What really happens at the end of the rainbow?

What really happens at the end of the rainbow?
Reducing uncertainty with reverse hydrology models Supervisors: Wlodek Tych Nick Chappell Keith Beven Ann Kretzschmar Lancaster Environment Centre

All these events happened somewhere in the world in 2015
Flooding is the most common natural disaster and is expected to get worse due to global warming. It is a worldwide problem affecting millions of lives and causing enormous economic impact

What can we do about it? Rainfall is the major input to most hydrological models but it is highly uncertain in both space and time. Reducing that uncertainty will help reduce the uncertainty in flood forecasts.

What can we do about it? Attempt to reduce the uncertainty in flood forecasts by reducing uncertainty in rainfall estimates Rainfall is the major input to most hydrological models but it is highly uncertain in both space and time. Reducing that uncertainty will help reduce the uncertainty in flood forecasts.

Rain falls over a catchment and is measured by a rain-gauge network however it doesn’t just rain over the gauges. We have no information about what is happening anywhere else in the catchment however all rainfall eventually becomes outflow which can be measured. If we can extract the information contained in the catchment outflow, we should be able to find out more about the rainfall dynamics over the whole catchment not just over the rain-gauges. Our network of gauges maybe sparse. One gauge could easily bias calculation of the catchment average rainfall time-series. How do we know how much information each gauge is providing? Can the outflow tell us? If so how? Traditional methods of modelling utilise only the information contained in the rainfall input however all rain that falls on a catchment will eventually become outflow. Catchment outflow contains information no only about the rainfall falling on the catchment including both its temporal and spatial variation but also about the processes which transform the rainfall into flow. If we could extract this information, it should be possible to reduce uncertainty not just by improving rainfall estimates but also by adding to understanding of the transformation processes involved.

Measured rainfall used to calculate catchment average
Rain gauge network: Measured rainfall used to calculate catchment average Rain falls over a catchment and is measured by a rain-gauge network however it doesn’t just rain over the gauges. We have no information about what is happening anywhere else in the catchment however all rainfall eventually becomes outflow which can be measured. If we can extract the information contained in the catchment outflow, we should be able to find out more about the rainfall dynamics over the whole catchment not just over the rain-gauges. Our network of gauges maybe sparse. One gauge could easily bias calculation of the catchment average rainfall time-series. How do we know how much information each gauge is providing? Can the outflow tell us? If so how? Traditional methods of modelling utilise only the information contained in the rainfall input however all rain that falls on a catchment will eventually become outflow. Catchment outflow contains information no only about the rainfall falling on the catchment including both its temporal and spatial variation but also about the processes which transform the rainfall into flow. If we could extract this information, it should be possible to reduce uncertainty not just by improving rainfall estimates but also by adding to understanding of the transformation processes involved.

Measured rainfall used to calculate catchment average ?
Rain gauge network: Measured rainfall used to calculate catchment average ? Rain falls over a catchment and is measured by a rain-gauge network however it doesn’t just rain over the gauges. We have no information about what is happening anywhere else in the catchment however all rainfall eventually becomes outflow which can be measured. If we can extract the information contained in the catchment outflow, we should be able to find out more about the rainfall dynamics over the whole catchment not just over the rain-gauges. Our network of gauges maybe sparse. One gauge could easily bias calculation of the catchment average rainfall time-series. How do we know how much information each gauge is providing? Can the outflow tell us? If so how? Traditional methods of modelling utilise only the information contained in the rainfall input however all rain that falls on a catchment will eventually become outflow. Catchment outflow contains information no only about the rainfall falling on the catchment including both its temporal and spatial variation but also about the processes which transform the rainfall into flow. If we could extract this information, it should be possible to reduce uncertainty not just by improving rainfall estimates but also by adding to understanding of the transformation processes involved.

Measured rainfall used to calculate catchment average ?
Rain gauge network: Measured rainfall used to calculate catchment average ? Catchment outflow: Measured Rain falls over a catchment and is measured by a rain-gauge network however it doesn’t just rain over the gauges. We have no information about what is happening anywhere else in the catchment however all rainfall eventually becomes outflow which can be measured. If we can extract the information contained in the catchment outflow, we should be able to find out more about the rainfall dynamics over the whole catchment not just over the rain-gauges. Our network of gauges maybe sparse. One gauge could easily bias calculation of the catchment average rainfall time-series. How do we know how much information each gauge is providing? Can the outflow tell us? If so how? Traditional methods of modelling utilise only the information contained in the rainfall input however all rain that falls on a catchment will eventually become outflow. Catchment outflow contains information no only about the rainfall falling on the catchment including both its temporal and spatial variation but also about the processes which transform the rainfall into flow. If we could extract this information, it should be possible to reduce uncertainty not just by improving rainfall estimates but also by adding to understanding of the transformation processes involved.

Can the catchment outflow tell us more?
Rain falls over a catchment and is measured by a rain-gauge network however it doesn’t just rain over the gauges. We have no information about what is happening anywhere else in the catchment however all rainfall eventually becomes outflow which can be measured. If we can extract the information contained in the catchment outflow, we should be able to find out more about the rainfall dynamics over the whole catchment not just over the rain-gauges. Our network of gauges maybe sparse. One gauge could easily bias calculation of the catchment average rainfall time-series. How do we know how much information each gauge is providing? Can the outflow tell us? If so how? Traditional methods of modelling utilise only the information contained in the rainfall input however all rain that falls on a catchment will eventually become outflow. Catchment outflow contains information no only about the rainfall falling on the catchment including both its temporal and spatial variation but also about the processes which transform the rainfall into flow. If we could extract this information, it should be possible to reduce uncertainty not just by improving rainfall estimates but also by adding to understanding of the transformation processes involved.

How much information is each gauge providing? Rain falls over a catchment and is measured by a rain-gauge network however it doesn’t just rain over the gauges. We have no information about what is happening anywhere else in the catchment however all rainfall eventually becomes outflow which can be measured. If we can extract the information contained in the catchment outflow, we should be able to find out more about the rainfall dynamics over the whole catchment not just over the rain-gauges. Our network of gauges maybe sparse. One gauge could easily bias calculation of the catchment average rainfall time-series. How do we know how much information each gauge is providing? Can the outflow tell us? If so how? Traditional methods of modelling utilise only the information contained in the rainfall input however all rain that falls on a catchment will eventually become outflow. Catchment outflow contains information no only about the rainfall falling on the catchment including both its temporal and spatial variation but also about the processes which transform the rainfall into flow. If we could extract this information, it should be possible to reduce uncertainty not just by improving rainfall estimates but also by adding to understanding of the transformation processes involved.

