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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

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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 137Cs 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 137Cs 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 137Cs, to estimate soil redistribution rates in agricultural environments. Here attention has focussed on excess 210Pb (referred as 210Pbex 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

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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 to 210Pbex and 7Be, provided key contrasts with 137Cs 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 ( 137Cs) 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 137Cs, 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, 10Pbex , and 7Be inventories to erosion and deposition rates 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 137Cs, 210Pbex, 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

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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 137 Cs • Proportional model *

• Simplified mass balance model* • Mass balance model • Mass balance model with tillage

• Profile shape model • Diffusion and migration

model 210 Pb • Mass balance model*

• Mass balance model with tillage*

• Diffusion and migration model*

7 Be • Profile shape model* • Profile shape model* * Models that have been further modified or developed

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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 Simplified mass balance model

Tillage depth, bulk density, year of tillage commencement

Mass balance model Tillage depth, year of tillage commencement, proportional factor, relaxation depth, annual fallout flux*

Mass balance model with tillage

Tillage depth, tillage constant, proportional factor, 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 * Only required for 137Cs 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 210Pbex and 7Be were adapted from those developed primarily for 137Cs, emphasis will be placed on the latter. The differences from 137Cs will be highlighted, when the models for 210Pbex and 7Be are introduced.

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3.1 Models for use with 137 Cs inventories 3.1.1 The Proportional Model The proportional model is based on the premise that 137Cs fallout inputs are completely mixed within the plough or cultivation layer and that the soil loss is directly proportional to the reduction in the 137Cs inventory due to loss of soil from the soil profile, since the beginning of 137Cs accumulation or the onset of cultivation, whichever is later. Thus, if half of the 137Cs 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 BdXTP=10

100 (1)

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 137Cs concentration of the eroded sediment remains constant through time. The 137Cs 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:

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′ =′′

Y BdXTP10

100 (2)

where: 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 137Cs 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 137Cs 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 137Cs 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 137Cs 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 137Cs fallout occurred in

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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 erosion rate R (m m-2 yr-1), the total 137Cs inventory (A , Bq m-2) for year t (yr) can be expressed as:

A t A P Rdref

t( ) ( )= −−1 1963 (3)

The above equation can be rearranged to derive the erosion rates as follows: Y dB

PX t

= − −

−10 1 1100

1 1963/( ) (4)

where: Aref = local reference inventory (Bq m-2);

Y = mean annual soil loss (t ha-1 yr-1); d = depth of 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); P = particle size correction factor. For a depositional 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 137Cs concentration of the deposited sediment Cd(t′) (Bq kg-1) according to:

′ =′ ′

=−

′ ′− − ′ − − ′∫ ∫R A t

C t e dt

A t A

C t e dt

ex

dt t

tref

dt t

t( )

( )

( )

( )( ) ( )λ λ

1963 1963

(5)

where:

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Aex(t) = the excess 137Cs 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 137Cs concentration Cd(t′) of deposited sediment can be assumed to be represented by the weighted mean 137Cs concentration of sediment mobilised from the upslope contributing area. Cd(t′) can therefore be calculated using the following equation: C t

RdSP C t RdSd

S

eS

( ) ( )′ = ′ ′∫ ∫1 (6)

where S (m2) is the upslope contributing area and Ce(t′) (Bq kg-1) is the 137Cs concentration in sediment mobilised from an eroding point, which can be calculated from Equation 3 according to:

C t P A td

PdA t P R

dPdA t e P R

de ref

t

reft t

t( ) ( ) ( ) ( ) ( )′ = ′ = ′ −

= −

′−− ′

′−

1 11963 1963

λ (7)

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 137Cs 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 137Cs fallout input occurs in 1963 is also an oversimplification.

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3.1.3 Mass Balance Model II A more comprehensive mass balance model requires consideration of the time-variant fallout 137Cs 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), the change in the total 137Cs inventory A(t) with time can be represented as:

dA tdt I t P R

d A t( ) ( ) ( ) ( ) ( )= − − +1 Γ λ (8)

where: A(t) = cumulative 137Cs activity per unit area (Bq m-2); R = erosion rate (kg m-2 yr-1); d = cumulative mass depth representing the average plough depth (kg m-2); λ = decay constant for 137Cs (yr-1); I(t) = annual 137Cs deposition flux (Bq m-2 yr-1); Γ = percentage of the freshly deposited 137Cs fallout removed by erosion before

being mixed into the plough layer; P = particle size correction factor. If an exponential distribution for the initial distribution of fresh 137Cs fallout at the surface of the soil profile can be assumed, following He and Walling (1997), Γ can be expressed as: Γ = − −P e R Hγ ( )/1 (9) where γ is the proportion of the annual 137Cs input susceptible to removal by erosion, and H (kg m-2) is the relaxation mass depth of the initial distribution of fresh fallout 137Cs at the surface of the soil profile.

