HBV lumped conceptual hydrological model

In a previous article, I started a series of articles to explain “Numerical models from black-box to physically based models” [link], and then I explained, “Four decisions you need to take to build a hydrological model” [Link] one of which was the model type, one of these model types is the lumped conceptual model. which is the base of most of the other model types. In this article, I will go through Bergström, 1992 HBV model, One of the versions of the HBV lumped conceptual hydrological model, I will list the hydrological processes simulated by each component of the model, the inputs, and outputs, and I will give a code snippet for each subroutine.

Schematic representation of distributed conceptual hydrological model MIKE SHE by DHI

Conceptual models

Generally, the term “conceptual” is used to refer to models that rely on a relatively small number of interlinked conceptual elements, each element represents a phase of the hydrological cycle and it is usually a storage component which has one input and one or more output, such as soil moisture and linear reservoir [1].

A large number of conceptual models are used nowadays in water resources applications such as NAM model (DHI), HBV model [2], and the tank model (Sugawara et al., 1983).

The HBV model was originally developed at the Swedish Meteorological and Hydrological Institute (SMHI) to simulate the rainfall-runoff relation and for hydrological forecasting, and it is used in more than 30 countries [4].

HBV model has a simple structure and allows to divide the catchment into sub-catchments, according to vegetation and elevation zones. The key advantage of conceptual models is that they are based on a reasonable representation of major hydrological behaviour, which gives an interpretation of the actual behaviour of the catchment.

The computational demand and data required for the HBV are very limited which give it the preference to be used in a wide range of application where data is a problem to obtain.

Schematic structure of HBV-96 model components

Model inputs

• The Input precipitation to the model is the mean areal precipitation, which can be calculated as a weighted mean of observation gauges around the catchment using the Thiessen polygon or isohyetal method.
• Temperature is required only in case of a catchment with snow. Potential Evapotranspiration is also required for calculations of actual evapotranspiration which can be calculated using the Penman formula. Runoff data at the outlet of the catchment is used for the calibration of the model, all of these components are related to each other through the water balance equation

𝑃−𝐸𝑎− 𝛥𝑆/𝛥𝑡=𝑄

• where P is precipitation (mm=day), EA is the actual evapotranspiration (mm=day), Q is the runoff(mm=day), and 𝛥S is the change in basin storage (mm), per time step 𝛥t (day)

Hydrological processes

• The basic structure of the HBV model includes snow melt, snow accumulation routine, soil moisture accounting routine, runoff generation & runoff routing. State variables are used and updated each time step to represent a specific hydrologic behaviour of the catchment.

Rainfall Subroutine

• The rainfall subroutine simulates the hydrological process of rainfall changes between water and snow state.
• To separate snow from precipitation, upper & lower temperature thresholds are used (utt, ltt),
• For temperature values higher than utt or less than ltt all rainfall is going to stay as rain, and all rainfall is going to be converted into snow respectively. While for temperature values in between utt & ltt a linear change based on the temperature from all to stay as rain at utt to all converted into snow at ltt.

Schematic representation for precipitation subroutine

• Two different correction factors for rainfall & snowfall are used to take into account errors in observations. These errors could be systematic errors or related to wind effects.

Snow subroutine

• The snow subroutine simulates two hydrological processes, snow melt and snow accumulation.
• Snow melt routine depends on a degree day relation which determines whether snow is melting or freezing based on a temperature threshold for melting (when temperature increase above the threshold for melting ttm)

𝑆𝑛𝑜𝑤 𝑚𝑒𝑙𝑡=𝐶𝑓𝑚𝑎𝑥(𝑡𝑒𝑚𝑝−𝑡𝑡𝑚)

• Snowpack retains melted water as long as the amount does not exceed a certain ratio (whc) of the snowpack. When the temperature decreases below ttm, melted water refreezes again based on a refreezing factor cfr[5].

𝑟𝑒𝑓𝑟𝑒𝑒𝑧𝑒𝑑 𝑚𝑒𝑙𝑡𝑒𝑑 𝑤𝑎𝑡𝑒𝑟=𝑐𝑓𝑟∗𝑐𝑓𝑚𝑎𝑥(𝑡𝑡𝑚−𝑡𝑒𝑚𝑝)

Soil moisture Subroutine

• The soil subroutine is the most important for controlling the runoff formation, as for each millimetre of rainfall or snowmelt it controls the contribution to the response function (ΔQ/ P) through β or the increase in soil moisture storage (1- ΔQ/ P), where P is the effective precipitation.

Soil Moisture Equation

• Where FC is the maximum soil moisture storage capacity (field capacity), SM is the soil moisture, and ΔQ is the recharge to the upper zone response bucket.

• Potential evapotranspiration is corrected using a correction factor E_corr, while actual evapotranspiration is calculated as a ratio of the soil moisture depending on Lp. Lp is a soil moisture value where if it is exceeded evapotranspiration reaches its potential value (Lp is used as a fraction of FC),

Evapotranspiration equation as a function in field capacity

Response Subroutine

• The response subroutine is responsible for transforming excess rainfall from the soil moisture zone into a runoff. It consists of a linear upper and lower reservoir representing a fast (subsurface) response and a slow (groundwater) response.

Routing function

• In order to get the right shape of the hydrograph at the outlet, the flow result from the response subroutine is routed using a triangular function which assumes

Triangular routing function (Maxbas)

• all of the subroutines are connected, where the output from one subroutine is the input for the next till we have the routed discharge value from the routing function (the triangular weighting function)
• As I mentioned before there are different versions of the HBV model, what I have shown in this article is the Bergström, 1992 model, however, there are different versions proposed by Bergström 1976. the main difference between these versions differs from the representation of the upper-zone response (linear/nonlinear) to the number of response reservoirs (upper/lower/interflow).
• Bergström, 1992 has also proposed two ways to simulate a lake which I will explain in a different article, in order to keep this article simple and small.
• You can check the Hapi python package where different HBV versions are included in a distributed framework.

GitHub - MAfarrag/Hapi: Hapi is a Python library for building Conceptual Distributed Model using…

References

[1] Jain, S. K. 1993. “Calibration of Conceptual Models for Rainfall-Runoff Simulation.” Hydrological Sciences Journal 38(5): 431–41.

[2] Bergström, S.: The HBV model — its structure and applications, Smhi Rh, 4, 35, 1992.

[3] Sugawara, M. Ozaki, E. Reference Manual for the Tank Model, National Research Center for Disaster Prevention, Tokyo, Japan

[4] Lindström, G., Johansson, B., Persson, M., Gardelin, M., and Bergström, S.: Development and test of the distributed HBV-96 hydrological model, J. Hydrol., 201, 272–288, https://doi.org/10.1016/S0022-1694(97)00041-3, 1997.

[5] Ihms: Integrated Hydrological Modelling System, 6.2, 2010.

[6] Bergström, Sten. 1976. “Development and Application of a Conceptual Runoff Model for Scandinavian Catchments.” Smhi RHO 7(November): 134.