How much information is each gauge providing? Is each gauge representative? Rain falls over a catchment and is measured by a rain-gauge network however it doesn’t just rain over the gauges. We have no information about what is happening anywhere else in the catchment however all rainfall eventually becomes outflow which can be measured. If we can extract the information contained in the catchment outflow, we should be able to find out more about the rainfall dynamics over the whole catchment not just over the rain-gauges. Our network of gauges maybe sparse. One gauge could easily bias calculation of the catchment average rainfall time-series. How do we know how much information each gauge is providing? Can the outflow tell us? If so how? Traditional methods of modelling utilise only the information contained in the rainfall input however all rain that falls on a catchment will eventually become outflow. Catchment outflow contains information no only about the rainfall falling on the catchment including both its temporal and spatial variation but also about the processes which transform the rainfall into flow. If we could extract this information, it should be possible to reduce uncertainty not just by improving rainfall estimates but also by adding to understanding of the transformation processes involved.

How much information is each gauge providing? Is each gauge representative? One disinformative gauge could bias the average Rain falls over a catchment and is measured by a rain-gauge network however it doesn’t just rain over the gauges. We have no information about what is happening anywhere else in the catchment however all rainfall eventually becomes outflow which can be measured. If we can extract the information contained in the catchment outflow, we should be able to find out more about the rainfall dynamics over the whole catchment not just over the rain-gauges. Our network of gauges maybe sparse. One gauge could easily bias calculation of the catchment average rainfall time-series. How do we know how much information each gauge is providing? Can the outflow tell us? If so how? Traditional methods of modelling utilise only the information contained in the rainfall input however all rain that falls on a catchment will eventually become outflow. Catchment outflow contains information no only about the rainfall falling on the catchment including both its temporal and spatial variation but also about the processes which transform the rainfall into flow. If we could extract this information, it should be possible to reduce uncertainty not just by improving rainfall estimates but also by adding to understanding of the transformation processes involved.

Sparse rain-gauge network How much information is each gauge providing? Is each gauge representative? One disinformative gauge could bias the average Catchment outflow contains information on Rainfall dynamics Catchment characteristics Rain falls over a catchment and is measured by a rain-gauge network however it doesn’t just rain over the gauges. We have no information about what is happening anywhere else in the catchment however all rainfall eventually becomes outflow which can be measured. If we can extract the information contained in the catchment outflow, we should be able to find out more about the rainfall dynamics over the whole catchment not just over the rain-gauges. Our network of gauges maybe sparse. One gauge could easily bias calculation of the catchment average rainfall time-series. How do we know how much information each gauge is providing? Can the outflow tell us? If so how? Traditional methods of modelling utilise only the information contained in the rainfall input however all rain that falls on a catchment will eventually become outflow. Catchment outflow contains information no only about the rainfall falling on the catchment including both its temporal and spatial variation but also about the processes which transform the rainfall into flow. If we could extract this information, it should be possible to reduce uncertainty not just by improving rainfall estimates but also by adding to understanding of the transformation processes involved.

Sparse rain-gauge network How much information is each gauge providing? One disinformative gauge could bias the average Catchment outflow contains information on Rainfall dynamics Catchment characteristics How can the information be extracted? Rain falls over a catchment and is measured by a rain-gauge network however it doesn’t just rain over the gauges. We have no information about what is happening anywhere else in the catchment however all rainfall eventually becomes outflow which can be measured. If we can extract the information contained in the catchment outflow, we should be able to find out more about the rainfall dynamics over the whole catchment not just over the rain-gauges. Our network of gauges maybe sparse. One gauge could easily bias calculation of the catchment average rainfall time-series. How do we know how much information each gauge is providing? Can the outflow tell us? If so how? Traditional methods of modelling utilise only the information contained in the rainfall input however all rain that falls on a catchment will eventually become outflow. Catchment outflow contains information no only about the rainfall falling on the catchment including both its temporal and spatial variation but also about the processes which transform the rainfall into flow. If we could extract this information, it should be possible to reduce uncertainty not just by improving rainfall estimates but also by adding to understanding of the transformation processes involved.

Sparse rain-gauge network How much information is each gauge providing? One disinformative gauge could bias the average Catchment outflow contains in formation on Rainfall dynamics Catchment characteristics How can the information be extracted? Rain falls over a catchment and is measured by a rain-gauge network however it doesn’t just rain over the gauges. We have no information about what is happening anywhere else in the catchment however all rainfall eventually becomes outflow which can be measured. If we can extract the information contained in the catchment outflow, we should be able to find out more about the rainfall dynamics over the whole catchment not just over the rain-gauges. Our network of gauges maybe sparse. One gauge could easily bias calculation of the catchment average rainfall time-series. How do we know how much information each gauge is providing? Can the outflow tell us? If so how? Traditional methods of modelling utilise only the information contained in the rainfall input however all rain that falls on a catchment will eventually become outflow. Catchment outflow contains information no only about the rainfall falling on the catchment including both its temporal and spatial variation but also about the processes which transform the rainfall into flow. If we could extract this information, it should be possible to reduce uncertainty not just by improving rainfall estimates but also by adding to understanding of the transformation processes involved. Reverse Hydrology may be the answer .....

What is reverse hydrology?
Traditionally, rainfall is used as the input to a model that purports to represent the catchment processes and outputs a simulated flow series which is an estimate of the measured flow. Reverse hydrology turns this idea on its head. A model is still fitted to the measured rainfall and flow but the parameters are then inverted and the flow used as input. The output from the model is an estimate of the catchment average rainfall.

Measured Rainfall Traditionally, rainfall is used as the input to a model that purports to represent the catchment processes and outputs a simulated flow series which is an estimate of the measured flow. Reverse hydrology turns this idea on its head. A model is still fitted to the measured rainfall and flow but the parameters are then inverted and the flow used as input. The output from the model is an estimate of the catchment average rainfall.

Catchment processes (dynamics) Measured Rainfall Traditionally, rainfall is used as the input to a model that purports to represent the catchment processes and outputs a simulated flow series which is an estimate of the measured flow. Reverse hydrology turns this idea on its head. A model is still fitted to the measured rainfall and flow but the parameters are then inverted and the flow used as input. The output from the model is an estimate of the catchment average rainfall.

Catchment processes (dynamics) Measured Rainfall Streamflow Traditionally, rainfall is used as the input to a model that purports to represent the catchment processes and outputs a simulated flow series which is an estimate of the measured flow. Reverse hydrology turns this idea on its head. A model is still fitted to the measured rainfall and flow but the parameters are then inverted and the flow used as input. The output from the model is an estimate of the catchment average rainfall.