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If t0 (yr) is the year when cultivation started, from Equations 8 and 9, the total 137Cs inventory A(t) at year t can be expressed as:

A t A t e P e I t e dtPR d t t R H PR d t t

t

t

( ) ( ) ( ( )) ( )( / )( ) / ( / )( )= + − − ′ ′− + − − − + − ′∫0 0

0

1 1λ λγ (10)

where A(t0) (Bq m-2) is the 137Cs inventory at t0 (yr):

A t I t e dtt tt

( ) ( ) ( )0

19540

0

= ′ ′− ′−∫ λ (11)

The erosion rate R can be estimated by solving Equation 10 numerically, when the 137Cs deposition flux and values of the relevant parameters are known. The 137Cs concentration of mobilised sediment Ce(t′) can be expressed as: C t I t

RP e P A t

deR H( ) ( ) ( ) ( )/′ = ′ − + ′−γ 1 (12)

For a depositional point (A(t)>Aref), assuming that the excess 137Cs inventory Aex (Bq m-

2) (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 137Cs associated with deposited sediment, the excess 137Cs inventory can be expressed as:

A R C t e dtex dt t

t

t

= ′ ′ ′− − ′∫ ( ) ( )λ

0

(13)

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 137Cs 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 137Cs, and the second is associated with erosion of the

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accumulated 137Cs stored within the plough layer. Again, Cd(t′) can be estimated from the 137Cs concentrations of the sediment mobilised from the upslope eroding area S:

C tRdS

P C t RdSd

S

eS

( ) ( )′ = ′ ′∫ ∫1 (14)

(same as equation 6) From Equations 13 and 14, the mean soil deposition rate R′ can be calculated from the following equation:

′ =′ ′− − ′∫

R A

C t e dt

ex

dt t

t

t

( ) ( )λ

0

(15)

Advantages and limitations: The mass balance model described here takes account of both the temporal variation of the 137Cs 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 137Cs contained in the soil will also be redistributed, and such redistribution needs to be taken into account when using the 137Cs measurements to derive estimates of rates of soil erosion by water. If the effects of tillage redistribution on 137Cs inventories can be quantified and taken into account, the remaining component of redistribution will reflect the impact of water erosion.

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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 βsin (16) 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: R F F L L R Rt Q out Q in i i i i t out t in= − = − = −

−( ) / (sin sin ) /, , , ,φ β β 1 (17)

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:

R LR L

t out i i

t in i i

,

,

sin /sin /

=

=−

φ βφ β 1

(18)

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 tdt

I t R C t R C t R C t A tt in t in t out t out w w out( ) ( ) ( ) ( ) ( ) ( ) ( )

, , , , ,

= − + − − −1 Γ λ (19)

where Ct,in, Ct,out and Cw,out (Bq kg-1) are the 137Cs 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 R R Rt out t in w= − +, ,

(20)

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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 tdt

I t R C t R C t R C t A tt in t in t out t out w w in( ) ( ) ( ) ( ) ( ) ( )

, , , , ,

= + − + ′ − λ (21)

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 R R Rt out t in w= − − ′, ,

(22) The 137Cs concentration of the soil within the plough layer Cs(t′) (Bq kg-1) can be expressed as:

C t A td

C td

A t Rd

A t e dtst

t

t

s for a net erosion site

for a net deposition site

( ) ( )

( ) [ ( ) ( ) ]

′ = ′

′ = ′ − ′′ ′′− ′′−∫10

1λ

(23) where |R| (R<0) is the net deposition rate. The relationships between Cs and Ct,in and Ct,out are as follows:

C t C t C t

C t PC t I tR

P e

t in t out s

w out sw

R Hw

, ,

,/

( ) ( ) ( )( ) ( ) ( ) ( )′ = ′ = ′

′ = ′ + ′ − −γ 1 (24)

while the 137Cs concentration of water-derived deposited sediment Cw,in(t′) (Bq kg-1) can be expressed as:

C tRdS

P C t RdSw in

S

w outS

, ,

( ) ( )′ = ′ ′∫ ∫1 (25)

