporates as an input variable the downstream flow length (Olivera et al. 1998) calculating distance along a flow path considering the elevation changes of the ground. These dynamic elements to estimate the effective precipitation constitute a notable difference with respect to the method proposed in studies such as Farias et al. (2020a).

With respect to the forecasting model, the previous discussion can be extended to this aspect, due to the traditional variable the flooding is expressed in terms of flow-depth, flow-velocity or flow-extent. In that sense, in this study, the forecast is on the flood risk variable defined by the exceedance probability of effective precipitation being modified by a parameter depending on the expected life of the hydraulic structures that could be proposed as a water-control measure.

In this study, the theoretical formulation for forecasting of flood risk in a tropical basin comprises as a main process the rainfall–runoff renamed as effective precipitation, which is estimated based on a spatially distributed conceptual model. The forecast effective precipitation is obtained from time series models. Subsequently, flood risk forecasting is supported by the exceedance probability of forecast effective precipitation in the study area.

The effective precipitation modeling is framed into the commonly used models of rainfall–runoff process (Márquez & Guevara 2013). The formulation of the model is developed by combining the fundamental equations that explain the motion of fluids and the advection–diffusion process for mass transport supported by the principle of mass conservation applied to an open channel.

In this study, one of the novelties consists of proposing a mathematical expression for explaining the unstable flow due to the attenuation of the flow depth provided by an external source as the precipitation associated with the term uh (Equation (5)) (Márquez & Guevara 2013). This term is an approximate expression for the discharge entering an open channel of unit width being decreased in a proportion specified by an attenuation coefficient that influences the flow depth consumption rate along the channel in the downstream direction x⁠.

Lumped models are proposed for forecasting of the components of the equation for unsteady flow in an open channel (CIHAM-UC-EP). The components involved are the precipitation and the parameters and t. ARIMA (AutoRegressive, Integrated, Moving Average) models (Box & Jenkins 1970) are used to evaluate the adjustment of the time series of components (Márquez & Guevara 2016).

The probabilistic functions that have been tested for adjusting to hydrological variables are the following (Guevara & Cartaya 2004): normal, exponential, extreme values, Weilbull, among others.

The proposed method to forecast spatial distribution of flood risk in a basin is developed in three stages: (1) calibration and validation of the effective precipitation model, (2) calibration of forecast models for components of the EP-CIHAM UC model and (3) the proposed model for forecasting of gridded flood risk (CIHAM-UC-FR).

The proposed model of effective precipitation (Equation (8)) is calibrated using four types of data: (a) time series of maximum precipitation for a duration of six hours for the period 1980–2018, (b) satellite images for the period 1979–2018, (c) map of soil types, (d) digital elevation model.

Time series of maximum precipitation for a duration of six hours: time series of precipitation were collected from three sources during three periods (Figure 1):

• (a)

  Hydrology and Meteorology Directorate (HMD) – Ministry of Environment and Natural Resources (MENR) collected precipitation data through an automatic hydrometeorological observation network at national scale during the period 1980–2000, using weighing–recording gauges with a strip-chart data logger. The MENR-HMD yielded records of maximum precipitation (P_max) for durations of 1 h, 3 h, 6 h, 12 h, 9 h and 24 h. In this study, the P_max time series from 48 gauges located in eight states is used (Figure 1) (Márquez et al. 2018; Farias et al. 2020b).

• (b)

  Service of Hydrology, Oceanography, Meteorology and Nautical Mapping (SHOMNM) – Bolivarian Army (BM) – Ministry for Defense (MD) generated P_max records applying the method of MENR-HMD for airports of Venezuela, and the P_max data are extracted from eleven gauges (Figure 1) (Infante et al. 2008).

• (c)

  National Institute of Meteorology and Hydrology (NIMH) – Ministry of Internal Relations, Justice and Peace (MIRJP) collected precipitation data by a telemetering automatic hydrometeorological observation network at national scale during the period 2015–2018, using tipping-bucket rain gauges with a telecommunication platform consisting of Ultra-High Frequency (UHF) and Extreme High Frequency (EHF) transmitters (Meza 2007). The records for precipitation measured each five minutes were obtained from 18 rain gauges (Figure 1) (Márquez et al. 2018; Farias et al. 2020b).

