Systematic and Random Errors in Dual GammaEnergy Soil Bulk Density and Water Content Measurements1

WALTER H. GARDNER, GAYLON S. CAMPBELL, AND C. CALISSENDORFF 2

ABSTRACT

Soil bulk density and water content may be obtained con-currently through measurement of the attenuation of gammaphotons from two different gamma ray sources and simultaneoussolution of the resulting attenuation equations. Errors in thesoil bulk density and water content measurements result fromrandom emission from the sources, random errors in attenuationcoefficients and soil column thickness measurements, presenceof a small, higher energy peak in the 241Am spectrum, andcounting equipment dead time. Using gamma photons from241Am at 0.060 Mev and from 137Cs at 0.662 Mev, attenuatedin 10-cm soil columns, the standard deviation in both watercontent and bulk density measurements is primarily due torandom emission and is about 0.007 g/cm3 for 106 countsmeasured in air and about 0.005 g/cm3 for 2.5 X 106 counts.However, as larger counts are used the precision of measure-ment of column thickness and soil and water attenuation coef-ficients becomes limiting. If, by obtaining large counts, thevariance due to random emission is reduced to the same mag-nitude as that associated with measurement of column thicknessor attenuation coefficients, precision limit would be about0.0035 g/cm3 at midrange values of water content and bulkdensity. At high count rates instrument dead time correctionsare made to get accurate counts. Gamma rays scattered downfrom the small 0.103 Mev peak in the 241Am spectrum result inattenuation dependent attenuation coefficients for which appro-priate corrections must be made. For narrow collimating slitsthe spatial resolution is only a little greater than slit thickness.Collimater and shielding requirements are shown.

Additional Index Words: soil water content, soil bulk density,gamma ray densitometry.

USE OF GAMMA RAY attenuation for nondestructive mea-surements of density has become a common industrial

technique. Its use for bulk density measurement in dry soil(van Bavel et al., 1957) and for sensing changes in watercontent in soil where bulk density may be presumed to

remain constant (Ashton, 1956; Ferguson and Gardner,1962; Gurr, 1962; Rawlins and Gardner, 1963; Davidsonet al., 1963; Gardner, 1965; and numerous others of morerecent origin) is well known. Almost from the start of suchapplications, numerous investigators have recognized that iftwo different gamma energies were used and the resultingattenuation equations were solved simultaneously, both soilbulk density and water content could 'be inferred concur-rently. It remained largely to select appropriate gammaenergies and to demonstrate techniques required (Gardner,W. H. and M. E. Fischer, 1966. Concurrent measurementof bulk density and water content of soil using two gammaray energies. Agron. Abstr. 46, ASA; Soane, 1967; Gardnerand Calissendorff, 1967; Corey et al., 1971). Limitations onprecision and resolution are imposed by the random natureof gamma ray emission from radioactive sources, the ab-sorption and scattering characteristics of these rays in thesoil and the lack of a gamma source with a suitably purepeak in a needed energy range. This paper is intended topresent the error analysis and design considerations re-quired to make use of the technique and will discuss someapplications. '

THEORETICAL

Accuracy and Precision Considerations

The equation required for calculating dry bulk density fromattenuation measurements in a wet sample is obtained by solv-ing simultaneously the two attenuation equations, one writtenfor each gamma energy used (Gardner and Calissendorff, 1967;Corey et al., 1971). Thus for energy a

Na = Noa exp[— S (^p + p.wae) — S'pcaPc]

and for energy b

Nb = Nob exp[— S (^p + fj,wbe) — S'/XcbPj

[1]

[2]

where N represents to total gamma count and ^ the massattenuation coefficient, with subscript o referring to a countin air, a and b the gamma energy used, and s, w, and c thesoil, water, or container; p and 6 are the dry soil bulk densityand water content both on a g/cm3 basis; and S and S' arethe thickness of the soil column and the combined thicknessesof the column container walls in cm. The simultaneous solu-

394 SOIL SCI. SOC. AMER. PROC., VOL. 36, 1972

4 6 8 1 0 1 2Column Thickness,S- cm.