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For a given point, the tillage-derived erosion or deposition rate (Rt,out-Rt,in) can be calculated from Equations 17 and the net soil erosion rate (R>0) or deposition rate (R<0) can be estimated by solving Equations 19, 20, 24 and 25 numerically. Advantages and limitations: The mass balance model described here represents an important improvement over the two mass balance models presented previously, in that it takes into account the effects of tillage-induced soil movement. The results from this model are likely to be closer to reality for cultivated soils than those from the other two mass balance models. However, to employ this model, additional information is needed. 3.1.5 The Profile Distribution Model (for uncultivated soils) For uncultivated soils, the depth distribution of 137Cs in the soil profile will be significantly different from that in cultivated soils, where the 137Cs is mixed within the plough or cultivation layer. In many situations, the depth distribution of 137Cs in an undisturbed stable soil will exhibit an exponential decline with depth that may be described by the following function (cf. Zhang et al., 1990; Walling & Quine, 1990): ′ = − −A x A eref

x h( ) ( )/1 0 (26) where: A′(x) = amount of 137Cs above the depth x (Bq m-2); Aref = 137Cs reference inventory (Bq m-2); x = depth from soil surface (kg m-2); h0 = coefficient describing profile shape (kg m-2) If it is assumed that the total 137Cs fallout occurred in 1963 and that the depth distribution of the 137Cs in the soil profile is independent of time, the erosion rate Y for an eroding point (with total 137Cs inventory Au (Bq m-2) less than the local reference inventory Aref (Bq m-2)) can be expressed as:

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Y t PX h=

−−

101963 1 100 0( ) ln( ) (27)

where: Y = annual soil loss (t ha-1 yr-1); t = year of sample collection (yr); X = percentage 137Cs loss in total inventory in respect to the local 137Cs reference

value (defined as (Aref-Au)/Aref×100); Au = measured total 137Cs inventory at the sampling point (Bq m-2). For a depositional location, the deposition rate R′ can be estimated from the excess 137Cs inventory Aex(t) (Bq m-2) (defined as Au-Aref) and the 137Cs concentration of deposited sediment Cd :

( )′ =′ ′

=−

′ −− − ′ −∫ ∫ ∫R A

C t e dt

A APRdS

A e dSex

dt t

t

tu ref

S

refR h

S( ) ( ) /λ

0

01 (28)

Advantages and limitations: The profile shape model is simple and easy to use. However, this model involves a number of simplifying assumptions and does not take account of the time-dependent nature of the 137Cs fallout input and the progressive evolution of the depth distribution of the 137Cs within the soil profile after deposition from the atmosphere. As such, it is likely to overestimate rates of soil loss. 3.1.6. The Diffusion and Migration Model (for uncultivated soils) Although the depth distribution model described above can be used to obtain approximate estimates of soil erosion or deposition rates for uncultivated soils, a more realistic approach needs to consider the time-dependent depth distribution behaviour, which will reflect the time-dependent 137Cs fallout input and the progressive redistribution of 137Cs in the soil profile after deposition from the atmosphere (e.g. Pegoyev & Fridman, 1978; Walling & He, 1993; He & Walling, 1997). In many situations, the redistribution of 137Cs in uncultivated soils can be described using a one-dimensional diffusion and migration

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model characterised by an effective diffusion coefficient and migration rate (cf. Pegoyev and Fridman, 1978; Reynolds et al., 1982; He and Walling, 1997). For example, in some situations, the 137Cs depth profile in uncultivated soils exhibits a broad concentration peak with the maximum concentration located below the soil surface. The variation of the 137Cs concentration Cu(t) (Bq kg-1) in surface soil with time t (yr) may be approximated as:

C t I tH

I t eD t t

e dtu

R HtV t t D t t( ) ( ) ( )

( )/

( )/( ) ( )≈ + ′− ′

′−−

− − ′ − − ′∫π

λ

0

142 (29)

where: D = diffusion coefficient (kg2 m-4 yr-1); V = downward migration rate of 137Cs in the soil profile (kg m-2 yr-1). For an eroding point, if sheet erosion is assumed to be the dominant process, then the erosion rate R may be estimated from the reduction in the 137Cs inventory Als(t) (Bq m-2) (defined as the 137Cs reference inventory Aref less the measured total 137Cs inventory Au (Bq m-2)) and the 137Cs concentration in the surface soil Cu(t′) from Equation 29 according to:

PRC t e dt A tut t

t

ls( ) ( )( )′ ′ =− − ′∫ λ

0 (30)

For a depositional location, the deposition rate R′ can be estimated from the 137Cs concentration of deposited sediment Cd(t′) and the excess 137Cs inventory Aex(t) (defined as the total measured 137Cs inventory Au less the local reference inventory Aref) using the following relationship:

′ =′ ′

=−

′ ′− − ′ − − ′∫ ∫R A

C t e dt

A A

C t e dt

ex

dt t

t

tu ref

dt t

t

t( ) ( )( ) ( )λ λ

0 0

(31)

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where Cd(t′) can be calculated from:

C tRdS

P PC t RdSd

S

uS

( ) ( )′ = ′ ′∫ ∫1 (32)

