Models for Converting Measurements of Environmental Radionuclide ...

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Models for Converting Measurements of Environmental Radionuclide Inventories (137Cs, Excess 210Pb, and 7Be) to Estimates of Soil Erosion

and Deposition Rates (Including Software for Model Implementation) D.E.Walling, Y. Zhang, and Q. He

Department of Geography, University of Exeter, Exeter, EX4 4RJ U.K.

1. Introduction

Environmental radionuclides, including caesium-137 (137Cs), excess lead-210 (210Pbex),

and Beryllium-7 (7Be), are being increasingly used to obtain information on soil redistribution rates for soil erosion and sediment budget investigations (cf. Ritchie &

Ritchie, 1995; Walling and Quine, 1995; Walling, 1998; Zapata, 2002). Work undertaken in a wide range of environments in different areas of the world has demonstrated that

their use, either independently or in combination, affords a valuable means of estimating rates of soil loss and sediment deposition, which possesses many advantages over

conventional monitoring techniques (cf. Loughran, 1989). These advantages include the

potential for deriving retrospective estimates of erosion and deposition rates based on a

single site visit and for assembling distributed information for individual points in the landscape, which can be used to study spatial patterns of soil redistribution.

Use of environmental radionuclide measurements to estimate rates of erosion and

deposition is founded on comparison of the inventories at individual sampling points with a reference inventory, representing the local fallout input and thus the inventory to be

expected at a site experiencing neither erosion nor deposition. A measured inventory for

an individual sampling point less than the reference value is indicative of erosion, whereas an inventory greater than the reference value is indicative of deposition. Although such comparisons of measured inventories with the local reference value provide useful qualitative information on the spatial distribution of erosion and

deposition in the landscape and on the relative magnitude of the values involved, in most instances quantitative estimates of erosion and deposition rates are required. The


derivation of quantitative estimates is heavily dependent upon the existence of a reliable

means of converting the magnitude of the measured inventory at a specific sampling

point, relative to the local reference inventory, to an estimate of the rate of erosion or deposition at that point.

Many different approaches have been used to convert 137Cs measurements to quantitative estimates of erosion and deposition rates (Walling and Quine, 2000; Walling and He,

1999, 2005). These methods include both empirical relationships, and theoretical models

and accounting procedures. In an effort to standardise the methods and procedures employed, Walling and He (2001) developed a PC-compatible software package that implemented a number of models (procedures) which appeared to provide meaningful results. The models varied in complexity from the simple proportional model to more

complex mass balance models and models which attempt to describe the key processes controlling the distribution of

Cs in the soil profile. Models applicable to both


cultivated and undisturbed (e.g. rangeland and permanent pasture) soils were included. This readily available standardised software has played an important role in promoting the use of


Cs in soil erosion and sedimentation-related studies across the world.

However, a number of problems have also been become apparent with the software

which potentially limit its applications. These problems range from the difficulties in specifying several of the model parameters, through rigid requirements for data structures within a file, to lack of error trapping and handling capacities.

Since the release of the software, further progress also has also been made in the use of other radionuclides, in addition to

Cs, to estimate soil redistribution rates in


agricultural environments. Here attention has focussed on excess 210


Pb (referred as

Pbex hereafter) and 7Be (e.g. Blake et al. 1999; Walling and He, 1999). These other

radionuclides share most of the assumptions associated with the 137Cs technique, e.g. 1) Wet deposition from rainfall is the dominant source

2) Strong affinity with soil particles, particularly the fine particles


3) Exponential decrease of mass concentration and inventory with depth down a undisturbed soil profile and a homogenised distribution within the plough layer for cultivated sites

4) Near uniform spatial distribution of inventories on undisturbed (uneroded) sites

5) Once absorbed by soil particles, subsequent movement will only occur in association with soil particles

It is therefore possible to adapt some of conversion procedures and models used for 137Cs



Pbex and 7Be, provided key contrasts with


Cs are take into account. It has been

shown that together these three radionuclides are able to provide information on soil redistribution over temporal scales ranging from a few days (7Be), through decades (


Cs) to around 100 years (210Pbex). Furthermore, use of the individual radionuclides in

combination offers potential to identify temporal trends in soil erosion and sedimentation rates and to elucidate the erosional history of a study site.

