GESTIÓN INTEGRAL DE RIESGOS 40

An alternative modelling strategy to that of Henriksen's is to use a systems-based approach. Models using this approach require two main sets of inputs (Figure 3.1), first a theoretical knowledge of the physical, chemical and biological processes occurring in a catchment and second detailed field observations with which to calibrate and validate the model (Whitehead et al. 1986). The relative extent to which these sets of inputs are known gives rise to two main systems- based modelling approaches, systems synthesis and systems analysis.

Systems synthesis models require a detailed knowledge of the major system processes and the theoretical behaviour of such processes. As the complexity of a catchment's hydrochemical processes prevents precise mathematical representation, at present systems synthesis models assume that these processes can be spatially and/or temporally aggregated. Despite this aggregation, however, the model still attempts to reproduce the deterministic behaviour of the catchment hydrochemical processes. Systems analysis models, in contrast, take a "black-box" approach with catchment processes represented by mathematical functions and do not attempt to replicate the complex deterministic behaviour of catchment processes. A major technique in this approach is the use of time series analysis.

Systems Analysis A posteriori Measurement Knowledge Systems Synthesis A priori Structural Knowledge

Figure 3.1: The combination of theoretical knowledge and field

data in the development of modelling strategies. (after Whitehead et al. 1985)

3.3.1 Time Series Modelling

The time series approach in systems analysis produces an empirical input-output or 'black box' model on the basis of statistical inference derived directly from the observed data. Such models are based on the assumption that a "law of large systems" (Young 1978) applies. As such, they are especially useful where system processes are particularly complex and where the overall input-output budget of the system is of major importance (Whitehead et al. 1986). Such criteria makes this modelling strategy especially useful for analysing the change in the pH of surface waters as a result of increased acid deposition.

The general form of a simple single input-output time series model consists of two parts, a deterministic process model and a stochastic noise model (Figure 3.2). The system output (Yt) is considered to result from the deterministic input (Ut), which is responsible for most of the output variation, together with a stochastic input (at) which accounts for the inevitable uncertainties in the relationship. Consequently the observed output at time, t, Yt is the sum of the outputs from the deterministic model (Xt) and the stochastic model (et) at the same instance, i.e:

Yt = Xt + et (3.5)

Both Xt and et are generated by autoregressive moving average (ARMA) discrete time series models (Box and Jenkins 1970) where the autoregressive term represents memory and correlation between the output data sequence and the moving average term represents the contribution of the input at each time step. The general form of the deterministic model is:

where z is a backward shift operator , so that, z-lXt = Xt-i and AI z-11 and B I z-11 are both polynomials in z-1. Consequently as a difference equation the model takes the form:

Xt - -aiXt-i - a2Xt-2—-anXt-n + boUt +....+ bnUt-n (3.7)

The stochastic model is of a similar form so that the full transfer function model is:

Yt = [A(z-l)/B(z-l)]Ut + [C(z-l)/D(z-l)]at (3.8)

The system output (Yt) is consequently seen as dependent on past values of the output (Yt) as well as past and present input values( Ut), together with past and present values of a hypothetical white noise input (at)

Such time series techniques have been applied to a variety of environmental situations, however, the models produced by Whitehead et al. (1986) are the first examples of the application of such techniques to the study of surface water acidification. In the models produced by Whitehead et al. (1986) the system output (Yt) represents the hydrogen ion concentration in streamwater and is related to the previous day’s hydrogen ion concentration and runoff by the two parameters A and B. The two parameters are estimated from the observed data by the use of a recursive instrumental variable algorithm. Consequently, variability in the parameter values through the analysis time period can be examined in order to investigate the catchment processes that influence acidification.

The application of time series techniques at Loch Dee in Scotland, Plynlimon in Wales and Birkenes in Norway indicates that such models can accurately simulate hourly, daily and weekly timescales. As the authors note, however, the accuracy is reduced as the data

sampling timescale increases relative to the catchment response time. A further weakness of time series models is due to the fact that the model parameters are constantly updated from the observed data. Consequently, the models are unable to simulate long term changes in streamwater chemistry in response to changes in deposition loadings or catchment land-use. However, the time series technique may be very useful in an initial analysis of a catchment, eg identifying the major processes that determine streamwater chemistry, prior to more detailed process-based modelling studies. This role is illustrated particularly well by the analysis of streamwater chemical data at Birkenes by Beck et al. (1987).