Advantages and limitations: The diffusion and migration model described here takes into account the time-dependent behaviour of both the 137Cs fallout input and its subsequent redistribution in the soil profile and therefore represents an improvement over the profile distribution model presented in the previous section. However, to use this model, more information on the behaviour of 137Cs in undisturbed soils is needed. 3.2 Models for use with 210Pbex and 7Be inventories Excess 210Pb has long been employed to determine sedimentation rates in depositional environments, such as reservoirs, lakes, floodplains, etc (cf. Appleby and Oldfield, 1992.). The presence of 7Be has also been used to assess the status of a sediment profile, i.e. whether or not it has been recently disturbed (cf. Bopp, et al. 1993). The use of 210Pbex and 7Be measurement for estimating soil redistribution rates is, however, is still in its infancy and not as well established as that of 137Cs. The behaviour of the two radionuclides in different soils still requires further investigation before their full potential can be exploited with confidence. The development of the conversion models for 210Pbex and 7Be presented here should be seen as an attempt to encourage the use of these two radionuclides in different environments and, in turn, to identify the shortcomings of the new models and to improve them accordingly. The similarities between the three radionuclides have already been emphasized in the introduction to these notes. They provide the theoretical basis for extending the 137Cs technique, as reflected by site selection, sampling procedures, sample treatment, etc., to the other two radionuclides. As a basis for the adaptation of the 137Cs conversion models to 210Pbex and 7Be, Table 3 presents a critical assessment of the key characteristics of the three radionuclides. A selection of conversion models used for 137Cs has been modified to accommodate the identified differences.

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Table 3 Comparison of the key features of the environmental radionuclides: 137Cs, 210Pbex and 7Be Radionuclide 137Cs Excess 210Pb 7Be Origin Weapons-testing Natural geogenic Natural

cosmogenic Half-life 30.2 years 22.3 years 53.3 days Covered time period Since 1954 > 100 years Days-months Temporal pattern of inputs

Main input commenced in 1954, peaked in 1963 and ceased in the 1980s*

Continuous input with limited inter-annual variations

Daily inputs need to be summed

Global patterns of reference inventory

High in northern hemisphere, low in southern hemisphere

Largely unknown Largely unknown

Depth distribution at eroding sites under (1) cultivation and (2) pasture

(1) Uniform distribution (2) Exponential decrease

(1) Uniform distribution, (2) Exponential decrease

Exponential decrease (both)

Tillage influence Possible Possible Not applicable

Time basis of estimated soil redistribution rates

Annual average Annual average Event(s) based

* 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 137Cs 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 137Cs within uncultivated sites, but with a more restricted depth distribution and therefore a much lower profile shape factor.

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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 (<20 kg m-2) is applicable; 4) Different decay constant.

Walling et al., (1999), Blake et al. (1999), and Schuller et al. (2006) provide further detail on the use of Be7 in soil erosion studies and the applications of conversion models using this approach. 3.2.2 Conversion Models for 210Pbex Inventories 210Pbex has a comparable half-life (22.3 years) to 137Cs. However, its natural origin and continuous input provide potential for deriving estimates of longer term erosion rates. Two mass balance models (one with a tillage component included and the other without) have been developed for use with the 210Pbex inventories from cultivated sites. They represent adaptations of mass balance model II and mass balance model III, respectively. The diffusion and migration model for 137Cs has also been modified for use with 210Pbex measurements obtained from uncultivated fields. The key modifications that have been made to the 137Cs conversion models are as follow:

1) The modelling period has been set to extend back 100 years from the sampling date. It is assumed that any 210Pbex fallout deposited before that will have become insignificant due to decay. In theory, only around 4% of the original radioactivity will remain after 100 years. The time of sampling is not used in the calculation. It is only relevant for interpreting the results obtained from applying the conversion model.

2) It is assumed that 210Pbex fallout input and the loss by decay represent a steady state and that the reference inventory will therefore remain the same through time. The deposition flux ( )(tI ) may be calculated intrinsically from the local reference inventory ( Aref, ) using Equation 32 :

3.22/)2ln(*)( reft AI = (32)