Against this background, it was judged necessary to update the software to rectify the

known problems, to incorporate new procedures or conversion models for environmental radionuclides other than


Cs, and to provide an integrated computational environment

that can convert radionuclide inventories to soil redistribution rates on a platform accessible to most researchers.

2. Development of an Excel add-in for the conversion of 137Cs,

and 7Be inventories to erosion and deposition rates

Pbex ,


Conversion of radionuclide inventories to estimates of erosion and deposition rates attempts to numerically infer the rate of removal or accretion of the radionuclide over a

specific timeframe. The complexities and uncertainties associated with the various soil redistribution processes mean that this commonly involves an iterative, subjective,

exploratory process. To meet the need outlined in the previous section, a research tool has been developed for converting




Pbex, and 7Be inventories to estimates of soil

erosion and deposition rates using VBA (Visual Basic Application). It is designed to deal with point data from a single transect that follows the flow line (direction of maximum


slope) down a slope, assuming that there is no sediment contribution from upslope areas or significant across slope soil redistribution.

As a standard add-in within Microsoft Excel, the updated software has the following advantages and characteristics:

1) It can take full advantages of the data management and data analysis functions available in Excel. The conversion results can be readily related to other

environmental variables or factors for further analysis.

2) To ensure meaningful model parameterisation, limits have been placed on the acceptable ranges for individual parameters and default values have been

provided. Procedures have also been included to derive or estimate several

parameters used in the models.

3) The conversion models for 137Cs, 210Pbex and 7Be can be accessed via a uniform, consistent, interactive interface in a user-friendly manner. The design of the

interface follows the logical flow of data analysis, involving input data source at the top, parameter specification in the middle, and storage of the results at the bottom.

4) Help information has been integrated into the software. Relevant information and guidance are provided at the appropriate time.

5) There are no restrictions on folder names / paths, data file locations and, thus, the user is given more flexibility in software installation and data management.

Table 1. Available models in the add-in

Cultivated Pasture • Proportional model * • Profile shape model • Simplified mass balance model* • Diffusion and migration model • Mass balance model • Mass balance model with tillage 210 Pb • Mass balance model* • Diffusion and migration model* • Mass balance model with tillage* 7 Be • Profile shape model* • Profile shape model* * Models that have been further modified or developed 137



A list of the models incorporated in the software for each radionuclide is provided in Table 1. For all models, the inventories for sampling points along the transect are

required and in most cases a particle size correction factor can be included if desired. Additional parameter requirements for the individual models are identified in Table 2. Table 2 A list of parameter requirements for individual models Model

Parameters required

Proportional model and Tillage depth, bulk density, year of tillage Simplified mass balance model commencement Mass balance model Tillage depth, year of tillage commencement, proportional factor, relaxation depth, annual fallout flux* Mass balance model with Tillage depth, tillage constant, proportional factor, tillage relaxation depth, slope length and slope gradient for each section of the transect, annual fallout flux* Diffusion and migration model Diffusion coefficient, relaxation depth, migration coefficient, annual fallout flux* Profile shape model Profile shape factor 137 * Only required for Cs models Each model has its specific set of parameters although some of these parameters are

common between models. It is important to recognise that the individual models are

different in their underlying assumptions, processes descriptions and representation of temporal variation. A sound understanding of the models and their parameters is an essential precursor to their applications. In order to avoid the possible misuse of the models, these issues will be addressed in the following sections.

3. Brief description of the models In this section, the theoretical basis of the models will be briefly discussed, along with their advantages and limitations. Since the conversion models provided for 7

Be were adapted from those developed primarily for



Pbex and

Cs, emphasis will be placed on

the latter. The differences from 137Cs will be highlighted, when the models for 210Pbex and


Be are introduced.


