a platform for evaluating the latest hydrological, climatic and water quality sensors and in parallel evaluating the latest dynamic models of these resultant high frequency data
The time-series derived from SMART sensors can be rich in information, and can help to develop understanding of watershed functioning. To achieve this we need to extract the dominant dynamics from these time-series using robust model identification tools that are subject to explicit uncertainty analysis.
We have published example applications using SMART models in hydrology and water chemistry using the RIVC algorithm (or ‘Refined Instrumental Variable Continuous-time Box-Jenkins identification algorithm’) of Young (2008). The algorithm is able to extract the dominant dynamics contained within the high-frequency hydrological and water quality time-series that we derived using SMART sensors. This algorithm is capable of identifying data-based models that incorporate dynamic delays and storage.
RIVC uses an iterative instrumental variable (IV) method for estimating general transfer functions capturing dynamic relationships between input (e.g., rainfall) and output (e.g., DOC concentration or load) variables using rational polynomial expressions in operator s (in continuous time), which directly translate to differential equations driven by the input variable. Full details are given in Jones et al. 2014 (ES&T).
Rationale for use of RIVC algorithm for model identification (adapted from Jones et al. 2014, ES&T SI):
A/ In one numerical experiment RIVC can identify the structure and parameters of a range of purely static models and models that include dynamic components,
B/ Dynamic Response Characteristics (DRCs) that have a physical interpretation can be derived from the identified model parameters,
C/ The covariance matrix for each identified model can be used directly to estimate uncertainty in the model parameter estimates,
D/ An optimal model structure can be selected using a combination of this uncertainty information, heuristic measures of the parsimony (to avoid over-parameterisation) and a feasible physical interpretation. This is the Data-Based Mechanistic (DBM) approach to modelling.
Conceptual systems diagram showing an identified two pathway response in the modelling of rainfall to DOC load.
E/ Model estimation involves iterative pre-filtering of the signals to remove the high frequency noise inherent within environmental data that affects identification of accurate parameter values.
F/ The routine can estimate transfer functions in continuous-time (CT-TF) that maybe more difficult to estimate compared to discrete-time transfer function models but are considered more accurate for systems with responses almost as fast as the monitoring time-step, and
G/ RIVC is computationally efficient, with its results permitting rapid Monte Carlo uncertainty analysis of the parameter ranges. The Instrumental Variable (IV) approach guarantees that the parameter estimates are asymptotically unbiased.
We have used the RIVC algorithm to identify models of stream hydrogen ion load from a rainfall input, and the first model of dissolved organic carbon (DOC) load from a rainfall input through a sequence of contiguous storms at four watersheds at Llyn Brianne. In both cases the optimal model identified is a purely linear second-order continuous-time transfer-function model. These are derived directly from the high-frequency (i.e. 15-min) rainfall, DOC and pH data collected using our SMART sensors. The dominant dynamics in these water quality variables through storm periods could not have been identified using infrequently sampled data (even 7 hourly sampling would have been insufficient).
Above: Observed DOC (red line) and simulated DOC (black line) in mg/L through three contiguous storms. Simulated DOC was modelled from rainfall (blue) in mm/15-min using the RIVC algorithm for the four instrumented watersheds at Llyn Brianne. The model is a purely linear, second-order transfer function. Source: Jones et al, 2014.
Above: The second-order transfer function model of DOC(LOAD) can be decomposed by partial fraction expansion into two parallel, first-order transfer functions representing different response pathways. 56% of DOC delivery followed a fast pathway at Nant-y-Craflwyn (LI3), and 44% of response followed a slow pathway. Source: presentation at SWIG meeting, Glasgow, 2014.
The model identification using the RIVC algorithm described above is implemented using the Computer-Aided Program for Time-Series Analysis and Identification of Noisy Systems (CAPTAIN) Toolbox for Matlab, developed at Lancaster University.
The CAPTAIN Toolbox contains algorithms for aspects of non stationary time series analysis, system identification, signal processing and forecasting, using unobserved components models, time variable parameter models, state dependent parameter models and multiple input transfer function models.