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3) Different decay constant. Most of the parameters used in the conversion models for 210Pbex inventories are very similar to those used for the 137Cs models. However, caution is needed in their specification, since the behaviour of 210Pbex in soils could still be different from that of 137Cs. More information related to the application of 210Pbex in soil erosion studies on cultivated land can be found in Walling and He (1999) and Walling et al. (2003). 4. Parameter specification As with many models, the reliability of the conversion models outlined in the previous section depends heavily on the specification of the relevant parameters. Some of these are more difficult to determine than others. Procedures for their estimation / determination are described in this section. 4.1 Reference inventory The reference inventory is a critical parameter for any study using 137Cs, 210Pbex, or 7Be. The value used determines whether a sampling point is designated as having undergone erosion or deposition, the intensity of those processes, and consequently the net erosion rate. For 137Cs, an algorithm has been developed to derive estimates of region-specific values, based on known relationships between the reference inventory, geographic location (longitude and latitude) and annual rainfall. However, it must be emphasized that the predicted value is only meant to provide a preliminary assessment of the likely reference inventory for a study site and the value should not be used in the conversion models. This is especially true if a study area has previously not been sampled for 137Cs, since the validity and strength of the relationship in such areas remains uncertain. Sampling of local undisturbed sites at a similar altitude to the study site is necessary to provide the reference value for use in the conversion models. There are currently no procedures to predict the reference inventories for 210Pbex, and 7Be. It is recommended that some sectioned samples should be collected from some sampled reference sites, to confirm their undisturbed status. These profile distributions can also be used to derive other parameters, such as the profile shape factor, migration rate and diffusion rate, etc.

22

4.2 Particle size correction factors for eroding and deposition sites The particle size correction factors aim to take account of the grain size selectivity of erosion and sedimentation processes. For an eroding site, the correction factor is a function of the ratio of the 137Cs concentration of mobilised sediment to that of the original soil (cf. He & Walling, 1996). Because the grain size composition of mobilised sediment is usually enriched in fines compared with the original soil, the correction factor is generally great than 1.0, due to the strong affinity of 137Cs for fine soil particles. Its value is therefore a function of the grain size composition of both mobilised sediment and the original soil. For a depositional site, the correction factor is a function of the ratio of the 137Cs concentration of deposited sediment to that of the mobilised sediment. Because the grain size composition of deposited sediment is frequently depleted in fine fractions compared with the mobilised sediment, the value of the correction factor is generally less than 1.0 In order to estimate values for the particle size correction factor for eroding sites (P) and depositional sites (P’), information on the grain size distribution of the soils and the mobilised sediment is needed. Values of P and P’ can be estimated using the procedures described by He and Walling (1996). If the specific surface area of mobilised sediment is Sms (m2 g-1), and that of the original soil is Ssl (m2 g-1), P can be calculated as: P

SSms

sl=

ν

(33) where ν is a constant with a value of ca. 0.65. If the specific surface area of deposited sediment is Sds (m2 g-1), P′ can be calculated as: ′ =

P

SS

ds

ms

ν

(34) The value of ν in Equation 34 is the same as that in Equation 33. The following relationship therefore exists:

23

PPSS

SS

SS

ms

sl

ds

ms

ds

sl′ =

=

ν ν ν

(35) 4.3 The proportion factor The value of the proportion factor will depend on the temporal distribution of local rainfall in relation to the timing of cultivation. In situations where high intensity rainfall events, which can generate surface runoff and thus erosion, occur shortly before the period of cultivation, the radionuclides already accumulated at the soil surface as well as input directly associated with these high intensity rainfall events will be susceptible to removal by erosion prior to their incorporation into the soil by tillage. In these circumstances, the value for the proportion factor can be assumed to be 1.0, if there is only one cultivation operation. In cases where the main period of high intensity rainfall events occurs immediately after cultivation has been completed and the remainder of the rainfall occurring during the year is unlike to generate surface runoff, the radionuclides accumulated at the soil surface before the occurrence of these high intensity events will have been incorporated into the plough layer, and only those directly associated with these rainfall events will be susceptible to removal by erosion. Under these circumstances, the value of proportion factor may be approximated by the ratio of the depth of rainfall associated with the period which produces surface runoff to the total annual rainfall. If there is more than one cultivation operation each year, the temporal pattern of precipitation in relation to each cultivation operation needs to be considered. 4.4 The tillage constant The significance of tillage translocation for within-field translocation has been emphasized in recent studies (Govers et al., 1996,1999; Quine, 1995). Its contribution to the redistribution of 137Cs and 210Pbex is quantified in some mass balance model by use of a constant that represents a slope-independent specific soil flux. It is a lumped value for the time period under investigations (about 40 years for 137Cs and 100 years for 210Pbex). No temporal and spatial variation is considered. While there are some data available, in terms of tillage constant, for specific implements, the direct application of such values in the conversion models is frequently complicated by the need to take account of the use of

24

different implements over a period of a few decades. It is suggested that the tillage constant should be estimated using the inventories of eroding sites, at the top of a slope where the contribution of water erosion and deposition can be assumed to be negligible. Assuming that there is no significant water erosion, the tillage erosion rate Rt (kg m-2 yr-1) can be estimated from the measured total 137Cs inventory A1(t) (Bq m-2) of a eroding point using the following Equation (derived from equation 10):

