The model that Neilson et al. [2010a] have developed for stream temperatures that incorporates surface heat fluxes is driven by and calibrated to in situ measurements and applied to the Virgin River in Utah, USA. Neilson et al. [2010a, 2010b] have provided numerical approximations demonstrating promising results, providing good predictions of downstream temperatures. Atmospheric exchanges are modeled using Chapra's expressions for various nodes of heat flux: long wave, shortwave, and back radiation, as well as evaporation and conduction [Chapra, 1997]. In a recent work by Heavilin and Neilson  a linearized version of this ADE is solved by means of Laplace transforms. The linearized ADE is given in equation (6) where TMC is the temperature of the main channel (°C), U is average velocity in the main channel (km d−1), D is longitudinal dispersion (km2 d−1), and where and constitute the linearization of the heat flux terms:
The solution resulting from Laplace transforms contains three terms, each representing the contribution to downstream temperatures from the boundary condition, the initial condition, and the exchange of heat with the surrounding environment. Equation (7) is the solution to equation (6) in Laplace space, where the Laplace transform of main channel and boundary condition temperatures are and , respectively. In both cases we have dropped the explicit dependency on s:
See Heavilin and Neilson  for details. As we mentioned previously, we would like to get more from the solution to the heat transport model than simply a good fit to downstream temperature profiles. Ultimately, we wish to use this solution to find a simple yet meaningful approximation for the contribution to downstream temperatures resulting from the boundary condition. This means an approximation in the original time domain to the first term in equation (7),
We apply the method described in section 2to the second factor on the right-hand side ofequation (8), , and solve for a lowest order rational polynomial expression that can be easily inverted. We begin by expanding the transformed function into a low-order Taylor series centered ats = 0 and construct the associated rational expression. The Taylor series is given by
where . We set this expansion equal to a rational polynomial. Assuming and noting the conditions placed on equation (2) from earlier, , we have and p = 0. Therefore, after normalizing , we have the relationship
Equating the Taylor series to ,
Multiplying by V1 and collecting terms in s
and equating like powers of s, we see that and . From this we construct the low-order rational function described inequation (10) that approximates , namely
Returning once again with this rational expression approximating the exponential factor in equation (8) we write
At this point we could compute the associated convolution since the term has an easily recognizable inverse, . Therefore the approximation is
For completeness we apply the same method to the remaining terms in the solution given in equation (7), and find the approximation for the contribution from the initial condition described by the second term in the solution (equation (7)) is
We can see that this term evaluates to the initial condition at t = 0, and decays to zero as time increases. The contribution from the term governing heat flux given by the third term in equation (7) is
A graph comparing these three approximations to their corresponding model results is given in Figure 2.
 However, the intent of this approach is to arrive at a nice approximation whose behavior is apparent from a closed form expression and that does not rely on additional computational power to describe the behavior of the system. Rather than wonder what the overall contribution from total boundary condition data is, we can ask what the influence of an additional temperature T0 at the boundary has on downstream temperatures. We could then assume and by the linearity of the integral, investigate the influence of a constant temperature T0 on the downstream temperatures. Substituting into equation (14) provides an expression for a constant temperature contribution at the upstream boundary. Once again we can decompose the resulting rational expression into a sum of two irreducible terms, and with some simplification
Again we see that the inverses are simply the exponential function and a constant, meaning our lowest order approximation under the assumption of a constant boundary temperature is . If we examine the approximation for A1 more closely, we notice that the term decays quickly in time, and since our question concerns downstream temperature (i.e., ), we let . This final simplification results in an expression for the influence of boundary conditions on downstream temperature,
It tells us that the boundary condition is decaying at a rate proportional to the distance from the boundary condition and at a rate approximately equal to . In Figure 3 we plot this approximation against the numerical inverse of the boundary condition term from the model in equation (12) using actual time series recorded at the most upstream site in the Virgin River by Bandaragoda and Neilson . For the purpose of illustration we will simply use the initial condition to represent a constant boundary condition temperature T0 = 25.9°C. For the convolution we use fast Fourier transforms and the Laplace inverse employs De Hoog's algorithm [De Hoog et al., 1982]. The result of this approximation effort is a very simple and concise expression that describes the decay of influence from the boundary condition on downstream temperatures. Moreover, the expression is very intuitive and clearly reveals the relationship between velocity dispersion, and those components from the surface heat flux function that contribute to exchange of energy from boundary heat.