Catchment processes (dynamics) Measured Rainfall Streamflow Traditionally, rainfall is used as the input to a model that purports to represent the catchment processes and outputs a simulated flow series which is an estimate of the measured flow. Reverse hydrology turns this idea on its head. A model is still fitted to the measured rainfall and flow but the parameters are then inverted and the flow used as input. The output from the model is an estimate of the catchment average rainfall. Streamflow

Catchment processes (dynamics) Measured Rainfall Streamflow Traditionally, rainfall is used as the input to a model that purports to represent the catchment processes and outputs a simulated flow series which is an estimate of the measured flow. Reverse hydrology turns this idea on its head. A model is still fitted to the measured rainfall and flow but the parameters are then inverted and the flow used as input. The output from the model is an estimate of the catchment average rainfall. Inverted dynamics Streamflow

Catchment processes (dynamics) Measured Rainfall Streamflow Inferred catchment rainfall Traditionally, rainfall is used as the input to a model that purports to represent the catchment processes and outputs a simulated flow series which is an estimate of the measured flow. Reverse hydrology turns this idea on its head. A model is still fitted to the measured rainfall and flow but the parameters are then inverted and the flow used as input. The output from the model is an estimate of the catchment average rainfall. Inverted dynamics Streamflow

Catchment processes (dynamics) Measured Rainfall Streamflow Inferred catchment rainfall Traditionally, rainfall is used as the input to a model that purports to represent the catchment processes and outputs a simulated flow series which is an estimate of the measured flow. Reverse hydrology turns this idea on its head. A model is still fitted to the measured rainfall and flow but the parameters are then inverted and the flow used as input. The output from the model is an estimate of the catchment average rainfall. Inverted dynamics Streamflow Reverse hydrology infers the rainfall time-series needed to generate streamflow

Objectives To show how reverse hydrology can be used:
To investigate how reverse hydrology might be used to assess the information content of individual rain-gauges and their affect on the estimation of a catchment average rainfall time-series leading to a reduction in uncertainty in flood flow forecasts. …… let’s see how it might work

to assess the information content of individual rain-gauges To investigate how reverse hydrology might be used to assess the information content of individual rain-gauges and their affect on the estimation of a catchment average rainfall time-series leading to a reduction in uncertainty in flood flow forecasts. …… let’s see how it might work

to assess the information content of individual rain-gauges to assess their affect on the estimation of a catchment average rainfall time-series To investigate how reverse hydrology might be used to assess the information content of individual rain-gauges and their affect on the estimation of a catchment average rainfall time-series leading to a reduction in uncertainty in flood flow forecasts. …… let’s see how it might work

Methodology The method uses and continuous time linear transform function to model the relationship between rainfall and catchment outflow. As the relationship is inherently non-linear, the non-linearity is modelled as a separate step before the model is identified. The parameters of the transfer function model are then inverted and regularisation applied as necessary to stabilise the numerical process. The output is a sequence of scaled rainfall which can then have the reverse of the non-linearity process applied to it to obtain an estimate of the catchment average rainfall. The following slides show a breakdown of the process assuming that the non-linear transform has already been applied to the rainfall……

Methodology Model Identification
The method uses and continuous time linear transform function to model the relationship between rainfall and catchment outflow. As the relationship is inherently non-linear, the non-linearity is modelled as a separate step before the model is identified. The parameters of the transfer function model are then inverted and regularisation applied as necessary to stabilise the numerical process. The output is a sequence of scaled rainfall which can then have the reverse of the non-linearity process applied to it to obtain an estimate of the catchment average rainfall. The following slides show a breakdown of the process assuming that the non-linear transform has already been applied to the rainfall……

Methodology P Pe Model Identification Power Law Observed rainfall
Non-linearity Power Law Pe Scaled rainfall The method uses and continuous time linear transform function to model the relationship between rainfall and catchment outflow. As the relationship is inherently non-linear, the non-linearity is modelled as a separate step before the model is identified. The parameters of the transfer function model are then inverted and regularisation applied as necessary to stabilise the numerical process. The output is a sequence of scaled rainfall which can then have the reverse of the non-linearity process applied to it to obtain an estimate of the catchment average rainfall. The following slides show a breakdown of the process assuming that the non-linear transform has already been applied to the rainfall……

Linear Transfer Function
Methodology Model Identification P Observed rainfall Non-linearity Power Law Pe Scaled rainfall Linear Transfer Function Continuous time Q Observed streamflow The method uses and continuous time linear transform function to model the relationship between rainfall and catchment outflow. As the relationship is inherently non-linear, the non-linearity is modelled as a separate step before the model is identified. The parameters of the transfer function model are then inverted and regularisation applied as necessary to stabilise the numerical process. The output is a sequence of scaled rainfall which can then have the reverse of the non-linearity process applied to it to obtain an estimate of the catchment average rainfall. The following slides show a breakdown of the process assuming that the non-linear transform has already been applied to the rainfall……

Methodology Model Identification P Observed rainfall Non-linearity Power Law Pe Scaled rainfall Linear Transfer Function Continuous time Q Observed streamflow Model Inversion The method uses and continuous time linear transform function to model the relationship between rainfall and catchment outflow. As the relationship is inherently non-linear, the non-linearity is modelled as a separate step before the model is identified. The parameters of the transfer function model are then inverted and regularisation applied as necessary to stabilise the numerical process. The output is a sequence of scaled rainfall which can then have the reverse of the non-linearity process applied to it to obtain an estimate of the catchment average rainfall. The following slides show a breakdown of the process assuming that the non-linear transform has already been applied to the rainfall……

Methodology Model Identification P Observed rainfall Non-linearity Power Law Pe Scaled rainfall Linear Transfer Function Continuous time Q Observed streamflow Model Inversion Q Observed streamflow RegDer method Inverse transfer function Peh Inferred scaled rainfall The method uses and continuous time linear transform function to model the relationship between rainfall and catchment outflow. As the relationship is inherently non-linear, the non-linearity is modelled as a separate step before the model is identified. The parameters of the transfer function model are then inverted and regularisation applied as necessary to stabilise the numerical process. The output is a sequence of scaled rainfall which can then have the reverse of the non-linearity process applied to it to obtain an estimate of the catchment average rainfall. The following slides show a breakdown of the process assuming that the non-linear transform has already been applied to the rainfall……

Methodology P Pe Q Q Peh Ph Model Identification Model Inversion
Observed rainfall Non-linearity Power Law Pe Scaled rainfall Linear Transfer Function * Continuous time Q Observed streamflow Model Inversion Q Observed streamflow RegDer method * Inverse transfer function Peh Inferred scaled rainfall Reverse non-linearity Reverse Power Law Ph Inferred catchment rainfall The method uses and continuous time linear transform function to model the relationship between rainfall and catchment outflow. As the relationship is inherently non-linear, the non-linearity is modelled as a separate step before the model is identified. The parameters of the transfer function model are then inverted and regularisation applied as necessary to stabilise the numerical process. The output is a sequence of scaled rainfall which can then have the reverse of the non-linearity process applied to it to obtain an estimate of the catchment average rainfall. The following slides show a breakdown of the process assuming that the non-linear transform has already been applied to the rainfall…… [*] Described in Kretzschmar et al (2014) Reversing hydrology: Estimation of sub-hourly rainfall time-series from streamflow. Environmental Modelling & Software 60:

Synthetic flow generation *
The inferred rainfall time series can then be used as input to the transfer function model whose output is a flow time series that is an estimate of the observed catchment outflow time series. * See Kretzschmar et al (2015) Reversing hydrology: quantifying the temporal aggregation effect of catchment rainfall estimation using sub-hourly data’, Hydrology Research

Synthetic flow generation *
Qobs Qsim and Qinv are both estimates of Qobs The inferred rainfall time series can then be used as input to the transfer function model whose output is a flow time series that is an estimate of the observed catchment outflow time series. * See Kretzschmar et al (2015) Reversing hydrology: quantifying the temporal aggregation effect of catchment rainfall estimation using sub-hourly data’, Hydrology Research

Application to an intensively instrumented catchment

Test catchment – Brue, SW UK
Area 135 km2 Crump weir and tipping bucket rain-gauge The Brue experimental catchment is in the SW of the UK in area subject to historic (and recent flooding). Its elevation ranges from 35m in the SW to 190m in the NE. There is a dense network of 49 rain-gauges from which 23 were selected for the current analysis as the proximity of some of the gauges lead to strong correlation between them. The 23 gauges maintain to geographical coverage. Spatial distribution of rainfall varies from event to event and winters are noticeably wetter than the summers.

Area 135 km2 Elevation m above sea level Crump weir and tipping bucket rain-gauge The Brue experimental catchment is in the SW of the UK in area subject to historic (and recent flooding). Its elevation ranges from 35m in the SW to 190m in the NE. There is a dense network of 49 rain-gauges from which 23 were selected for the current analysis as the proximity of some of the gauges lead to strong correlation between them. The 23 gauges maintain to geographical coverage. Spatial distribution of rainfall varies from event to event and winters are noticeably wetter than the summers.

Area 135 km2 Elevation m above sea level 23 gauge network used Crump weir and tipping bucket rain-gauge The Brue experimental catchment is in the SW of the UK in area subject to historic (and recent flooding). Its elevation ranges from 35m in the SW to 190m in the NE. There is a dense network of 49 rain-gauges from which 23 were selected for the current analysis as the proximity of some of the gauges lead to strong correlation between them. The 23 gauges maintain to geographical coverage. Spatial distribution of rainfall varies from event to event and winters are noticeably wetter than the summers.

Area 135 km2 Elevation m above sea level 23 gauge network used Winters wetter than summers Crump weir and tipping bucket rain-gauge The Brue experimental catchment is in the SW of the UK in area subject to historic (and recent flooding). Its elevation ranges from 35m in the SW to 190m in the NE. There is a dense network of 49 rain-gauges from which 23 were selected for the current analysis as the proximity of some of the gauges lead to strong correlation between them. The 23 gauges maintain to geographical coverage. Spatial distribution of rainfall varies from event to event and winters are noticeably wetter than the summers.

Area 135 km2 Elevation m above sea level 23 gauge network used Winters wetter than summers Spatial rainfall distribution varies from event to event Crump weir and tipping bucket rain-gauge The Brue experimental catchment is in the SW of the UK in area subject to historic (and recent flooding). Its elevation ranges from 35m in the SW to 190m in the NE. There is a dense network of 49 rain-gauges from which 23 were selected for the current analysis as the proximity of some of the gauges lead to strong correlation between them. The 23 gauges maintain to geographical coverage. Spatial distribution of rainfall varies from event to event and winters are noticeably wetter than the summers.

Location of 23 selected gauges
Location of rain-gauges and associated Thiessen Polygons (used as weighting) Theissen polygons were calculated using ARCGIS and used to calculate the catchment average rainfall. Analysis was carried out for all gauges however the 4 highlighted have been used as examples. The analysis was carried out as follows: Location of 23 selected gauges

Analysis Thiessen polygon weighted catchment average

Analysis – Thiessen polygon weighted catchment average
Thiessen Polygon catchment average rainfall – estimate of ‘True’ rainfall PTP The first step in the analysis uses the catchment average rainfall and catchment outflow. The model fit is assessed and the outflow estimates compared using Rt2 (the coefficient of determination) where rt2=1 would show a perfect fit between the observed and simulated data. Fit of model gives first indication of how good representative the calculated average (our best available estimate of the TRUE rainfall) might be. Stage 1 Catchment average rainfall calculated using Thiessen polygon method (Pav) Pav and observed catchment outflow (Qobs) used to identify model Flow simulated from model (Qsim) estimate of Qobs Model inverted using the RegDer method Qobs used as input Output inferred rainfall series (Peh) Peh used to generate inferred estimate of flow (Qinv) Qobs, Qsim and Qinv compared using Rt2

Thiessen Polygon catchment average rainfall – estimate of ‘True’ rainfall Transfer function model identified from catchment average rainfall and catchment outflow – simulated flow time-series generated PTP Qobs Qsim MAV The first step in the analysis uses the catchment average rainfall and catchment outflow. The model fit is assessed and the outflow estimates compared using Rt2 (the coefficient of determination) where rt2=1 would show a perfect fit between the observed and simulated data. Fit of model gives first indication of how good representative the calculated average (our best available estimate of the TRUE rainfall) might be. Stage 1 Catchment average rainfall calculated using Thiessen polygon method (Pav) Pav and observed catchment outflow (Qobs) used to identify model Flow simulated from model (Qsim) estimate of Qobs Model inverted using the RegDer method Qobs used as input Output inferred rainfall series (Peh) Peh used to generate inferred estimate of flow (Qinv) Qobs, Qsim and Qinv compared using Rt2

Thiessen Polygon catchment average rainfall – estimate of ‘True’ rainfall Transfer function model identified from catchment average rainfall and catchment outflow – simulated flow time-series generated PTP RegDer inversion method applied and estimate of catchment rainfall inferred from outflow Qobs Qsim MAV RegDer The first step in the analysis uses the catchment average rainfall and catchment outflow. The model fit is assessed and the outflow estimates compared using Rt2 (the coefficient of determination) where rt2=1 would show a perfect fit between the observed and simulated data. Fit of model gives first indication of how good representative the calculated average (our best available estimate of the TRUE rainfall) might be. Stage 1 Catchment average rainfall calculated using Thiessen polygon method (Pav) Pav and observed catchment outflow (Qobs) used to identify model Flow simulated from model (Qsim) estimate of Qobs Model inverted using the RegDer method Qobs used as input Output inferred rainfall series (Peh) Peh used to generate inferred estimate of flow (Qinv) Qobs, Qsim and Qinv compared using Rt2 Minv Pinf