3.1 Models for use with


Cs inventories

3.1.1 The Proportional Model

The proportional model is based on the premise that


Cs fallout inputs are completely

mixed within the plough or cultivation layer and that the soil loss is directly proportional to the reduction in the

beginning of

half of the




Cs inventory due to loss of soil from the soil profile, since the

Cs accumulation or the onset of cultivation, whichever is later. Thus, if

Cs input has been removed, the total soil loss over the period is assumed to

be 50% of the plough depth. The model can be represented as follows: Y = 10

BdX 100TP


Where: Y = mean annual soil loss (t ha-1 yr-1);

d = depth of the plough or cultivation layer (m);

B = bulk density of soil (kg m-3);

X = percentage reduction in total 137Cs inventory (defined as (Aref-A)/Aref×100);

T = time elapsed since the initiation of 137Cs accumulation or the commencement of cultivation, whichever is later (yr);

Aref = local 137Cs reference inventory (Bq m-2);

A = measured total 137Cs inventory at the sampling point (Bq m-2); P = particle size correction factor for erosion.

An inference from the assumptions of the proportional model is that the

concentration of the eroded sediment remains constant through time. The

137 137

Cs Cs

concentration of deposited sediment at a depositional point may therefore be assumed to be constant. In cases where the 137Cs inventory A for a sampling point is greater than the

local reference inventory Aref, deposition of sediment may be assumed and the annual

deposition rate Y′ (t ha-1 yr-1) may be estimated using the following equation:


Y ′ = 10


BdX ′ 100TP ′


X′ = percentage increase in total 137Cs inventory (defined as (A-Aref)/Aref×100);

P′ = particle size correction factor for deposition.

Advantages and limitations: The proportional model requires only information on plough depth, in addition to the values of

Cs inventory for the sampling points and the local


reference inventory, and it is therefore easy to apply. However, the assumptions of this model represent a considerable oversimplification of reality in terms of the accumulation of 137Cs in the soil. The accumulation of 137Cs takes place over several years and some of

the fallout input will remain at the soil surface prior to incorporation into the soil profile by cultivation. If some of the


Cs accumulated on the surface is removed by erosion

prior to incorporation into the profile the estimates of soil loss provided by the model will overestimate actual rates of soil loss. Perhaps more importantly the model does not take into account the progressive dilution of 137Cs concentrations in the soil within the plough

layer, due to the incorporation of soil from below the original plough depth, as a result of surface lowering by erosion. As a result, the estimates of erosion rates obtained are likely

to underestimate the rates of soil loss. Equally, deposition rates estimated using this

procedure will be underestimated because the model fails to take into account in progressive reduction in


Cs activity of the mobilised sediment that is subsequently

deposited, as erosion proceed. For this reason, the proportional model is unlikely to

provide reliable estimates of soil redistribution rates and its use is not recommended. It is included in this software package to permit comparison of the results obtained with those provided by other more reliable models.

3.1.2 A Simplified Mass Balance Model (Mass Balance Model I)

Mass balance models attempt to overcome some of the limitations of the simple proportional model by taking account of both inputs and losses of


Cs to and from the

profile over the period since the onset of 137Cs fallout. Zhang et al. (1990) have proposed a simplified mass balance model, which assumes that the total

Cs fallout occurred in



1963 instead of over a longer period extending from the mid 1950s to the mid 1970s. In its original form this simplified mass balance model did not take account of particle size effects but a correction factor P has been included here.