Satellite images: 20 Landsat images were acquired for the period 1979–2018, corresponding to a single scene, where the Pao River basin is contained. The temporal series of images from the four Landsat satellites were grouped according to the type of Landsat satellite (Márquez et al. 2019): (1) L2 MSS (1979), (2) L4TM (1987, 1988, and 1989), (2) L5TM (1984, 1986, 1990, 1991, 1996, 1997, 1998, 1999, 2000 and 2001), (3) L7ETM (2002 and 2003) and (4) L8OLI (2014, 2015, 2016, 2017, and 2018).

Raster map of soil type for the study unit was obtained by vectorization and subsequent rasterization processes from the source map made for Venezuela, setting the cell size to 30 m (Farias et al. 2020a).

Digital elevation model (DEM): four ASTER (Advance Spaceborne Thermal Emission and Reflection Radiometer) Global DEMs covering the study area, the Pao River basin, were acquired from the Earthexplorer website, identified as ASTGDEMV2_0N09W068, ASTGDEMV2_0N09W069, ASTGDEMV2_0N10W068, ASTGDEMV2_0N10W069.

The data were acquired and estimated applying three methods that allow the spatial distribution to be obtained (Figure 2): (a) maximum precipitation, (b) land use and land cover factor, and (c) flow length.

Raster map of maximum precipitation (Figure 2): the monthly spatial maximum precipitation for a duration of 6 h prediction was obtained applying Equation (10) constituted by two terms. The first term, the global mean of precipitation, was calculated based on the number of rain gauges within the study area (Figure 1). The second term is the spatially correlated stochastic part of the variation predicted using an experimental variogram fitted to a J-Bessel function (Márquez et al. 2019). A total of 456 monthly maximum rainfall raster maps were estimated for a duration of 6 h in the study area (Figure 3(a)), the cell size being 500 m.

Raster map of land use and land cover factor (Figure 2): the raster map of land use and land cover (LULC) factor is defined as the curve number (CN) (Soil Conservation Service 1986), which is obtained by superimposing the soil type and land use and land cover (LULC) raster maps. The soils of the Pao River basin have predominantly particles from fine and finest, being classified as class C (moderately high runoff) and D (high runoff) according to the hydrologic classification system of US-SCS. The land cover/land use raster maps in the study area were estimated using the Landsat satellites finding the following results (Farias et al. 2018; Márquez et al. 2018, 2019; Farias et al. 2020b): (1) urban (5%), (2) agricultural (0.6%), (3) rangeland (26%), (4) vegetation (40.5%), (5) degraded soil (24.4%) and (6) water bodies (1.5%).

Raster map of downstream flow length (Figure 2): the hydrologically corrected terrain data (DEM) were used with a cell size of 30 m as input (Fleming & Doan 2013). The flow length was derived in Geographical Information System (GIS). The downstream flow length to the watershed outlet was calculated as the difference between the flow length value of the cell and the flow length value of its corresponding outlet cell (Olivera et al. 1998).

The calibration of the CIHAM-UC effective precipitation (CIHAM-UC-EP) model was made using statistical software to solve the non-linear equation by applying a steepest descent algorithm for the fitting of the parameters. The 456 mean values of the two fitted parameters, and t, are represented in Figure 3(b) and 3(c).

This was carried out by comparing raster maps of the CIHAM-UC-EP with the US-SCS-EP models (Farias et al. 2020a).

The stage of forecasting of CIHAM-UC-EP implies two sequential steps: (a) fitting of forecasting models for components of CIHAM-UC-EP and (b) forecasting of the components of CIHAM-UC-EP for future events.

• (a)

  Fitting of forecasting models for components of CIHAM-UC-EP (Figure 3): five ARIMA models are evaluated for mathematical fitting to the components of the CIHAM-UC-EP model (⁠ extracted from the processing of 456 maps for the period 1980–2018. The forecasting models for the components of the CIHAM-UC-EP model are chosen by comparing the results of applying the statistical errors.

