Quoted from: Grayson, Rodger B., Ian D. Moore, and Thomas A. Mcmahon. "Physically based hydrologic modeling: 1. A terrain‐based model for investigative purposes." Water resources research 28, no. 10 (1992): 2639-2658. https://doi.org/10.1029/92WR01258 

      THALES incorporates both the Hortonian mechanism of surface runoff as well as a representation of variable source area runoff and exfiltration of subsurface flow. Surface runoff can be modeled as sheet，rill or channelized flow. Evaporation is not represented since the purpose of the model is to simulate individual events.

      TAPES-C (Topographic Analysis Programs for the Environmental Sciences: Contour)

      TAPES-C is a set of computer programs that generate a network of interconnected elements on a surface defined by elevation isolines. The analysis has developed from the original work of O'Loughlin [1986] in which an automated method for deriving streamlines on a catchment was presented, but this was only applicable to steady state analysis. Moore et al. [1988b] describe a development of this work using the "stream tube" approach of Onstad and Brakensiek[1968]. The most recent version of TAPES-C derives a series of interconnected elements which are an ideal structure for the dynamic simulation of catchment processes and is described by Moore and Grayson [1991].

      An application of TAPES-C to the 7-ha catchment at Wagga Wagga in New South Wales, Australia, is presented in figure 1. For each element bounded by adjacent stream-lines and contours, the following attributes are calculated: element area, total upslope contributing area, element num-ber of the upslope and downslope connecting elements,x, y,z coordinates of the element centroid,x,y,z coordinates of the midpoint on the downslope boundary of the element, average slope of an element, the width of the upslope and downslope boundaries of the element，the length of the element, the aspect or azimuth of the element and the plan curvature.

      The model requires either a file of digitized contours or access to a digitizing table to provide the input information. Boundaries of the catchment can be digitized by the user or the program can calculate the boundary based on trajectories from points at the catchment outlet.

      Hydrologic Component

      Separate surface and subsurface models based on the TAPES-C analysis were presented by Moore and Grayson[1991]. THALES is a combination of those two models and is named after Thales of Miletos, a Greek philosopher who recognized the influence of topography on runoff generation.It enables a wide range of hydrologic processes to berepresented，allowing the characteristics of the particular catchment and storm to determine the actual response. By modeling the subsurface movement of water as well as the infiltration process, the continuum of hydrologic responses, excluding base flow from deep aquifers, can be simulated. An early version of THALES, containing different algorithms but the same basic structure as the version presented here, was described by Grayson et al.[1988].

      THALES is a relatively simple model. More complex and computationally demanding approaches based on Richards'sequation (e.g., SHE[Abbott et al., 1986a,b]) could be used but as Beven [1989] stated, the uncertainty of the process descriptions and parameter values discount any advantage in model applications to real catchments. The elements and attributes calculated in TAPES-C define the area within which model parameters are assumed to be constant and provide the element network for the routing of water. The model is based on the assumption that infiltrated water flows downslope in a saturated layer overlying an impermeable base and is modeled as kinematic subsurface flow. If the subsurface flow rate exceeds the capacity of the soil profile to transmit the water, surface saturation occurs and rain falling on the saturated areas becomes direct runoff. Runoff can also be generated by Hortonian overland flow and the infiltrated water is assumed to flow vertically through an unsaturated zone to become part of the saturated layer from where it can flow downslope.

      The element structure of THALES, derived from TAPES-C, allows each element to have different infiltration, surface and subsurface flow parameters although this is usually not practical because of the large data requirements. In most applications, the parameters are measured for each soil type or region of different surface condition and it is assumed that these parameters do not vary within each region or soil type. A map of the regions of different parameters can then be numerically overlaid on the element map of the catchment. This approach is based on the assumption that the variation in parameter values within a region is less than that between regions and that the within-element characteristics are uniform. Neither of these assumptions is necessarily true [e.g., Beven,1989; Sharma et al., 1980; Loague and Gander,1990].

The infiltration rate and volume as well as the rainfall excess are computed for each element using either a relationship based on an equivalence between the Green and Ampt and the Horton infiltration capacity equations [Morel-Seytoux，1988a,b]or the relationship of Smith and Parlange [1978]. Variable rainfall intensity infiltration is simulated, but the rainfall intensity in any tine increment t is assumed constant. The Morel-Seytoux formulation was described by Moore and Grayson [1991]. The Smith and Parlange model is applied in a similar way and can be expressed in the form

where f is the infiltration rate, K, is the saturated hydraulic conductivity, F is the volume of water infiltrated and B =Hc(0s- 0i) where H is the capillary drive，0, is the saturated water content and 0; is the initial water content.