Thiessen Polygon catchment average rainfall – estimate of ‘True’ rainfall Transfer function model identified from catchment average rainfall and catchment outflow – simulated flow time-series generated PTP RegDer inversion method applied and estimate of catchment rainfall inferred from outflow Qobs Qsim MAV Inferred rainfall used to generate inferred flow time-series – an estimate of the catchment outflow RegDer The first step in the analysis uses the catchment average rainfall and catchment outflow. The model fit is assessed and the outflow estimates compared using Rt2 (the coefficient of determination) where rt2=1 would show a perfect fit between the observed and simulated data. Fit of model gives first indication of how good representative the calculated average (our best available estimate of the TRUE rainfall) might be. Stage 1 Catchment average rainfall calculated using Thiessen polygon method (Pav) Pav and observed catchment outflow (Qobs) used to identify model Flow simulated from model (Qsim) estimate of Qobs Model inverted using the RegDer method Qobs used as input Output inferred rainfall series (Peh) Peh used to generate inferred estimate of flow (Qinv) Qobs, Qsim and Qinv compared using Rt2 Minv Pinv Qinv MAV

Thiessen Polygon catchment average rainfall – estimate of ‘True’ rainfall Transfer function model identified from catchment average rainfall and catchment outflow – simulated flow time-series generated PTP RegDer inversion method applied and estimate of catchment rainfall inferred from outflow Qobs Qsim MAV Inferred rainfall used to generate inferred flow time-series – an estimate of the catchment outflow RegDer The first step in the analysis uses the catchment average rainfall and catchment outflow. The model fit is assessed and the outflow estimates compared using Rt2 (the coefficient of determination) where rt2=1 would show a perfect fit between the observed and simulated data. Fit of model gives first indication of how good representative the calculated average (our best available estimate of the TRUE rainfall) might be. Stage 1 Catchment average rainfall calculated using Thiessen polygon method (Pav) Pav and observed catchment outflow (Qobs) used to identify model Flow simulated from model (Qsim) estimate of Qobs Model inverted using the RegDer method Qobs used as input Output inferred rainfall series (Peh) Peh used to generate inferred estimate of flow (Qinv) Qobs, Qsim and Qinv compared using Rt2 Minv Observed, simulated and inferred flow time-series compared Pinv Qinv Qobs MAV

Analysis Individual gauges

Analysis – individual gauges
M1 Using the rainfall input from each individual gauge and the catchment outflow, identify a series of models Models are fitted to the rainfall measured at each individual gauge and the catchment outflow. Inverting these models and using the inferred rainfall to generate an estimate of the catchment outflow allows the contribution of each gauge to be assessed by comparing Rt2 values and assessing the hydro and hyetographs by eye. Stage 2 Measured rainfall (Pobs 1:23) from individual gauges and observed catchment outflow (Qobs) used to identify a series of models Flow simulated from each model (Qsim 1:23) are estimates of Qobs Models inverted using the RegDer method Qobs used as input to each model Outputs are inferred rainfall series (Peh 1:23) Peh 1:23 used to generate inferred estimates of flow (Qinv 1:23) Qobs, Qsim (1:23) and Qinv(1:23) compared using Rt2

M1 to Using the rainfall input from each individual gauge and the catchment outflow, identify a series of models Models are fitted to the rainfall measured at each individual gauge and the catchment outflow. Inverting these models and using the inferred rainfall to generate an estimate of the catchment outflow allows the contribution of each gauge to be assessed by comparing Rt2 values and assessing the hydro and hyetographs by eye. Stage 2 Measured rainfall (Pobs 1:23) from individual gauges and observed catchment outflow (Qobs) used to identify a series of models Flow simulated from each model (Qsim 1:23) are estimates of Qobs Models inverted using the RegDer method Qobs used as input to each model Outputs are inferred rainfall series (Peh 1:23) Peh 1:23 used to generate inferred estimates of flow (Qinv 1:23) Qobs, Qsim (1:23) and Qinv(1:23) compared using Rt2

M1 to Using the rainfall input from each individual gauge and the catchment outflow, identify a series of models Invert each model and use to infer estimates of the catchment average rainfall time series for each gauge RegDer Minv1 RegDer Minv23 to Models are fitted to the rainfall measured at each individual gauge and the catchment outflow. Inverting these models and using the inferred rainfall to generate an estimate of the catchment outflow allows the contribution of each gauge to be assessed by comparing Rt2 values and assessing the hydro and hyetographs by eye. Stage 2 Measured rainfall (Pobs 1:23) from individual gauges and observed catchment outflow (Qobs) used to identify a series of models Flow simulated from each model (Qsim 1:23) are estimates of Qobs Models inverted using the RegDer method Qobs used as input to each model Outputs are inferred rainfall series (Peh 1:23) Peh 1:23 used to generate inferred estimates of flow (Qinv 1:23) Qobs, Qsim (1:23) and Qinv(1:23) compared using Rt2

M1 to Using the rainfall input from each individual gauge and the catchment outflow, identify a series of models Invert each model and use to infer estimates of the catchment average rainfall time series for each gauge RegDer Minv1 RegDer Minv23 Infer outflow estimates for each gauge and compare with simulated and observed outflow time-series to Qi1 M1 Qi23 to Models are fitted to the rainfall measured at each individual gauge and the catchment outflow. Inverting these models and using the inferred rainfall to generate an estimate of the catchment outflow allows the contribution of each gauge to be assessed by comparing Rt2 values and assessing the hydro and hyetographs by eye. Stage 2 Measured rainfall (Pobs 1:23) from individual gauges and observed catchment outflow (Qobs) used to identify a series of models Flow simulated from each model (Qsim 1:23) are estimates of Qobs Models inverted using the RegDer method Qobs used as input to each model Outputs are inferred rainfall series (Peh 1:23) Peh 1:23 used to generate inferred estimates of flow (Qinv 1:23) Qobs, Qsim (1:23) and Qinv(1:23) compared using Rt2

M1 to Using the rainfall input from each individual gauge and the catchment outflow, identify a series of models Invert each model and use to infer estimates of the catchment average rainfall time series for each gauge RegDer Minv1 RegDer Minv23 Infer outflow estimates for each gauge and compare with simulated and observed outflow time-series to Qi1 M1 Qi23 to Qobs Qsim 1:23 Qi 1:23 Models are fitted to the rainfall measured at each individual gauge and the catchment outflow. Inverting these models and using the inferred rainfall to generate an estimate of the catchment outflow allows the contribution of each gauge to be assessed by comparing Rt2 values and assessing the hydro and hyetographs by eye. Stage 2 Measured rainfall (Pobs 1:23) from individual gauges and observed catchment outflow (Qobs) used to identify a series of models Flow simulated from each model (Qsim 1:23) are estimates of Qobs Models inverted using the RegDer method Qobs used as input to each model Outputs are inferred rainfall series (Peh 1:23) Peh 1:23 used to generate inferred estimates of flow (Qinv 1:23) Qobs, Qsim (1:23) and Qinv(1:23) compared using Rt2

