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Put another way: the volume of water within a reach changes at a rate equal to the difference between the inflow and outflow rates to and from the reach.
For streamflow flow routing Source uses MUSICX uses an implicit Euler scheme which can be expressed as (Clark and Kavetski, 2010):
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To make use of the continuity equation (Equation 1) it is necessary to relate the storage in the reach to the flow entering and leaving that reach. Source MUSICX uses the Muskingum storage function (Koussis, 2009):
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From which the Muskingum storage function (Equation 4) can be rewritten as:
| Equation 6 |  |
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Source offers MUSICX offers a form of variable parameter Muskingum routing in which the value of x is a constant but the value of K is allowed to vary with flow (Koussis, 1978). From the Kleitz-Seddon rule (Brutsaert, 2005: p.190) the speed that a small monoclinal wave will move through a reach is:
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For the convenience of the model users (it being easier to think in terms of the wave travel times) Source MUSICX defines a variable K which uses the wave travel time interpretation. Combined with the assumption that the flow rate in the reach is represented by the index flow q:
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The routing formulation implemented in Source uses MUSICX uses an implicit Euler scheme with the assumptions that:
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Information on evaluating k (or travel time or K), x and m is given in a number of sources such as Pilgrim (1987). It is worth noting that when discussing storage-discharge relationships, Pilgrim (1987: eg equation 7.4) refers to the term 
being a "representative discharge for a reach" and this is the same as the index flow used here. Pilgrim (1987: Figure 7.21) also recasts the wave speed-discharge relationship developed by Wong and Laurenson (1983, 1984) into a relationship between time of travel of flood peak and discharge, which is more relevant to the way routing in Source MUSICX would be typically applied. However, note that the discussion of aspects such as stability criteria by Pilgrim (1987) applies to the implicit Heun scheme and not the implicit Euler scheme used in SourceMUSICX.
Output data
Outputs include time series of link outflows and modelled fluxes.
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