A t A t e I t e dtR d t t R d t t

t

t

1 1 0 1 0 1

0

( ) ( ) ( )( / )( ) ( / )( )= + ′ ′− + − − + − ′∫λ λ (36)

The tillage constant can be estimated from the erosion rate:

φ β β= =R L R Lt out, ,sin sin

1 1

1

1 1

1 (37)

A separate add-in has been developed to solve the above equations numerically for the tillage constant, using the measured inventories for 137Cs or 210Pbex and taking account of site-specific values for the reference inventory, tillage depth, slope gradient and soil bulk density. For 137Cs, the year of tillage commencement and the sampling year can also be specified. This add-in is provided as an optional component of the software. 4.5 The profile shape factor, migration rate and diffusion coefficient The profile shape factor (h0) describes the rate of exponential decrease in inventory or radioactivity with depth for a soil profile from an uncultivated site. The larger the value of the profile shape factor, the deeper the 137Cs penetration into the soil profile. If 137Cs measurements have been made on depth-incremental samples collected from a reference site, h0 can be estimated using trend analysis in Excel via curve-fitting. To estimate the value of h0, an exponential function in the form of f z f e z h( ) ( ) /

=−0 0 can be fitted to the

relationship between sampling depth and mass concentration or inventory, where f(0) is the fallout concentration (inventory) at the surface and z is the sampling depth expressed as a cumulative mass above the given depth.

25

The diffusion coefficient (D) and the migration rate (V) are used to characterize the evolution of the shape of the 137Cs profile with time. High values of D and V will imply a deeper penetration of 137Cs into the soil profile. Although more precise values of D and V can be obtained through solving the one-dimensional transport equation (cf He & Walling, 1997), they may be approximated using the following equations:

V Wt

p≈

−1963 (38)

( )D

N Wtp p

≈−

−

2

2 1963( ) (39)

where: t = the year when the soil core was collected (yr);

Wp = mass depth of the maximum 137Cs concentration (kg m-2); Np = distance between the depth of the maximum 137Cs concentration and the

point where the 137Cs concentration reduces to 1/e of the maximum concentration (kg m-2)

For 137Cs, D (kg2 m-4 yr-1) and V (kg m-2 yr-1) are normally in the range of 30-50 kg2 m-4 yr-1 and 0.2-1.0 kg m-2 yr-1, respectively. Since the maximum 210Pbex concentration usually occurs at the soil surface, it is convenient to represent the evolution of the depth distribution as a diffusion process and to assume that V = 0. D can be calculated using Equation 40 (He and Walling, 1997).

)4(*5.012

2

0 DV

DDV

h−+=

λ (40)

Routines have been developed for estimating h0, D, and V for reference sites where values of depth incremental sample mass and corresponding values of mass concentration are available. These can be accessed when a migration and diffusion model is selected within the software. It is important to note that the estimated values will depend on the

26

thickness of the depth increments used in sectioning the core. Since soil erosion is a surface processes, it is vital to obtain detailed information near the top of the profile, in order to better characterize the upper part of the depth distribution. 4.6 Annual deposition flux file With the exception of the simplified mass balance model, all other mass balance models use the annual deposition flux in their calculations. For 210Pbex, the deposition flux is calculated by the programme itself, using the given reference inventory and assuming a continuous steady-state annual input. For 137Cs, the annual fallout deposition since 1954 has to be provided. Few, if any, studies will have this kind of data available for specific sites. The software therefore uses the given reference inventory value and generalised information on the temporal distribution of annual fallout to synthesise the annual series for a study site. Assuming that the study site has the same relative annual variation as that represented by the generalised patterns for the reference stations, although different in absolute magnitude, the local 137Cs deposition flux I(t) can be calculated from the following equation:

)()( tItI nα= (41) where: In(t) = the 137Cs deposition flux for the reference station (Bq m-2 yr-1); α = a scaling factor which can be calculated as follows:

n

reft

ttn

refAA

detI

A=

′′=

∫ ′−−

1954

)()( λα (42)

where An (Bq m-2) is the present total atmospheric fallout inventory for the 137Cs deposition at the reference station. The reference station file is a plain text file that contains a single column of numbers listing the annual flux in Bq m-2 from 1954. The default value for years after 1983 is zero. Representative station files for both the northern and south hemisphere without Chernobyl-related 137Cs inputs are included in the help documentation. They can be copied, saved and modified. A customised annual

27

deposition flux file can also be generated for those areas where the Chernobyl- related 137Cs input is known to be significant, if the deposition flux at the time (1986) is known. 5. Model choice for 137Cs and 210Pbex