Results – example events

Summer event (SED4) – 2/8/1994 – 7/8/1994
Highly variable in space and flashy in time (darker colours, more rain) Average over 23 gauges Summer event – 5 days August 1994 Occurred in the summer when flow is low. It is localised in time but widespread in space varying in intensity across the catchment as can be seen from the rainfall histogram and 3D plot. The single event produces a single flow event. The Thiessen polygons are coloured by total rainfall that fell during the event (darker colour means more rain) and indicates that the more intense rain fell on the E of the catchment. Column chart shows totals by gauge in graphical form and variation from the mean (Error bars are 1 std deviation in length)

Autumn event (SS5) – 9/9/1995 – 13/9/1995
Variable in space and time (darker colours, more rain) Average over 23 gauges Autumn event – 5 days September 1995 Occurred in early Autumn and shows a longer period of rainfall patchy in intensity over the catchment. Flow remains low and despite the longer period of rain, only one main flow event is triggered.

Winter event (JANS) – 26/1/1995 – 29/1/1995
Persistent in time and space (darker colours, more rain) Average over 23 gauges Winter event – 4 days January 1995 Occurred in the winter when flow is high. Rainfall is more sustained although it can be seen to vary in both space and time though less than the previous events. Rain falls over most of the catchment showing a spatial variation in intensity with the highest amounts of rain fall on the higher ground. The flow is responsive to the rainfall events and shows several peaks.

Example events Catchment average rainfall and catchment outflow

Catchment average rainfall and outflow - Example events
Observed and simulated outflow (catchment rainfall) Summer Event 0.962 0.722 Autumn Event The plots in the first column show the observed catchment average outflow (in blue) over plotted by the flow simulated using the identified model (in red). The simulated flow shows spikes that do not occur in the observed but do occur in the average rainfall. The model is using information from the rainfall not all of which is required to generate the flow. The second column of plots shows the observed catchment outflow (in blue) over plotted by the flow simulated using the identified model with inferred catchment rainfall as the input. In 2 cases the fit is almost perfect and in the 3rd is slightly improved over the original simulation using rainfall alone however the patchy rainfall distribution of event Autumn event has affected the estimate of the catchment average to extent that even reverse hydrology cannot extract enough information to give a good fit. (spatial influence on the estimate of the catchment average) Inferred rainfall is been generated by inverting the model and using streamflow as the input and thus contains information on rainfall over the whole catchment. Column 3 shows the catchment average rainfall (blue line) over-plotted by the inferred rainfall (red line). The rainfall inferred from the catchment outflow follows a much smoother profile than the observed rainfall. It makes use of the information contained in the streamflow as well as the information in the rainfall to extract the characteristics of the rainfall pattern that are essential for generating the streamflow. It is at a lower sampling resolution than the original observations but we have shown in a previous paper (Kretzschmar et al 2015 and conference publication) that the dynamics of the system are being maintained. This loss of resolution is the price that must be paid for the numerical stability of the RegDer method. The poorer fit of Autumn event is a result of the model parameter estimation being influenced by calculating the average using of individual gauges (perhaps several in this case) that do not represent the catchment rainfall dynamics sufficiently well i.e. they are providing disinformation. The other events may also have gauges that influence the average adversely. Perhaps reverse hydrology can provide us with a means of identifying them….. Winter Event 0.944

Observed and simulated outflow (catchment rainfall) Observed and inferred catchment rainfall Summer Event 0.962 0.722 Autumn Event The plots in the first column show the observed catchment average outflow (in blue) over plotted by the flow simulated using the identified model (in red). The simulated flow shows spikes that do not occur in the observed but do occur in the average rainfall. The model is using information from the rainfall not all of which is required to generate the flow. The second column of plots shows the observed catchment outflow (in blue) over plotted by the flow simulated using the identified model with inferred catchment rainfall as the input. In 2 cases the fit is almost perfect and in the 3rd is slightly improved over the original simulation using rainfall alone however the patchy rainfall distribution of event Autumn event has affected the estimate of the catchment average to extent that even reverse hydrology cannot extract enough information to give a good fit. (spatial influence on the estimate of the catchment average) Inferred rainfall is been generated by inverting the model and using streamflow as the input and thus contains information on rainfall over the whole catchment. Column 3 shows the catchment average rainfall (blue line) over-plotted by the inferred rainfall (red line). The rainfall inferred from the catchment outflow follows a much smoother profile than the observed rainfall. It makes use of the information contained in the streamflow as well as the information in the rainfall to extract the characteristics of the rainfall pattern that are essential for generating the streamflow. It is at a lower sampling resolution than the original observations but we have shown in a previous paper (Kretzschmar et al 2015 and conference publication) that the dynamics of the system are being maintained. This loss of resolution is the price that must be paid for the numerical stability of the RegDer method. The poorer fit of Autumn event is a result of the model parameter estimation being influenced by calculating the average using of individual gauges (perhaps several in this case) that do not represent the catchment rainfall dynamics sufficiently well i.e. they are providing disinformation. The other events may also have gauges that influence the average adversely. Perhaps reverse hydrology can provide us with a means of identifying them….. Winter Event 0.944

Observed and simulated outflow (catchment rainfall) Observed and simulated outflow (inferred rainfall) Observed and inferred catchment rainfall Summer Event 0.962 0.996 0.722 Autumn Event 0.752 The plots in the first column show the observed catchment average outflow (in blue) over plotted by the flow simulated using the identified model (in red). The simulated flow shows spikes that do not occur in the observed but do occur in the average rainfall. The model is using information from the rainfall not all of which is required to generate the flow. The second column of plots shows the observed catchment outflow (in blue) over plotted by the flow simulated using the identified model with inferred catchment rainfall as the input. In 2 cases the fit is almost perfect and in the 3rd is slightly improved over the original simulation using rainfall alone however the patchy rainfall distribution of event Autumn event has affected the estimate of the catchment average to extent that even reverse hydrology cannot extract enough information to give a good fit. (spatial influence on the estimate of the catchment average) Inferred rainfall is been generated by inverting the model and using streamflow as the input and thus contains information on rainfall over the whole catchment. Column 3 shows the catchment average rainfall (blue line) over-plotted by the inferred rainfall (red line). The rainfall inferred from the catchment outflow follows a much smoother profile than the observed rainfall. It makes use of the information contained in the streamflow as well as the information in the rainfall to extract the characteristics of the rainfall pattern that are essential for generating the streamflow. It is at a lower sampling resolution than the original observations but we have shown in a previous paper (Kretzschmar et al 2015 and conference publication) that the dynamics of the system are being maintained. This loss of resolution is the price that must be paid for the numerical stability of the RegDer method. The poorer fit of Autumn event is a result of the model parameter estimation being influenced by calculating the average using of individual gauges (perhaps several in this case) that do not represent the catchment rainfall dynamics sufficiently well i.e. they are providing disinformation. The other events may also have gauges that influence the average adversely. Perhaps reverse hydrology can provide us with a means of identifying them….. Winter Event 0.944 0.999