3.3.2 Systems Synthesis Models

The vast majority of acidification models are based on the system synthesis approach, such models include the "Birkenes*’ model (Christophersen et al. 1982, Seip et al. 1986), "MAGIC" (Cosby et al. 1985a, 1985b), "ILWAS" (Chen et al. 1983, Goldstein et al. 1984), "Trickle Down" ( Schnoor et al. 1984 Schnoor and Stumm 1985;), "Pulse" (Bergstrom et al. 1985) and "RAINS" (Alcamo et al. 1985). Whilst these models show considerable variation in modelling philosophy and structure there is, however, a considerable convergence of thought concerning the key processes. Thus to a greater or lesser extent all the models accept the importance, through the charge balance principle, of the anion mobility concept (Seip et al. 1980), cation exchange and the dissolution of aluminium in determining short-term solution chemistry together with the importance of mineral weathering in determining the longer term response. This section of the review will focus on three such models

namely, "MAGIC", "Birkenes" and "ILWAS". The "MAGIC" and "Birkenes" models will be examined initially as they are not only the two most widely adopted acidification models, but are also the only systems synthesis models to have been applied to catchments in upland Britain. The "ILWAS" model will then be examined in considerable detail as an evaluation of this model, as a tool for studying acidification at Loch Dee in Southwest Scotland, forms the bulk of the work in this thesis.

3.3.2.1 "MAGIC" (Model of Acidification of Groundwater In Catchments)

The main objectives in the development of this model were twofold. The first objective was to develop a relatively simple process-orientated model which could accurately simulate the chemical behaviour of catchments so as to enable long-term predictions of the effects of acid deposition to be made. The second objective was to determine the extent to which spatially distributed physical and chemical processes in a catchment can be "averaged" or lumped without affecting a model's ability to reproduce a catchments behaviour (Cosby et al. 1985a). The conceptual basis of the model lies in the research reported by Reuss and Johnson (1985) whereby soil system processes are described by equilibrium reactions involving the dissolved and adsorbed phases of ions in the soil-soilwater system, together with the effects of carbonic acid resulting from the elevated carbon dioxide levels in the soil. Soil chemical processes simulated in the "MAGIC" chemical module thus comprise the following; i)cation exchange in soils, ii)aluminium mobilisation and solubility,

iii)carbonic acid dissociation, iv)anion retention in soils and v)mineral weathering.

Cation exchange reactions between the soil and soilwater are assumed to involve only the four base cations (sodium, potassium, calcium and magnesium) and aluminium. In these reactions the total cation charge in solution is determined by the anions in solution through the charge balance principle, whilst the relative amount of each cation on the exchange complex is determined by the use of ion exchange equilibria. The general form of such exchange equilibria is:

Ka,b = (Aa+ )b I (B b+ )a (3.9)

where Ka/b is the activity ratio for two positively charged ions A and B with valences a+ and b+ respectively and the brackets represent ion activities in solution (Bolt 1967). As the value of Ka,b is dependent on the fraction of each ion on the exchange complex, this particular form of exchange equilibria restricts simulations to the short term (ie weeks to months) where changes in the exchange fraction will not be significant. Consequently, in order to evaluate the effect of changes in the amount of an ion on the exchange complex, "MAGIC" makes use of a modified form of the Gaines-Thomas exchange coefficient (Gaines and Thomas 1953), for example the equilibrium expression for the calcium - sodium exchange reaction:

{Na+)2 Eca

ScaNa = --- (3-10)

{Ca2+} ENa2

where the braces represent the aqueous activities, E^a and Eca the equivalent fraction of the exchange site occupied by the respective ions, and S is the selectivity coefficient. As the selectivity coefficients represent cation exchange reactions "lumped" over the entire

catchment, their values may be estimated either from field data or via model calibration.

Aluminium solubility reactions in the model are based on the assumptions that there is an equilibrium between Al3+ in the soil solution and a solid phase of A1(OH)3 and that as solution concentrations of H+ and Al3+ vary the reversible reaction

3H+ +A1(OH)3 = Al3+ + 3H2O (3.11)

occurs instantaneously. The equilibrium expression for this reaction

is; (Al3+)/(H+)3 = KAi (3.12)

As with the cation selectivity coefficients, Kai is a "lumped" value

assumed to be representative of the entire catchment, and consequently is also a model parameter estimated with each application. The model also includes solution phase reactions of A13+ with sulphate and fluoride ions, the equilibrium constants for these reactions are, however, seen as unvarying in different applications.