• (b)

  Forecasting of the components of CIHAM-UC-EP for future events (Figure 3): the forecasting models are applied to obtain the results for a future date. The forecast components are substituted in the CIHAM-UC-EP model to predict the forecast raster map of CIHAM-UC-EP.

The generation of the forecast flood risk map involves two steps (Figure 2): (a) fitting of a probabilistic distribution model of the forecast CIHAM-UC-EP, (b) determining of the exceedance probability of the forecasted CIHAM-UC-EP grid and (c) generation of the forecast flood risk map:

• (a)

  Fitting of probabilistic distribution model of forecast CIHAM-UC-EP: 13 probabilistic distribution models were tested for adjusting to the CIHAM-UC-EP selecting exponential function, and the best fitting to the forecasted CIHAM-UC-EP was log likelihood.

• (b)

  Determining of exceedance probability of forecast CIHAM-UC-EP grid: the curve of exceedance probability of the forecast CIHAM-UC-EP dataset was obtained from statistical software. The raster map of exceedance probability of the forecast CIHAM-UC-EP was produced by applying a conversion function from the polygon layer in GIS.

• (c)

  Proposed model for forecasting of gridded flood risk (CIHAM-UC-FR): the generation of the forecast flood risk map was carried out by applying Equation (11) used as a unique input to the raster map of exceedance probability of forecast CIHAM-UC-EP setting four values to the parameter n from 1 to 100 (Figure 4).

The results of applying the method for the forecasting of the spatial distribution of flood risk in the study area are presented according to three stages (Figure 2): (1) calibration and validation of the effective precipitation model, (2) forecasting of effective precipitation and (3) estimating of the spatial distribution of forecast flood risk.

The trend cycle (TC) is estimated by smoothing the P_max time series data represented by the continuous trace (Figure 3(a)), using a moving average, which is distanced by observed data in a difference equal to the seasonality varying between 4 and 15 during the period 1980–2018.

The midpoint in the trend cycle is at 4,150 m² h⁻¹ (Figure 3(b)), and the flow attenuation parameter is associated with the infiltration process in natural channels and terrain. The rate of infiltration can be estimated from parameter assuming that the absorption of flow along the flow length occurs in areas with natural coverage such as forests and degraded soil. In the study area, the flow length varies between 1,586 and 3,984 m, with the result that the infiltration rate varies from 2.6 m h⁻¹ to 1 m h⁻¹ (2,610 to 1,000 mm h⁻¹).

Tests of infiltration rate were made taking measures in time intervals from 2 and 137 minutes ranging in values less than 400 mm h⁻¹ for a cumulative probability of 75%, and atypical values between 600 and 2,400 mm h⁻¹ (Márquez 2009; Márquez & Guevara 2011a; Guevara & Márquez 2012). By comparing, the field measurements of infiltration rate with the values found evaluating the magnitude of coefficient for a determined flow length, similar results have been found.

The quasi-physically measured concentration time is estimated by comparing values by applying Manning's equation referred to as one of the main components of concentration time using GIS commands (Feldman 2000). The estimated velocities through Manning's equation range between 1 and 7 m s⁻¹ for a cumulative frequency of 79%. Therefore, the channel travel times resulting ranged between 1.5 and 7 h. The values meaning a fraction of concentration time are a reference that represent an order of 1:1.5 greater than the trend cycle of the flow attenuation time (Figure 3(c)). The flow attenuation time should run in the initial stage of precipitation event occurrence in natural areas of the watershed because of the descent trend of the exponential pattern associated with the infiltration rate process, explained as due to the aerated spaces in the soil matrix and the water saturation subsequent of micro-conduits after an elapsed time greater than one hour.

The results of the US-SCS-EP model found by Farias et al. (2020a) give a spatial distribution where EP has a small deviation between the maximum and minimum values because of the equation structure being a ratio based on water balance, overestimating EP by comparing with the spatial distribution obtained by applying the CIHAM-UC-EP-model on the same study area.