Identifying disinformative gauges
Example events Individual gauges Each gauge used to calculate the catchment average provides information which varies from event to event due to the spatial variation of the rainfall field. As each gauge only provides part of the information used in the previous analysis, following the same procedure using the observed rainfall measured at each gauge allows us to examine their contribution to the catchment average. Comparing the simulated outflow using each gauge should give a measure of the amount of information the gauge is supplying given that ONLY the rainfall that gauge is being used as an estimate of the catchment average.

Example events, individual example gauges
Summer event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.962 0.958 0.957 0.965 0.961 Qinv 0.996 0.995 0.997 BATC KILK CRAW Catchment average Catchment average FRAN FRAN

Summer event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.962 0.958 0.957 0.965 0.961 Qinv 0.996 0.995 0.997 BATC KILK - Observed and simulated outflow (measured rainfall) KILK CRAW Catchment average Catchment average FRAN FRAN

Summer event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.962 0.958 0.957 0.965 0.961 Qinv 0.996 0.995 0.997 BATC KILK - Observed and simulated outflow (measured rainfall) KILK CRAW KILK - Observed catchment rainfall and inferred catchment rainfall Catchment average Catchment average FRAN FRAN

Summer event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.962 0.958 0.957 0.965 0.961 Qinv 0.996 0.995 0.997 BATC KILK - Observed and simulated outflow (inferred rainfall) KILK CRAW KILK - Observed catchment rainfall and inferred catchment rainfall Catchment average FRAN

Winter event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.944 0.905 0.954 0.934 0.738 Qinv 0.999 0.998 0.913 BATC Average a good fit All gauges except FRAN show good fit KILK CRAW Catchment average FRAN

Winter event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.944 0.905 0.954 0.934 0.738 Qinv 0.999 0.998 0.913 BATC KILK - Observed and simulated outflow (measured rainfall) KILK CRAW Catchment average FRAN

Winter event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.944 0.905 0.954 0.934 0.738 Qinv 0.999 0.998 0.913 BATC KILK - Observed and simulated outflow (measured rainfall) KILK CRAW KILK - Observed catchment rainfall and observed gauge rainfall Catchment average FRAN

Winter event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.944 0.905 0.954 0.934 0.738 Qinv 0.999 0.998 0.913 BATC FRAN - Observed and simulated outflow (measured rainfall) KILK CRAW Catchment average FRAN

Winter event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.944 0.905 0.954 0.934 0.738 Qinv 0.999 0.998 0.913 BATC FRAN - Observed and simulated outflow (measured rainfall) KILK CRAW FRAN - Observed catchment rainfall and observed gauge rainfall Catchment average FRAN

Winter event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.944 0.905 0.954 0.934 0.738 Qinv 0.999 0.998 0.913 BATC KILK - Observed and simulated outflow (measured rainfall) KILK CRAW Catchment average FRAN

Winter event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.944 0.905 0.954 0.934 0.738 Qinv 0.999 0.998 0.913 BATC KILK - Observed and simulated outflow (measured rainfall) KILK CRAW KILK - Observed catchment rainfall and inferred catchment rainfall Catchment average FRAN

Winter event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.944 0.905 0.954 0.934 0.738 Qinv 0.999 0.998 0.913 BATC FRAN - Observed and simulated outflow (measured rainfall) KILK CRAW Catchment average FRAN

Winter event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.944 0.905 0.954 0.934 0.738 Qinv 0.999 0.998 0.913 BATC FRAN - Observed and simulated outflow (measured rainfall) KILK CRAW FRAN - Observed catchment rainfall and inferred catchment rainfall Catchment average FRAN

Autumn event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.722 0.647 0.645 0.881 0.863 Qinv 0.754 0.718 0.704 0.998 BATC CRAW - Observed and simulated outflow (measured rainfall) KILK CRAW Catchment average FRAN

Autumn event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.722 0.647 0.645 0.881 0.863 Qinv 0.754 0.718 0.704 0.998 BATC CRAW - Observed and simulated outflow (measured rainfall) KILK CRAW CRAW - Observed catchment rainfall and observed gauge rainfall Catchment average FRAN

Autumn event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.722 0.647 0.645 0.881 0.863 Qinv 0.754 0.718 0.704 0.998 BATC KILK - Observed and simulated outflow (measured rainfall) KILK CRAW Catchment average FRAN

Autumn event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.722 0.647 0.645 0.881 0.863 Qinv 0.754 0.718 0.704 0.998 BATC KILK - Observed and simulated outflow (measured rainfall) KILK CRAW KILK - Observed catchment rainfall and observed gauge rainfall Catchment average FRAN

Autumn event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.722 0.647 0.645 0.881 0.863 Qinv 0.754 0.718 0.704 0.998 BATC CRAW - Observed and inferred outflow (inferred rainfall) KILK CRAW Catchment average FRAN

Autumn event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.722 0.647 0.645 0.881 0.863 Qinv 0.754 0.718 0.704 0.998 BATC CRAW - Observed and inferred outflow (inferred rainfall) KILK CRAW CRAW - Observed catchment rainfall and inferred catchment rainfall Catchment average FRAN

Autumn event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.722 0.647 0.645 0.881 0.863 Qinv 0.754 0.718 0.704 0.998 BATC KILK - Observed and inferred outflow (inferred rainfall) KILK CRAW Catchment average FRAN

Autumn event – Qsim and Qinv fit to Qobs (Rt2) Example rain-gauges Catchment average KILK BATC CRAW FRAN Qsim 0.722 0.647 0.645 0.881 0.863 Qinv 0.754 0.718 0.704 0.998 BATC KILK - Observed and inferred outflow (inferred rainfall) KILK CRAW KILK - Observed catchment rainfall and inferred catchment rainfall Catchment average FRAN

To sum up ….. In general, reverse hydrology
In practice it is unlikely that a dense gauge network such as the Brue will exist to enable a good catchment average to be obtained. We will only have rainfall records from a few gauges and the catchment outflow to work with. The fewer the gauges, the more important it becomes to identify any misrepresentative gauges as they will carry much more weight and bias the estimate of the average (TRUE rainfall) Fitting a model to the gauge rainfall and catchment outflow will give an indication about how representative the gauge might be from the Rt2 value. Inverting the model and comparing the simulated outflow with the catchment outflow indicates whether the gauge can be safely incorporated in the calculation of the catchment average. If the inferred rainfall captures the essential flow generating dynamics of the catchment it can be safely included. Inversion is able to do this because it utilises information contained in the catchment outflow pertaining to the rainfall over the whole catchment and characteristics of the catchment itself that transform the rainfall into flow.