For an eroding site (A(t)Aref), assuming a constant deposition rate R′ (kg m-2 yr-1) at

the site, the sediment deposition rate can be estimated from the excess inventory relative

to the reference inventory and the

Cs concentration of the deposited sediment Cd(t′)


(Bq kg-1) according to: R′ =


Aex ( t )




( t ′ )e − λ ( t − t ′ ) dt ′


∫C t


A( t ) − Aref d

( t ′ )e − λ ( t − t ′ ) dt ′




Aex(t) = the excess


Cs inventory of the sampling point over the reference

inventory at year t (defined as the measured inventory less the local reference inventory) (Bq m-2);

Cd(t′) = 137Cs concentration of deposited sediment at year t′ (Bq kg-1); λ = decay constant for 137Cs (yr-1);

P′ = particle size correction factor. Generally, the


Cs concentration Cd(t′) of deposited sediment can be assumed to be

represented by the weighted mean

Cs concentration of sediment mobilised from the


upslope contributing area. Cd(t′) can therefore be calculated using the following equation: Cd ( t ′ ) =


∫ RdS

∫ P ′C ( t ′) RdS S




where S (m2) is the upslope contributing area and Ce(t′) (Bq kg-1) is the



concentration in sediment mobilised from an eroding point, which can be calculated from Equation 3 according to: Ce ( t ′ ) = P

A( t ′ ) P  = Aref ( t ′ )1 − P  d d

R  d

t ′−1963


P R  Aref ( t )e λ ( t − t ′ ) 1 − P   d d

t ′−1963


where Aref(t)=Aref. Advantages and limitations: The simplified mass balance model takes into account the

progressive reduction in the 137Cs concentration of the soil within the plough layer due to

the incorporation of soil containing negligible


Cs from below the original plough

depth. It represents an improvement over the proportional model. This model is also easy

to use and requires only information on plough depth. However, this model does not take

into account the possible removal of freshly deposited 137Cs fallout by erosion before its incorporation into the plough layer by cultivation. The assumption that the total

fallout input occurs in 1963 is also an oversimplification.




3.1.3 Mass Balance Model II

A more comprehensive mass balance model requires consideration of the time-variant fallout


Cs input and the fate of the freshly deposited fallout before its incorporation

into the plough layer by cultivation.

For an eroding point (A(t)Aref), assuming that the excess 137Cs inventory Aex (Bq m2

) (defined as the measured total inventory A(t) less the local direct fallout input Aref) at

an aggrading point is due to the accumulation of


sediment, the excess 137Cs inventory can be expressed as:

A ex = ∫ R ′Cd ( t ′ )e − λ ( t − t ′ ) dt ′ t


Cs associated with deposited


where R′ (kg m-2 yr-1) is the deposition rate and Cd(t′) (Bq kg-1) is the 137Cs concentration

of deposited sediment. Cd(t′) will reflect the mixing of sediment and its associated



concentration mobilised from all the eroding areas that converge on the aggrading point. Cd(t′) essentially comprises two components, the first of which is associated with the removal of the freshly deposited


Cs, and the second is associated with erosion of the 11

accumulated 137Cs stored within the plough layer. Again, Cd(t′) can be estimated from the 137

Cs concentrations of the sediment mobilised from the upslope eroding area S: Cd ( t ′ ) =


∫ RdS

∫ P ′C ( t ′) RdS S




(same as equation 6)

From Equations 13 and 14, the mean soil deposition rate R′ can be calculated from the following equation: R′ =







( t ′ )e − λ ( t − t ′ ) dt ′

Advantages and limitations: The mass balance model described here takes account of

both the temporal variation of the


Cs fallout input and the initial distribution of fresh

fallout in the surface soil. Results from this model are likely to be more realistic than

those provided by the simplified mass balance model I presented in the previous section. However, information on the plough depth, the relaxation mass depth H and parameter γ

is required in order to use this model.

3.1.4. A Mass Balance Model Incorporating Soil Movement by Tillage (Mass Balance Model III) The mass balance models described previously do not take account of soil redistribution introduced by tillage. As tillage results in the redistribution of soil in a field, the



contained in the soil will also be redistributed, and such redistribution needs to be taken into account when using the

Cs measurements to derive estimates of rates of soil


erosion by water. If the effects of tillage redistribution on


Cs inventories can be

quantified and taken into account, the remaining component of redistribution will reflect the impact of water erosion.