The errors in the time series of ⁠, and t in the period 1980–2000 are similar for the calibration and validation stages, indicating that the model passes five statistical tests related to excessive runs up and down, above and below median, autocorrelation, difference in mean and variance. The influence of residual autocorrelations is non-significant due to the model describing all of the dynamic structure in a time series.

For the time series during the period 1980–2018, there is high variance in values measured during the period 2015–2018 (Figure 3(a)), which explains the difference between errors in the calibration and validation stages (Table 1). The influence of the AR(1) and MA(1) coefficients on the independent variable for the time series 1980–2000 is lower in the order of two times than those values for the time series 1980–2000. The future prediction predominantly reproduces the observed pattern in during the period 1980–2015.

The prediction intervals of mean values for ⁠, and t in the period 1980–2000 are made based on the 68% confidence level that includes a standard deviation from the mean value, which represents the upper and lower limits shown in Figure 3, where it can be seen that the prediction is close to the forecast mean value.

The exceedance probability of the forecast CIHAM-UC-EP grid shows that the highest exceedance probability varying between 56% and 96%, occurs where the natural coverage is mostly distributed (forested area, degraded soil). The lowest exceedance probability is found in the urban area (25%–56%).

The proposed equation for forecasting of flood risk can be expressed by substituting the equation associated with the exceedance probability in Equation (11).

A novel method for forecasting gridded flood risk has been created from the Center of Hydrological and Environmental Research of the University of Carabobo (CIHAM-UC, in Spanish) called CIHAM-UC-FFR, obtaining appropriate results for water management in vulnerable areas to flooding events such as the urban fraction of land use and land cover (LULC) in a study area at a watershed scale, because of the occurrence of the rainfall–runoff process. The main purpose of the CIHAM-UC-FFR method is to provide information to make decisions for contributing to the prevention of the consequences of flooding by protecting human beings and goods.

The proposed method might be classified as a hybrid, which can be applied for obtaining lumped or distributed values in the forecast main variables of interest, estimated using models proposed for the effective precipitation (CIHAM-UC-EP) and flood risk (CIHAM-UC-FR) depending on availability of local information. In general, the CIHAM-UC-FFR method includes various model categories for its components: (1) CIHAM-UC-EP (event, conceptual/quasi-conceptual, lumped/distributed, measured/fitted parameters), (2) precipitation (event, distributed/lumped, empirical, stochastic, fitted parameter), (3) parameters of CIHAM-UC EP (event, lumped, empirical, stochastic, fitted parameter), (4) probabilistic function for CIHAM-UC EP (event, lumped/distributed, empirical, stochastic, fitted parameter), (5) CIHAM-UC-FR (event, distributed, empirical, deterministic, fitted parameter).

The advantages of applying the proposed method are based on the following. (1) The update of the calibration of the equations due to it offering an easy incorporation of new information for making continuous recalibration of the components for the CIHAM-UC-EP model by using time series of precipitation measured from rain gauges and time series of satellite images for the LULC factor. (2) The adaptability of the three parameters included in the CIHAM-UC-EP model to be derived using algorithms for mathematical adjustment or obtained from field measurements, which means that CIHAM-UC-EP can be classified as a conceptual or quasi-conceptual model. The flow attenuation parameter might be obtained measuring infiltration rate for a determined flow length, expressed in m² h⁻¹. The flow attenuation time can be obtained from the quasi-physically measured concentration time. (3) The capability of the model to explain the temporal variation of flow depth depending on the terrain elevations, flow path and the flow length in the terrain establishing a strong difference with other models, whose results tend to be homogeneous in watersheds. (4) The capability to adapt the components of the CIHAM-UC-EP model to multiple forecasting models for time series. (5) The perspective of change in the response obtained from the CIHAM-UC-FR model supported on the exceedance probability of the EP and its association with the flood risk process, being adapted to various scenarios in periods from short (monthly) to long term (annually) according to the increase of expected life of the hydraulic structure.

It is recommended for future evaluations of the CIHAM-UC-EP and CIHAM-UC-FR models to increase the spatio-temporal resolution of the involved parameters and variables and to compare with field measurements to validate the obtained results.