      Application of this equation requires an implicit solution if the time interval used in the model is large. For short time intervals (10-30 s) a rapid algorithm for the calculation of infiltration rates is provided by an explicit solution based on the infiltrated volume at the previous time step.

      During periods of no rain, the soil profile drains, and for the next rainfall period, a new value of e; is used to determine the value of B in (1). During drainage, the soil water profile is assumed to be rectangular and the hydraulic conductivity is related to soil water content by the Brooks and Corey [1964] relationship that has the hydraulic conductivity (K(0)), the soil water content (0), the residual soil-water content (0,) and the pore size index as parameters.

      The water that enters the soil profile is added to the unsaturated zone which then discharges into the underlying saturated zone; i.e., only vertical flow is modeled in the unsaturated zone. The rate at which water enters the saturated zone is given by the hydraulic conductivity computed from the Brooks and Corey [1964] relationship, i.e., it is assumed that the hydraulic gradient is unity. The soil water content is determined from a volume balance of the unsaturated zone. The total unsaturated soil volume for an element changes over time as the saturated depth varies and a water balance is used to redistribute the unsaturated soil water content as a result of a change in total unsaturated soil volume. Lateral flow enters and exits the element via the saturated zone. If the saturated zone reaches the surface, saturated source area runoff occurs and exfiltration of sub-surface flow is possible. This exfiltrated water adds to the rainfall excess and is routed using the four-point finite-difference scheme of Brakensiek [1967].

      Surface runoff may continue after rainfall ceases and this surface runoff can infiltrate in a downslope element if the soil profile of the downslope element is not saturated, i.e, where Hortonian overland flow is occurring. This runoff-run-on problem is likely to be important where broadsheet flow occurs, but relatively unimportant when the flow concern-trates in defined rills, channels and streams. The input parameters required by the model are water content at field capacity, drainable porosity, effective hydraulic conductivity of the soil profile and the coefficient and exponent of the flow area -discharge relationship (given, for example, by Manning's equation or the Darcy-Weisbach equation) as well as the infiltration parameters [Moore and Grayson, 1991].

      The solution scheme begins at the uppermost element, progressing along a contour and. then down to the next element on the lower contour. This continues until the last element is reached, after which the calculations for the next time step begin at the uppermost element. For each element the streamlines define no-flow boundaries, the upper contour is an inflow boundary while the lower outflow boundary is the computational “node." The simulation calculates the surface and subsurface flow, infiltration rate, exfiltration rate, and rainfall excess for every element at every time step. Any of these data can be written to output files for later analysis. The elements where the conditions for kinematic overland flow are violated, based on the kinematic flow number [Woolhiser and Ligett, 1967] are identified and checks are made of local (element) and global (catchment) water balances. The convergence criteria for the flow routing algorithms as well as the time step, start and finish times of the simulation, are chosen by the user.

      THALES includes the option of applying a nonuniform distribution of antecedent soil water conditions. The distribution scales the initial water content of an element from the element's value of In (A/b tan B) where A/b is the specific catchment area (the upslope contributing area per unit width) and B is the surface slope angle [Beven and Kirkby, 1979]. The initial soil water content for a particular element is input by the user and the algorithm scales the appropriate value to all other elements. If the distribution of In (Aib tan B) is accurate in representing the variation in soil water content within a catchment, use of this option reduces the running time of the model because there is no need to establish initial soil water conditions by simulating long periods prior to the events of interest. In some cases, it is preferable to use a distribution that includes the effect on antecedent soil water conditions of evaporation between rainfall events. For example, Moore et al. [1988a] demonstrated that a combination of In (A/b tan B) and aspect (representing evaporation) was a useful predictor of surface soil water content. It is a simple matter to include such algorithms if required.

      Several other options are possible that enable the exploration of common assumptions used in hydrologic modeling. These relate mainly to the representation of overland flow, which can be represented as broadsheet flow, rill flow using the equations such as those described by Moore and Burch [1986], or channelized flow if the location and properties of the channel are known.