The effect of tillage in redistributing soil can be represented by a downslope sediment flux. Following Govers et al. (1996), the downslope sediment flux FQ (kg m-1 yr-1) from a unit contour length may be expressed as: FQ = φ sin β


where β (°) is the slope angle, and φ ( kg m-1 yr-1) is a site-specific constant.

If a flow line down a slope is divided into several sections and each section can be

approximated as a straight line, then for the ith section (from the hilltop), the net soil redistribution induced by tillage Rt (kg m-2 yr-1) can be expressed as:

Rt = ( FQ ,out − FQ , in ) / Li = φ (sin βi − sin βi −1 ) / Li = Rt , out − Rt , in


where Li (m) is the slope length of the ith segment, and Rt,out (kg m-2 yr-1) and Rt,in (kg m-2 yr-1) are defined as:

Rt , out = φ sin βi / Li


Rt , in = φ sin βi −1 / Li

For a point experiencing water erosion (rate Rw (kg m-2 yr-1)), variation of the total 137Cs

inventory A(t) (Bq m-2) with time t can be expressed as:

dA( t ) = (1 − Γ ) I ( t ) + Rt ,in Ct ,in ( t ) − Rt , out Ct ,out ( t ) − Rw Cw ,out ( t ) − λA( t ) dt

where Ct,in, Ct,out and Cw,out (Bq kg-1) are the



Cs concentrations of the sediment

associated with tillage input, tillage output and water output respectively. The net erosion rate R (kg m-2 yr-1) is:

R = Rt , out − Rt , in + Rw



For a point experiencing water-induced deposition (rate R′w, (kg m-2 yr-1)), variation of the total 137Cs inventory with time can be expressed as:

dA( t ) = I ( t ) + R t ,in C t ,in ( t ) − Rt , out C t , out ( t ) + R w′ C w , in ( t ) − λA( t ) dt


where Cw,in (Bq kg-1) is the 137Cs concentration of the sediment input from water-induced

deposition. The net erosion rate R is:


R = Rt ,out − Rt ,in − Rw′ The


Cs concentration of the soil within the plough layer Cs(t′) (Bq kg-1) can be

expressed as:

A( t ′ ) d R 1 C s ( t ′ ) = [ A( t ′ ) − d d Cs ( t ′ ) =

∫ A( t ′′ )e

t −1

for a net erosion site − λt ′′


dt ′′]

for a net deposition site (23)

where |R| (R0) or deposition rate (R 100 years Days-months Continuous input Daily inputs with limited inter- need to be annual variations summed Largely unknown

Largely unknown

(1) Uniform distribution, (2) Exponential decrease Possible

Exponential decrease (both) Not applicable Event(s) based

Time basis of Annual average Annual average estimated soil redistribution rates * An exception is the Chernobyl incident which caused 137Cs deposition in 1986. However, Chernobyl fallout had a limited spatial distribution. 3.2.1 A Conversion Model for 7Be Inventories

The 7Be radionuclide has a much shorter half-life than

Cs and, therefore, provides a


valuable tracer for examining short-term soil redistribution processes. Its penetration depth into the soil will be shallow (less than 2 cm in most cases), since its short half-time means that there will be limited time for downward migration and diffusion. Tillage

operations between 7Be deposition and the time of sampling will invalidate its use, because tillage operation will mix the 7Be into the plough layer and make the 7Be concentration in the soil too low to be detectable. The 7Be depth distribution encountered

by 7Be studies on agricultural land are likely to be similar to the profiles associated with 137

Cs within uncultivated sites, but with a more restricted depth distribution and therefore

a much lower profile shape factor.


To convert 7Be inventories to estimates of erosion and deposition rates along a transect, the profile distribution model for 137Cs was modified as follow:

1) The modelling period is changed from decades to a single event;

2) Annual natural decay is no longer important;

3) A much lower value for the profile shape factor (

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