For 137Cs and 210Pbex, there is a choice of several conversion models. The discussion of their advantages and limitations provided in the previous sections should help users to make an appropriate choice, according to the intended use for the estimated erosion and deposition rates, the land use history of the study sites, and the availability of the relevant parameters and the feasibility of obtaining them if they are not available. While it is impossible to indicate which model is more suitable for any specific project, some general guidelines are provided in the following diagram (see Figure 1). Apart from data availability, it is shown that a knowledge of the site-specific radionuclide redistribution processes is a key prerequisite for the choice of the correct model. Collection of sectioned cores from some critical points / sites will not only help to refine the sampling strategy but also yield valuable information about the ongoing soil redistribution processes. For example, the depth of the maximum concentration down the profile for a reference site provides an indication of the migration rate for 137Cs. The inventory deficit at the very top of a slope can be used as an indication of the intensity of tillage translocation. 6. Management of the add-in and its use 6.1 System requirement and installation The software distributed is a standard Microsoft Excel add-in. It consists of one add-in application file (named radiocalc.xla), one compiled html help file (named radiocalc.chm) and one rainfall data file if the approximate local reference inventory has to be estimated. The minimum software environment is a copy of Excel 97 operating on Windows 98 and Internet Explorer 5.0. It was developed on a PC running Windows 98 and Excel 2000 and subsequently tested on the following Window’s operating systems and Excel 2003. User's feedbacks on its performance on other system combinations are most welcome.

28

Figure 1. General guidelines for the choice of conversion models Add-in installation To install the programme, one only has to move the files to a user-preferred folder. It is strongly recommended that all three files should be saved in one folder. The ideal

29

location is the 'Add-ins' folder created by the Windows setup programme. One way to find the location of this folder on your PC is to try saving a workbook as an add-in and note down the full path to the prompted default folder provided by Excel. When you use the add-in for the first time, the following steps must be followed to prepare it for use: 1) Start Excel 2) Navigate the menu system as follow: Tools | Add-ins. If the files have been saved in the recommended folder, the add-in named 'radiocalc.xla' should be in the list of available add-ins. Otherwise, click the 'Browse' button, find the file 'radiocalc.xla' and open it. It should, then, appear on the list. Make sure the leading box in front of the 'radiocalc.xla' is checked by clicking it if necessary. 3) Close the add-in window and a new menu item named 'radionuclide inventories conversion' should be added to the 'Tools' menu. For the latest version of the Excel (2007), it will appear on the tab for the add-ins. Then, you can use the add-in by simply clicking newly added menu item. 4)) If you have to save help files separately from the add-in file itself, you need to specify the correct path via the 'options' button on the main menu before the online help for the add-in can function properly. 5) Apart from adding one entry to the Window’s registry, no other files or modifications are to be made to the user's system files and configurations.

How to remove the add-in Removal of the add-in from your computer involves two steps: 1) Remove the added menu item from the menu system: navigate the add-in list via Tools | Add-ins. Then, remove the add-in by clicking on the leading box. The programme will also delete its entry to the Windows's registry. 2) Remove the programme files: go to the folder and delete the files as usual. When you go to the add-in list next time, you will be prompted by the Excel to delete the add-in from the list of available add-ins. It is advised that users should avoid deleting the programme associated files before removing the added menu item.

30

6.2 How to use the add-in Input data management The data for the sampling points should be entered into an Excel worksheet and listed in columns which run from the top to the bottom of the slope. The same numbers of entries will be expected for sample inventory, particle size correction factor, etc. An active worksheet is required for it to run. All input data are expected to be in this worksheet and the output results will also be saved in it. Interaction with the interface The interface of the add-in should be familiar to any Microsoft Window user. Designed as a tool for research purpose, it is assumed that the user knows how to deal with command buttons, option buttons, input boxes, etc., in a Window operating system. With this add-in, users also have to select a particular set of values or range (multiple cells in a single column) for data input and output. This can be done by selecting the first cell in the range, holding down the mouse, dragging over the cells, and releasing the mouse when you reach the last cell. If a mistake is made, one can simply click the first cell and start all over again. Where a range is expected, the text input box is locked (no keyboard entry will be allowed) to avoid error in its specification. Instead, an arrow-labelled button to its right is provided. The user simply clicks the button to select a range interactively from the active worksheet. How to get help The add-in has online help included. It is a compiled html help file that can be viewed while the programme is run or independently as an ordinary file as long as you have Microsoft Internet Explorer (version 5 and above) installed. To obtain further technical helps, you can email the contact author at D.E.Walling@ex.ac.uk.

