The Concept of Runoff Elements As a Basis of Scale-Free Approach To Runoff Formation Modelling the experience of “Hydrograph” model development and implementation Yu.B. Vinogradov, O. Semenova State Hydrological Institute St. Petersburg, Russia

To describe the water dynamics in river basin - slope and channel transformation Scientific objective Modelling objective To estimate the water income to channel system and finally to the basin outlet Main tasks 1.Infiltration, water movement in the soil layer, formation of classical surface and subsurface flow 2. Slope (surface, subsurface and underground) inflow to channel network 3. Channel flow and lag time

Main modelling laws Adequacy to natural processes The parameters of the equations should not depend on the argument Use of only available data (otherwise, simplification of the model)

The basic equations of water dynamics in a “physically-based” models… violation of the main modelling rules? Water dynamics in soil – the Richard’s equation Slope and channel flow – the equation of Saint- Venant or kinematic wave Underground flow – the Boussinesq equation

The Richard’s equation (moisture diffusion/conductivity)  – volume moisture D() – moisture diffusion coefficient K( ) – hydraulic conductivity t – time, z – soil depth Parallel between the “diffusion of soil water” and heat conductivity – suspended moisture existence is impossible? The Hallaire's effect Nonlinearly dependence of D and K on  Change of D in 10 4 and K in times corresponds to the range of natural variation of 

t - time, x – distance, H – depth, V – velocity, q – inflow, C – Chezy coefficient, The equation of Saint-Venant (kinematic wave, etc…) Water movement is represented by thin continuous water layer Disagreement between units of flow depth (mm) and grid sizes (km) Requirement for absent information (morphology, roughness) No possibility to evaluate the described process except by runoff in the outlet Need of calibration – different methods – different parameters Exaggeration of slope distances and underestimation of slope inclination in spatial schematization – coefficients

The Boussinesq equation True structure of the underground aquifers is unknown Information on quantity and stratigraphy of impervious beds and aquifers (capacity, inclinations, connection with underground and external channel system, filtration and water yield coefficients) is required and ABSENT The fact of great groundwater storage is not taken into account K,  – filtration and water yield coefficients of rock, H – level of ground waters over the surface of impervious bed or piezometric head in the case of heady movement, t – time, x – distance

What fee do we pay for the “physical base” of such models? Nonlinearity Uniqueness Uncertainty Equifinality Scale [Beven, 2001] Is not it a time for some other methods? Should we stop to pretend for our knowledge of real runoff processes?

The concept of runoff elements Basin – elementary slopes or watersheds – runoff elements system Runoff element: Natural formation Part of elementary slope limited by micro-divides directed with its open part to the slope non- channel or underground drainage system The size depends on natural conditions but mainly inclination (the underground runoff elements are much larger than surface ones)

For each runoff element there is a balance ratio (1) W – water volume (m 3 ), s and q – inflow and outflow (m 3 s -1 ); t – time (s). (2) ,  - hydraulic parameters of a runoff element. a = /n and b = n, where n is a number of runoff elements. a = a * /F and b = b * F, T = 1/(a* b*), H = ln(q/b*+1)/a* a * [m -1 ], b * [ms -1 ] – standardized hydraulic coefficients, F – area, T – typical outflow time, H – water storage Q from all runoff elements of the given level to channel system is: (3) Q  - initial value of runoff Q and S is runoff formation intensity (m 3 s -1 ); t - computation time interval (sec) during which S is constant.

Assumptions with the depth… The infiltration capacity of water-holding rocks naturally decreases The outflow rate decreases The water storage increases

System of runoff elements Level Type of runoff a*a* τWater storage, mm Outflow intensity, mm/min - Surface min Soil hours Rapid ground 10 – 1 1.2–11.6 days 69.3– 195 (1 – 0.22)*6* Ground0.32 – months – 1 year 301– 674 (1 – 0.22)*6* Upper underground 0.01 – –32 years 995– 2152 (1 – 0.22)*6* Deep underground 3.2*10 -4 – 3.2* –1000 years 3161– 6812 (1 – 0.22)*6* Historical underground – – years 10000– (1 – 0.22)*6*10 -6 parameter b* is constant

It is a schematization in the conditions of full uncertainty. It doesn’t contradict the known hydro-geological items (structure and water storage). It corresponds to observed curves of runoff depletion for the basins of various sizes. It corresponds to the materials of isotope hydrology on residence time. It works for different scales and conditions…! The concept of runoff elements – what is that?

Some results… Observed (black) against simulated (red) hydrograps

Study area

LARGE-SCALE BASINS Lena at Kusur basin area 2,4 million km 2

Aldan at Verkhoyansky Perevoz basin area km 2

Kolyma at Kolymskoye, basin area km 2

Indigirka at Vorontsovo basin area km 2

Yana at Dgangky, basin area km 2

MIDDLE-SCALE BASINS Vitim at Bodaybo, basin area km 2

Uchur at Chyul’bu, basin area km 2

SMALL-SCALE BASINS Suntar at Sakharynia river mouth, basin area 7680 km 2

Timpton at Nagorny, basin area 613 km 2

Katyryk at Toko, basin area 40.2 km 2

Statistics on observed vs simulated flow (averaged for all basins in Eastern Siberia) DailyYear Nash-Sutcliffe Relative error (in absolute value) 36 %10 %

to state the fact of specific complexity and unsolveness of the problem to state the weakness of traditional “physically- based” approaches to arise the critique and following discussion on the problemto arise the critique and following discussion on the problem We aimed: And finally, we are looking for: serious cooperation in order to test our approach on isotope data

Thank you for your attention!