Marian Scott David O’Donnell, Alastair Rushworth, Kelly Gallacher, Adrian Bowman, Claire Miller

1. Modelling river networks Marian Scott David O’Donnell, Alastair Rushworth, Kelly Gallacher, Adrian Bowman, Claire Miller Statistics, University of Glasgow

2. Sensing the natural world • The Water Framework Directive makes very explicit that when we consider rivers, we need to consider the river basin as a whole • This means that we need to think about the flow connections that exist across the rivers and streams. • There have been some recent developments in terms of R packages to allow this to be more widely implemented, but it is still a research challenge.

3. Sensing the natural world • on river networks, dependence in water properties between two locations is likely to arise from the distance the water must travel between them, and whether or not other tributaries contribute en route. In this case a non-Euclidean `stream distance' must be used models must take account of the lack of dependence between pairs of locations for which one does not contribute water to the other.

4. Sensing the natural world • From a statistical modelling point of view we want to: • Develop statistical approaches to compare trends and seasonality of water quality determinands in river water at different spatial scales and accounting for spatial and temporal correlation. Specifically, we need to find appropriate spatial correlation models

5. Spatial patterns of change- at river basin scale

6. Spatial patterns of change- at river basin scale the circles represent the stations on the network, clearly not spatially representative Joint work with David O’Donnell, Mark Hallard (SEPA), Adrian Bowman, Alastair Rushworth

7. Spatial patterns of change- at river basin scale The diagram shows large hydrological areas in England and Wales and the monitoring locations More than 8000 monitoring locations. The spatial structure is becoming complex Kelly Gallacher and Claire Miller, with EA

8. Distance measures • The classical spatial model uses Euclidean distance (“as the crow flies”) • But for a river network, we might need a different measure of distance such as a stream distance Stream distance is literally the shortest distance between two locations, computed along the stream network But if we use stream distance, theoretically it has been shown that standard covariance models are not generally valid

9. Covariance models • Peterson and Ver Hoef have defined some new models: Tail-up and tail-down Tail up: the value at location s on a stream segment i depends on upstream locations and weights of the different upstream segments (often based on flow) and locations that are not flow-connected have zero covariance. Tail down: they allow correlation between flow unconnected locations- often used for fish modelling

10. Covariance models • It is possible to build covariance models that are a mixture of stream distance based and Euclidean. • The choice depends at least partly on the context that you are modelling e.g diffuse pollution, a Euclidean component would seem sensible.

11. Spatial patterns of change- at river basin scale

12. What do you need? • An enumeration of every stream segment in the catchment. A stream segment is defined as a stretch from a `source' to a confluence, a stretch between two confluences or the stretch from the lowest altitude confluence and the river mouth. So that every location on the river network can be assigned a number representing the stretch on which it lies. • Connectivity matrix Cnn, a binary matrix denoting the flow connectivity of all of the stream segments represented in the catchment, for example Cij = 1 implies that two segments i and j are connected by flow.

13. What do you need? • Distance matrix Dnn, a real matrix defining the distance along the river between every pair of elements in the catchment • High resolution set of coordinates with segment numbers, the first two columns of M give the longitude and latitude of the n `new' points on the river catchment, and the third column contains values giving the segment membership of each point.

14. What do you need? • (Optionally)Stream weightings sn, a vector describing the flow capacity of the stream segments in e (could be average flows) • Pollutant data X, which must include the columns: measured concentrations at a set of sites, dates, the stream segment number that each site corresponds to and the distance upstream of the site location

15. Development and application of spatio-temporal models respecting the river structure Flexible regression models over river networks, O’Donnell, Rushworth, Bowman, Scott and Hallard, 2014, Data from SEPA, Tweed river network, approx 80 stations, data are roughly monthly. Space- locations on the river network (s) , Time- in years (t) , and within year (z) (for the seasonal pattern) additive model including interaction terms for space and time, with residual correlation The interaction terms capture the adjustment required to explain how the time and seasonal components vary over space

16. Development and application of spatio-temporal models respecting the river structure Flexible regression models over river networks, O’Donnell, Rushworth, Bowman, Scott and Hallard, 2014, The correlation structure for the residuals is presumed separable in space and time. d captures the network separation in terms of number of stream units, t captures the time lag, w captures the n stream structure between i and j

17. Spatial patterns of change- at river basin scale Joint work with David O’Donnell, Mark Hallard (SEPA), Adrian Bowman, Alastair Rushworth

18. Spatial patterns of change- at river basin scale Joint work with David O’Donnell, Mark Hallard (SEPA), Adrian Bowman, Alastair Rushworth

19. Spatial patterns of change- at river basin scale Joint work with David O’Donnell, Mark Hallard (SEPA), Adrian Bowman, Alastair Rushworth

20. Spatial patterns of change- at river basin scale Joint work with David O’Donnell, Mark Hallard (SEPA), Adrian Bowman, Alastair Rushworth

21. Spatial patterns of change- at river basin scale- clustering Joint work with Ruth Haggarty, Claire Miller, Mark Hallard (SEPA)

22. Acknowledgements • Scottish sensors systems centre (SSSC), for funding and also GU Sensors initiative, Carbon Landscapes and Drainage (CLAD) • Colleagues in Glasgow: Adrian Bowman, Ruth Haggarty, David O’Donnell, Alastair Rushworth, Martin Coleman • Colleagues at Scottish EPA: Mark Hallard, Fiona Wylie, Malcolm Smith, Campbell Gemmell

23. References • O’Donnell et al, 2014. Flexible regression models over river networks , Appl Statist, 63(1), 47-63 • Peterson and Ver Hoef (2014). STARS an ArcGIS toolset used to calculate the spatial information needed…. J Statist Soft 56(2) • Peterson and Ver Hoef (2014) SSN: an R package for spatial statistical modelling in stream networks. J Statist Soft 56(3)

24. Video link • http://onlinelibrary.wiley.com/doi/10.1111/rssc.12024/suppinfo