Improves the estimate of the rainfall generating the catchment outflow by utilising information it contains In practice it is unlikely that a dense gauge network such as the Brue will exist to enable a good catchment average to be obtained. We will only have rainfall records from a few gauges and the catchment outflow to work with. The fewer the gauges, the more important it becomes to identify any misrepresentative gauges as they will carry much more weight and bias the estimate of the average (TRUE rainfall) Fitting a model to the gauge rainfall and catchment outflow will give an indication about how representative the gauge might be from the Rt2 value. Inverting the model and comparing the simulated outflow with the catchment outflow indicates whether the gauge can be safely incorporated in the calculation of the catchment average. If the inferred rainfall captures the essential flow generating dynamics of the catchment it can be safely included. Inversion is able to do this because it utilises information contained in the catchment outflow pertaining to the rainfall over the whole catchment and characteristics of the catchment itself that transform the rainfall into flow.

Improves the estimate of the rainfall generating the catchment outflow by utilising information it contains Can help identify inconsistent or disinformative data In practice it is unlikely that a dense gauge network such as the Brue will exist to enable a good catchment average to be obtained. We will only have rainfall records from a few gauges and the catchment outflow to work with. The fewer the gauges, the more important it becomes to identify any misrepresentative gauges as they will carry much more weight and bias the estimate of the average (TRUE rainfall) Fitting a model to the gauge rainfall and catchment outflow will give an indication about how representative the gauge might be from the Rt2 value. Inverting the model and comparing the simulated outflow with the catchment outflow indicates whether the gauge can be safely incorporated in the calculation of the catchment average. If the inferred rainfall captures the essential flow generating dynamics of the catchment it can be safely included. Inversion is able to do this because it utilises information contained in the catchment outflow pertaining to the rainfall over the whole catchment and characteristics of the catchment itself that transform the rainfall into flow.

Improves the estimate of the rainfall generating the catchment outflow by utilising information it contains Can help identify inconsistent or disinformative data Extract the aspects of the rainfall time-series that generate floods from noisy raw rainfall data In practice it is unlikely that a dense gauge network such as the Brue will exist to enable a good catchment average to be obtained. We will only have rainfall records from a few gauges and the catchment outflow to work with. The fewer the gauges, the more important it becomes to identify any misrepresentative gauges as they will carry much more weight and bias the estimate of the average (TRUE rainfall) Fitting a model to the gauge rainfall and catchment outflow will give an indication about how representative the gauge might be from the Rt2 value. Inverting the model and comparing the simulated outflow with the catchment outflow indicates whether the gauge can be safely incorporated in the calculation of the catchment average. If the inferred rainfall captures the essential flow generating dynamics of the catchment it can be safely included. Inversion is able to do this because it utilises information contained in the catchment outflow pertaining to the rainfall over the whole catchment and characteristics of the catchment itself that transform the rainfall into flow.

To sum up ….. In general, reverse hydrology However
Improves the estimate of the rainfall generating the catchment outflow by utilising information it contains Can help identify inconsistent or disinformative data Extract the aspects of the rainfall time-series that generate floods from noisy raw rainfall data However In some cases, the disinformation from a single gauge is too great for even inversion to extract the essential dynamics In practice it is unlikely that a dense gauge network such as the Brue will exist to enable a good catchment average to be obtained. We will only have rainfall records from a few gauges and the catchment outflow to work with. The fewer the gauges, the more important it becomes to identify any misrepresentative gauges as they will carry much more weight and bias the estimate of the average (TRUE rainfall) Fitting a model to the gauge rainfall and catchment outflow will give an indication about how representative the gauge might be from the Rt2 value. Inverting the model and comparing the simulated outflow with the catchment outflow indicates whether the gauge can be safely incorporated in the calculation of the catchment average. If the inferred rainfall captures the essential flow generating dynamics of the catchment it can be safely included. Inversion is able to do this because it utilises information contained in the catchment outflow pertaining to the rainfall over the whole catchment and characteristics of the catchment itself that transform the rainfall into flow.

Conclusions The RegDer inversion method extracts the essential rainfall signal required to generate streamflow from the noisy input

Conclusions The RegDer inversion method extracts the essential rainfall signal required to generate streamflow from the noisy input Inversion provides a way of assessing whether a rain- gauge is supplying disinformation to the catchment average

Limitations and further work
When flow is low, models do not fit or invert well The information supplied by each gauge varies from event to event

Limitations and further work
When flow is low, models do not fit or invert well The information supplied by each gauge varies from event to event Future work Test on a wider range of events and catchments Look at longer periods of data rather than individual events Investigate the effect of rain-gauge network density Investigate the potential for gap filling and record extension

Take home message …. Although a network of gauges is preferable, if only a single gauge (or a very sparse network) is available, it is generally better to use inferred rainfall to generate streamflow than observed rainfall from a single gauge

Take home message …. Although a network of gauges is preferable, it is generally better to use inferred rainfall than observed rainfall from a single gauge to generate catchment outflow …… Although a network of gauges is preferable, if only a single gauge (or a very sparse network) is available, it is generally better to use inferred rainfall to generate streamflow than observed rainfall from a single gauge

Acknowledgements and Thanks
Funding: NERC CREDIBLE project British Hydrological Society Lancaster University Faculty of Science and Technology Lancaster University Graduate College

Acknowledgements and Thanks
Funding: NERC CREDIBLE project British Hydrological Society Lancaster University Faculty of Science and Technology Lancaster University Graduate College Support: My long suffering supervisors Wlodek Tych, Nick Chappell and Keith Beven from Lancaster Environment Centre

Thank you for listening 들어 주셔서 감사합니다

Thank you for listening 들어 주셔서 감사합니다 Any questions? 질문?

References [1] Taylor, C.J., Pedregal, D.J., Young, P.C., Tych, W., (2007). Environmental time series analysis and forecasting with the captain toolbox. Environmental Modelling & Software 22 (6), 797–814. [2] Kretzschmar, A., Tych, W and Chappell, N. A. (2014) Reversing hydrology: Estimation of sub-hourly rainfall time-series from streamflow. Environmental Modelling & Software 60: [3] Kretzschmar, A, Tych, W, Chappell, N & Beven, K (2015), 'Reversing hydrology: quantifying the temporal aggregation effect of catchment rainfall estimation using sub-hourly data' Hydrology Research /nh

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