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References Appleby, P.G. and Oldfield, F. (1992). Application of 210Pb to sedimentation studies. In

M. Ivanovich and R.S. Harmon (eds): Uranium Series Disequilibrium. Oxford University Press, 731-778

Blake, W.H., D.E. Walling, and Q. He. (1999). Fallout beryllium-7 as a tracer in soil erosion investigations. Applied Radiation and Isotopes 51(5):599-605.

Bopp, R.F., H. J. Simpson, S. N. Chillrud, D. W. Robinson. (1993). Sediment-derived chronologies of persistent contamitants in Jamaica Bay. Estuaries: 16(38):606-616.

Govers, G., Quine, T. A., Desmet P. J. J and Walling, D. E. (1996). The relative contribution of soil tillage and overland flow erosion to soil redistribution on agricultural land. Earth Sur. Proc. Landforms 21: 929-946.

Govers, G., D.A. Lobb, and T.A. Quine. 1999. Preface - Tillage erosion and translocation: emergence of a new paradigm in soil erosion research. Soil and Tillage Research 51(3-4):167-174.

He, Q. and Walling, D.E. (1996). Interpreting the particle size effect in the adsorption of 137Cs and unsupported 210Pb by mineral soils and sediments. J. Environ. Radiact. 30: 117-137.

He, Q. and Walling, D.E. (1997). The distribution of fallout 137Cs and 210Pb in undisturbed and cultivated soils. Appl. Radiat. Isotopes. 48: 677-690.

Pegoyev, A. N. and Fridman, Sh. D. (1978). Vertical profiles of cesium-137 in soils (translation). Pochvovedeniye 8: 77-81.

Quine, T.A. 1995. Estimation of erosion rates from caesium-137 data: the calibration question, pp: 307-329. In: I.D.L. Foster, A.M. Gurnell, and B.W. Webb (eds.), Sediment and Water Quality in River Catchments, John Wiley, London.

Reynolds, W. D., Gillham, R. W. and Cherry, J. A. (1982). Evaluation of distribution coefficients for the prediction of strontium and cesium migration in a uniform sand. Can. Geotech. J. 19: 92-103.

Ritchie, J. C. and Ritchie, C. A. (1995). 137Cs use in erosion and sediment deposition studies: promises and problems. In: IAEA, Use of Nuclear Techniques in Studying Soil Erosion and Siltation. IAEA-TECDOC-828, 111-201.

Quine, T. A. (1989). Use of a simple model to estimate rates of soil erosion from caesium-137 data. J. Water. Resources 8: 54-81.

Schuller, P., Iroume, A., Walling, D.E., Mancilla, B., Castillo, A., and Trumper, R.E. (2006). Use of Berryllium-7 to document soil redistribution following forest harvest operations. J. Environ. Qual. 35: 1756-1763

Walling, D. E. and He, Q. (1993). Towards improved interpretation of 137Cs profiles in lake sediments. In: Geomorphology and Sedimentology of Lakes and Reservoirs (ed. J. McManus & R. W. Duck), 31-53. Wiley, Chichester, UK.

Walling, D. E. and He, Q. (1999). Using fallout lead-210 measurements to estimate soil erosion on cultivated land. Soio Sci. Am. J. 63:1404-1412

Walling, D. E. and He, Q. (2000). The global distribution of bomb-derived 137Cs reference inventories. Final Report on IAEA Technical Contract 10361/RO-R1. University of Exeter.

Walling, D. E. and Quine, T. A. (1993). Use of caesium-137 as a tracer of erosion and sedimentation: Handbook for the application of the caesium-137 technique. University of Exeter, UK.

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Walling, D. E. and Quine, T. A. (1995). The use of fallout radionuclide in soil erosion investigations. In: Nuclear Techniques in Soil-Plant Studies for Sustainable Agriculture and Environmental Preservation. IAEA Publ. ST1/PUB/947: 597-619.

Walling, D. E., He, Q. and Blake, W. H. (1999). Use of 7Be and 137Cs measurements to document short-term and medium-term rates of water-induced soil erosion on agricultural land. Water Resource Research 35: 3865-3874.

Walling, D.E., Collins, A.L. and Sichingabula, H.M. (2003). ‘Using unsupported lead- 210 measurements to investigate soil erosion and sediment delivery in a small Zambian catchment.’ Geomorphology 52, 193-213.

Zapata, F, 2002. Handbook for the assessment of soil erosion and sedimentation using environmental radinuclides. Kluwer Ac. Publ., The Netherlands

Zhang, X. B., Higgitt, D. L. and Walling, D. E. (1990): A preliminary assessment of the potential for using caesium-137 to estimate rates of soil erosion in the Loess Plateau of China. Hydrol. Sci. J. 35: 267-276.