The routines are implemented following a Data-Based Mechanistic (DBM) approach to modelling dynamic systems, following these steps:
1/ Identify as many potential model structures as possible, minimising prior assumptions about processes – that are often unknown,
2/ Reject most model structures using objective statistical & mathematical criteria (incorporating ‘Principles of Parsimony’ via heuristic measures),
3/ Reject further models that have no physical (e.g., hydrological) interpretation – giving models for testing against independent observations (e.g., dynamics within component flow paths).
Explicit uncertainty analysis is a key part of SMART modelling. The widely used GLUE methodology provides a method for accounting for calibration and uncertainty estimation of distributed models through the use of generalised likelihood methods. The framework includes the possibility of different sets of parameter values being equally likely simulators of watersheds within a given model structure and errors in definition of boundary conditions and observed variables (Beven and Binley, 1992).
Software for GLUE is available here:
TOPMODEL is a rainfall-runoff model (Beven and Kirkby, 1979) that uses topographic information (specific watershed area and wetness index) relating to runoff generation. It is a physically based, distributed watershed model, and can be used with the GLUE methodology of uncertainty estimation, thereby making it a SMART model. Follow this link for implementation in R.
A new version of TOPMODEL that allows simulation of dynamically variable upslope contributing areas was presented by Beven and Freer (2001).
Kretzschmar, A., Tych, W., Chappell, N.A. and Beven, K.J. 2016. Reversing hydrology: quantifying the temporal aggregation effect of catchment rainfall estimation using sub-hourly data. Hydrology Research, 47: in press. DOI: 10.2166/nh.2015.076 view online.
Jones, T.D., Chappell, N.A. and Tych, W. 2014. First dynamic model of dissolved organic carbon derived directly from high frequency observations through contiguous storms. Environmental Science & Technology, 48: 13289-13297. view online.
Jones, T.D. and Chappell, N.A. 2014. Streamflow and hydrogen ion interrelationships identified using Data-Based Mechanistic modelling of high frequency observations through contiguous storms. Hydrology Research, 45(6): 868-892. view online.
Kretzschmar, A., Tych, W., and Chappell, N.A. 2014. Reversing Hydrology: estimation of sub-hourly rainfall time-series from streamflow. Environmental Modelling and Software, 60: 290-301. doi: 10.1016/j.envsoft.2014.06.017 view online.
Beven, K.J. and Binley, A.M. 2014. GLUE: 20 years on. Hydrological Processes, 28: 5897-5918. view online.
Leedal, D., Weerts, A.H., Smith, P.J. and Beven, K.J. 2013. Application of data-based mechanistic modelling for flood forecasting at multiple locations in the Eden catchment in the National Flood Forecasting System (England and Wales). Hydrology and Earth System Sciences, 17: 177-185. view online.
Chappell, N.A., Bonell, M., Barnes, C.J., and Tych, W. 2012. Tropical cyclone effects on rapid runoff responses: quantifying with new continuous-time transfer function models. In: Revisiting Experimental Catchment Studies in Forest Hydrology, Webb, A.A., Bonell, M. Bren, L. Lane, P.N.J., McGuire, D., Neary, D.G., Nettles, J., Scott, D.F., Stednik J. & Wang, Y. (eds) IAHS Publication 353, Wallingford, IAHS Press. 82-93. view online.
Chappell, N.A., and Tych, W. 2012. Identifying step changes in single streamflow and evaporation records due to forest cover change. Hydrological Processes, 26, 100-116 doi:10.1002/hyp.8115. view online.
Ockenden M.C. and Chappell, N.A. 2011. Identification of the dominant runoff pathways from the data-based mechanistic modelling of nested catchments in temperate UK. Journal of Hydrology, 402, 71-79. view online.
Beven, K. and Freer, J. 2001. A dynamic TOPMODEL. Hydrological Processes, 15: 1992-2011. view online.
Taylor, C.J., Pedregal, D.J., Young, P.C. and Tych, W. 2007. Environmental time series analysis and forecasting with the Captain Toolbox. Environmental Modelling and Software, 22: 797-814. view online.
Presentation made at the British Hydrological Society National Symposium (Sept 2014) on ‘Modelling water quality in UK upland streams using high-frequency observations’, including RIVC and CAPTAIN click here to access.
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