Inorganic carbon reactions included in the model involve the hydration of carbon dioxide to produce carbonic acid together with the subsequent breakdown of the carbonic acid and the production of carbonate, bicarbonate and hydrogen ions. The model also assumes that the soil water is in equilibrium with carbon dioxide in the soil-air with the partial pressure of carbon dioxide being a model calibration parameter.

The above three processes form the equilibrium part of the model (Cosby et al. 1985a). Whilst this is a useful tool for short term predictions, (i.e. weeks), the aim of "MAGIC" is to predict the long term effects (i.e. decades) of acid deposition on streamwater

chemistry. Consequently, this equilibrium model is coupled to a dynamic model (Cosby et al. 1985b) based on the input-output mass balance of ions in the catchment.

The mass balance is applied for each major base cation and strong acid anion in the form of a dynamic mass balance equation;

DXt / dt = Fx + Wx - Q*(X) (3.13)

where Xt is the total amount of the ion (x) in the catchment; Fx is the atmospheric flux to the catchment; Wx is the mineral weathering flux; (X) is the ion concentration in streamwater and Q is stream discharge. The atmospheric and mineral weathering fluxes are both model inputs. An upper limit to the weathering rates is provided by the observed net catchment losses. The actual rates used in the model are chosen by a trial and error procedure until the model accurately reproduces observed stream concentrations and adsorbed values of base cations. The strong acid anions, with the exception of sulphate, are assumed to have no adsorbed phase. Sulphate adsorption is determined by a further sub-model (Cosby et al. 1984), based on a simple non-linear Langmuir isotherm of the form:

Es = E™ * (SO42-) / (C + (SO42-)) (3.14)

where Es is adsorbed sulphate, (SO42-) is the dissolved sulphate concentration, Emx is the maximum adsorption capacity of the soils and C is the half-saturation constant. Such a non-linear process leads to a decrease in the response time of the system to changes in atmospheric deposition, as the amount of adsorbed sulphate is increased.

In contrast to the complexity with which catchment chemistry is described the model's description of catchment hydrology is very

simple using average hydrological conditions such as mean annual precipitation and streamwater flow volumes. In its original form as applied at White Oak Run in Virginia U.S.A., a catchment is represented in the model by two compartments, one describing soil processes and the other stream processes (Figure 3.3). However, if a catchment's soils show a distinct separation of A and B horizons, with different processes in the two horizons, the model may be expanded to include two soil compartments. In the same manner, in regions where snowmelt is an important hydrological component, another compartment may be added to represent snow accumulation and melt processes.

At White Oak Run the model faithfully reproduced present-day water chemistry and gave plausible values for the model parameters. Over the longer term the model also indicated the importance of sulphate saturation in soils in determining the response of streamwaters to changes in sulphate deposition. As the authors noted, however, verification of the model would require years/decades of catchment monitoring. Alternatively model verification maybe accomplished through the reconstruction of past water chemistry and comparing the results with historical records.

Accurate data of past water chemistry is in general severely limited. At Loch Dee in Galloway, however, a palaeoecological analysis of sediment cores (Battarbee et al. 1985, Flower et al. 1987) has enabled the pH record of the surface waters to be reconstructed for several hundred years. Consequently "MAGIC" was applied to one of the sub­ catchments at Loch Dee, the moorland Dargall Lane, by Cosby et al. (1986a). This location was also studied in order to assess how well the

Atmospheric Deposition

v

Figure 3.3 Schematic view of MAGIC (Model for Acidification of Groundwater in Catchments).

model would perform in an area subject to high seasalt inputs. The simulations indicate that the conceptual basis of the model is well founded with the model predictions of past pH closely following the historical reconstruction of Battarbee et al. (1987). Furthermore, the model was also able successfully to cope with the high levels of seasalt inputs. Later work at Loch Dee (Neal et al. 1986) involved the use of "MAGIC" as a tool to investigate the effects of coniferous afforestation and/or acidic deposition on streamwater acidity. In this work the influence of the trees was confined to increased dry and occult deposition and no allowance was made for biomass uptake and changes in hydrological pathways. Despite these limitations the simulations indicated that coniferous afforestation can have a significant deleterious effect on streamwater acidity. Furthermore, the results indicated that any reduction in acidic deposition would have to be significantly greater for forested than moorland catchments, if any increase in streamwater acidity is to be prevented.

