How do catchment structure and event parameters influence the shape of transit time distributions?

Abstract:

Introduction

The way catchments store and release incoming water can be characterized by transit time distributions (TTD) that describe how much water of a specific precipitation event leaves the catchment after how much time. TTDs are utilized as transfer functions in transfer function-convolution approaches that convert incoming tracer signatures into outgoing tracer signatures. The mean of these TTDs, the mean transit time (mTT) provides valuable information on hydrochemical catchment response and biogeochemical cycling. Mostly the distributions have been assumed to be continuous, smooth and unimodal functions (e.g. gamma, advection-dispersion) that are also time-invariant. More recently there have been attempts to make the distributions variable in time and to loosen the functional shape to allow for multimodality and discontinuity. This paper is aimed at further exploring the shapes of TTDs. How they change with varying antecedent moisture conditions, varying precipitation event characteristics and event patterns, varying storage properties and varying transport parameters. Also, we use an indicator (a dimensionless number) that enables the combination of the information we obtained to allow us to predict the shapes of TTDs. This information is necessary if we want to make modeling with the transfer function-convolution approach more realistic.

Methods

In order to gain more information on TTDs we used HYDRUS-3D (a distributed physically-based hydrologic model) to track a marked precipitation event through a hillslope and record both the hydrologic response that it causes and its transit through the hillslope. Using the HYDRUS software a three-dimensional hillslope (200 m long, 45 m wide and 40 m high at the highest point) with a constant slope of 11 degrees was created (Figure a). The hillslope contained two layers, a soil layer at the surface and a bedrock layer at depth. The depth of the soil layer D_soil was either 0.5 m or 1 m to explore how soil storage capacity influences TTDs. The bedrock layer was thickest at the back of the slope (39 m) and shallowest at the outlet (1 m); it was characterized by low saturated hydraulic conductivity (10^-5 m/day). For the soil layer we varied the saturated hydraulic conductivity k_sat using both an intermediate value of 10^-2 m/day as well as a high value of almost 2 m/day. Additionally, we used a dual-porosity model to include some of the effects caused by macropore and other preferential flow processes. We assumed a free drainage boundary condition for the bedrock layer at the outlet of the hillslope and a seepage face boundary condition for the soil layer. All other boundaries except the soil surface were considered no-flow boundaries. Furthermore the model was initiated with two different moisture states theta_ant, a dry one (soil water content 9% = effective saturation 6%) and a wetter one (soil water content 18% = effective saturation 27%). Precipitation was distributed evenly over the entire hillslope. The precipitation event that was tracked through the system lasted 1 day and delivered 10 mm of water. The subsequent precipitation pattern (i.e. the precipitation that fell after the tracked event had ceased) differed for each test run. On the one hand a constant flux was assumed that lasted for the entire modeling period. The intensity of this subsequent precipitation varied from 50 mm/day to 0.5 mm/day to 0.005 mm/day. On the other hand we tested the effect of three distinct and consecutive precipitation events (all lasting one day with an increasing gap without precipitation in between them - one day, two days, four days) on the shape of the TTD of the tracked event. Combining all the variable parameters results in 48 distinct scenarios that produce 48 distinct TTDs (for an overview of the variable parameters see Figure b and Table 1).

Table 1:Variable modeling parameters that result in 48 distinct response scenarios.

Soil Depth (m)	Antecedent Effective Saturation (%)	Soil Conductivity (mm/day)	Subsequent Precipitation Intensity (mm/day)	Subsequent Precipitation Pattern
0.5	6	20	0.005	1 Continuous Event
1	27	2000	0.5	3 Consecutive Events
-	-	-	50	-

The resulting TTDs of the model runs were compared to potential transfer functions with varying shape parameters. The three types of distributions we considered are the exponential-piston flow model (EPF), the gamma model (GAM) and the advection-dispersion model (ADM) (Figure c).

Results

Looking first at the scenarios with high intensity P_sub, we saw that generally the shallow soil produces TTDs with sharper peaks given that all other parameters are identical. This was expressed by larger dispersion parameters (ADM) or smaller shape parameters (GAM). TTDs through initially dry soils are best reproduced by the ADM which is an expression of a high fraction of event water in the hydrologic response. Since there is only little preevent water in the soil, the hydraulic conductivity of the soils has to increase first before event water starts to leave the system. During wet conditions, however, the immediate response of event water is much larger since event water is able to move rapidly through the system with the velocity set by the saturated hydraulic conductivity. Therefore GAM functions are a better fit than ADM type distributions. If P_sub is less intense the peaks of the TTDs decrease and the distributions are more stretched out in time (the dispersion parameters of the ADM decrease, the shape parameters of GAM functions increase to values between 0.9 and 1.8). In case an event is followed by low intensity P_sub, the resulting TTDs do not show a clear reaction into one direction. GAM functions with shape parameters around 1 are common for most scenarios, also the ADM with values much larger than 1 competes with GAM distributions for the high initial peaks.

Discussion

There is no doubt that TTDs vary greatly in time. The TTDs produced by our model ranged from distributions with high initial peaks (GAM with shape parameter 0.06) to more damped distributions (GAM with shape parameter 1.8). These variations are caused by a combination of controlling factors. It follows that if we want to improve our modeling approaches that use transfer functions to determine time-variable mTTs it is not enough trying to relate single controls to the shape of the transfer functions. Instead, we propose to use a combined index parameter that is able to inform us on both the state of the catchment at the time that the precipitation event occurs and the subsequent precipitation event conditions that drive (or do not drive) water flow. Based on the parameters that were used in our study, we developed a dimensionless number (the flow path number F) that relates available storage space in a catchment to incoming and outgoing water fluxes in order to indicate when storage thresholds are exceeded. These storage thresholds are supposed to be more or less sudden changes in dominant flow paths that cause differences in mean flow velocities and dispersion properties that eventually affect the shape of TTDs. The flow paths number consists of soil depth D_soil and porosity n as indicators of total available soil water storage, antecedent moisture content theta_ant as an indicator of how much of the available storage is already occupied, the subsequent precipitation index P_sub as an indication of how much water is added to the soil storage in the days after an event (P_sub is the amount of precipitation falling on the hillslope during the average inter-event time t_i) and a modified version of the saturated hydraulic conductivity K_rem as an indicator of how much water can be transported away within the soils during the average event duration t_e:

K_rem = k_eff * t_e * A_out/A_in

Where k_eff is the effective hydraulic conductivity, A_out is the cross-sectional area in the soil layer at the bottom of the hillslope through which water can leave the hillslope and A_in is the total surface area of the hillslope through which water enters the system. A_out/A_inaccounts for the fact that lateral flow accumulates along the hillslope gradient, therefore soils are not able to conduct water laterally at the same rate it is being applied vertically. The flow path number (F) is hence defined as:

F = (P_sub - K_rem) / (D_soil * (n - theta_ant))

In this study the patterns of TTD shapes can be divided into two main response classes. On the one hand there are responses in systems with high antecedent moisture contents, on the other hand there are responses in systems with low antecedent moisture contents. For high antecedent moisture contents, GAM type TTDs dominate, for low antecedent moisture contents ADM type TTDs are more common. Interestingly, EPF type TTDs cannot be found anywhere. After this initial distinction by antecedent moisture content, the shape parameters of the TTDs can be narrowed down by looking at the F values. F values smaller than 0.3 correspond to ADM type TTDs with shape parameters between 1 and 4 (damped response). F values larger than 0.3 correspond to ADM type TTDs with shape parameters larger than 20 (high initial response). F values larger than 0.3 correspond to GAM type TTDs with shape parameters between 0.5 and 1 (high initial response). F values between 0.3 and 0.1 indicate GAM type TTDs with shape parameters between 1 and 2. F values smaller than 0.1 relate to GAM type TTDs with shape parameters >2 (Figure d).

One of the drawbacks of this modeling approach is the fact that HYDRUS does not account for overland flow, neither does it contain a dual permeability routine that lets you model flow in macropores and flow in the matrix simultaneously. That means that rapid responses that would be caused by these processes once a certain storage threshold is exceeded are not captured. This is especially true for the scenarios with low k_sat and high P_sub. To tackle this problem we are going to repeat this modeling exercise with another distributed hydrologic model (CATHY) that is able to include overland flow processes.

Conclusion

We found that shapes of TTDs vary considerable both in space (due to differences in soil depth and saturated hydraulic conductivity) and time (in response to different antecedent moisture contents and subsequent precipitation amounts). In order to improve the modeling of time-variable catchment response we propose to compute the flow path number F before the use of a transfer function-convolution model and use it in combination with information on antecedent moisture content to decide on the expected type and shape of the TTD for each time step. This way, this approach will make time-variable transit time modeling more realistic.

Figure a):Three-dimensional hillslope model used to generate transit time distributions in HYDRUS

Figure b): Potential response controls varied between the model runs. Soil depth D_soil, antecedent moisture content theta_ant, saturated hydraulic conductivity K_s and subsequent precipitation P_sub.

Figure c):Examples of potential transfer functions that were matched to modeled catchment responses with varying shape parameters: exponential-piston flow model (EPF), gamma model (GAM) and advection-dispersion model (ADM).

Figure d): Classification of TTDs by means of antecedent moisture conditions and the flow path number F. Generally, dry antecedent moisture conditions produce ADM type TTDs; wet antecedent moisture conditions produce GAM type TTDs. Larger Fs are related to higher initial peaks, smaller Fs cause more damping of the signal.