A further recent extension of "MAGIC" has been the use of Monte- Carlo techniques to reduce the uncertainty in parameter estimation (Cosby et al. 1986b, Musgrove et al. 1988a). The use of such techniques indicates that "MAGIC" maybe a useful tool for examining long-term regional scale as well as catchment scale responses to acid deposition (Musgrove et al. 1988b).

3.3.2.2 "The Birkenes model"

Whereas "MAGIC" is primarily concerned with the long term response of catchments to acid deposition, the "Birkenes" model, (Christophersen and Wright 1981, Christophersen et al. 1982), was

developed in order to determine if the short-term response of streamwater chemistry to acid deposition could be adequately explained by a small number of physical processes.

The basis of the model is the hydrological submodel (Figure 3.4) which is a modification of the two reservoir system developed by Lundquist (1977). In this model the upper reservoir may be considered as pipeflow or fast throughflow which enters the streams directly and consists of water that is mainly in contact with the humus and upper soil horizons. The lower reservoir, in turn, provides baseflow and represents water that is mainly in contact with the deeper mineral soil.

The chemistry component of the model is built on top of the hydrological model and consists in turn of two sub-models for sulphate dynamics and cation partitioning. There are two major assumptions in the chemistry model; firstly that sodium and chloride follow each other through the system and take no part in determining the concentrations of other species in the stream and secondly that concentrations of potassium, ammonium, nitrate, bicarbonate and organic anions can be ignored as they account for only a few percent of the ionic sum at Birkenes (Christophersen et al. 1982). The model chemistry is consequently implicitly based on the mobile anion concept (Seip 1980), with concentrations of the remaining cations hydrogen, calcium, magnesium, and aluminium balanced by concentrations of sulphate.

The sulphate sub-model (Christophersen and Wright 1981, Christophersen et al. 1982) attaches chemical processes involving

P - Precipitation E - Evapotranspiration A - Upper Reservoir E E A B

sulphate to the two compartment hydrological model. In the upper reservoir it is assumed that all sulphate of atmospheric origin remains water soluble. It is further assumed that sulphate concentrations in drainage waters from the reservoir (Ca) are directly proportional to the amount of water-soluble sulphate on the solid phase (Fa), such that essentially a linear sulphate adsorption isotherm is obtained i.e.

Ca = Ka Fa (3.15)

Mineralisation of sulphate in the upper reservoir during dry periods is also included in the model by allowing Fa to increase by a fixed amount for each day the reservoir is empty.

In the lower reservoir the only process considered is adsorbtion/desorbtion. In contrast to the upper reservoir a more complex non-linear adsorb tion isotherm is used of the form;

dCb/dt = kb(Ceq-Cb) (3.16)

where kb is the adsorbtion rate coefficient, Ceq is the equilibrium concentration and Cb is the concentration of sulphate in drainage water from the lower reservoir. Consequently, concentrations are assumed to fall exponentially towards a fixed equilibrium value. Also in order to prevent unrealistic buildup of sulphate concentrations in dry periods, when evapotranspiration occurs, sulphate is removed from solution and transferred to the solid phase.

The cation sub-model is linked to the sulphate sub-model by the assumption that the sulphate concentration in each reservoir is balanced by the total concentration of hydrogen, calcium, magnesium, and aluminium and that the reservoirs are ideally mixed.

Aluminium concentrations in the model are produced by a gibbsite equilibrium equation of the type used in "MAGIC" eq.(3.12), together with a correction term to account for aluminium complexes with fluoride and organic compounds. The divalent cations calcium and magnesium are assumed to vary together and consequently are summed and lumped together as M+. The relationship between [H+] and [M+] in the upper reservoir is, as in "MAGIC", governed by cation exchange and is modelled by an equation of type eq.(3.9). The adsorbed ion concentrations in the soil, however, are assumed to be constant thereby leading to a constant selectivity coefficient or lime