14

Fig. 1 — Contribution to variance in bulk density or water con-tent from random emission of radiation sources for doublegamma measurements. Mass attenuation coefficients have thefollowing values: for 241Am /tsa = 0.305 cm2/g, tiwa =0.2036; for i37Cs Msb = 0.0769, ftwb = 0.0858; for thermalneutrons (TN) nsn — 0.4130 and nwn = 2.632.

tions for 6 and p are

,ln(Na/Nca)-nsaln(Nb/Ncb)Sk

aln(Na/Nca)-/,wb\rl(Nb/Ncb)Sk

[3]

[4]

where k = /x,sayu.wb ancl tne container count, Nc,replaces N0 to eliminate the container terms from equations[1] and [2]. In general we are interested in the precision ofboth dry bulk density and water content measurements. If oneassumes that the variances are independent, contributions tovariance of the various components of equations [3] and [4]may be summed to give variance in dry bulk density or watercontent using standard error propagation formulae (Young,1962). Written for water content, the equation is

2 / 09 -.2 2 , 36B = (———) °\r + (•B \APJ ' Na T \ *\7

30 -.2

"Jycb

de ,2

96

30 N2

86 .2

dOV-wb

[5]

where <r2 is the variance in the mean value of the componentindicated by the subscript. A similar equation is obtained forap

z by replacing the 0's by p's.Attainable precision is limited by random emission of the

radioactive sources, therefore the limit in the absolute preci-sion of the method is obtained from the first four terms ofequation [5], Carrying out the partial differentiation of equa-tions [3] and [4] required by equation [5] and replacing<rN

2 by N, which may be done for the Poisson distribution ofrandom emission, yields (Gardner, 1965; Gardner and .Calis-sendorff, 1967):

CT0Cre) = ' ^ + ̂ 1 + ̂ + ̂ ) [6]/V A/ A/ A/ /1 * n * * />« * * h ' ~ /iJi

(T -,_ -. ̂ •

2 ii 2'tob ^mb

where the subscript (re) is used to designate the contributionto variance from random emission. Taking Nca = Ncb = Nc,equations [6] and [7] become:

{ 1 + exp [S(/j.wb 6 + [8]

2Tp(re)

2

{ 1 + exp [S(ij.wa e + /j,sap)]}

[9]

where 2Va and Nb have been replaced by their equivalentsfrom equations [1] and [2] written in terms of JVC ratherthan N0. Often the exponential terms in equations [8] and[9] are much greater than 1, either from container countsbeing taken for several times as long as soil counts or fromstrong attenuation of low energy beams. We will thereforeassume that the contribution to <rp

2 and <re2 from the random

emission errors in the container counts is negligibly small. Fromequations [8] and [9], we note that the variance due to ran-dom emission goes as the inverse of the container count.

We note also that o^oe)2 and ffp( re)2 are functions of S,the column thickness. Figure 1 shows o^cre)2 and "^(re)2 asa function of S for midrange values of 8 and p and a containercount of 106 for both the cesium-americium and the americium-thermal neutron systems. Contributions to variance in e andp due to random error in measuring column thickness andattenuation coefficients are given in Table 1. Percentageerror in each component parameter which would result inerrors in water content and mineral bulk density of 0.001 and0.01 g/cm3 is also shown in Table 1.

Assuming JVC = 106, p = 1.2 g/cm3, 0 = 0.15 g/cm3, andS — 10 cm, the limiting precision in measurements of watercontent or mineral bulk density due only to random emission,is about a = 0.007 g/cm3. If soil column thickness is knownto about the nearest 0.01 cm and mass attenuation coefficientsare known to four places the combined variance is about 0.05X 10~4 which leads to a total standard deviation of about <reor <rp = 0.0073 g/cm3.

In many experiments, greater precision in water contentmeasurements is desirable. With increased counting time, beamcross section, or unit cross sectional source strength (limitedwith 241Am), Nc may be increased and <re(re)2 decreased.The total error in water content, due both to random emissionand to random errors involved in initial measurement of theparameters, for the example of Table 1, can be reduced toffg = 0.005 g/cm3 if the variance due to random emission isreduced to 0.2 X 10~4. Such an increase would require Nc =2.5 X 106 which could be obtained by counting 2.5 times aslong or using a collimating slit increased in cross, section by2.5 times. As variance due to random emission is further re-duced by increasing Nc, errors in the other parameters, whichhave a relatively minor effect at 106 counts, become increas-

GARDNER ET AL.: ERRORS IN SOIL BULK DENSITY AND WATER MEASUREMENTS 395

Table 1—Variance coefficient (partial derivatives of equation[5]), their numerical values for the 241Am-137Cs system, and

errors in water content 6 and mineral bulk density p result-ing from the indicated error in the specified component

where midrange values of 9 = 0.15 g/cm3, p = 1.2g/cm3, and S = 10 cm have been used."

Bulk Density - g/cms

0.5 1.0 1.5 2.0

Component

S

IVb

fab

9 (0 or p)9 (component;

(e/sf(^ #/lt)2

GZgH0/k)2

<MsaP/k)2

Numericali value

0.000231.136

18.3072.71

1,171

Error in component producingindicated error

0. 001 g/cm3 0.01 g/cm3

a

0.67 6.70.46 4.60.27 2.70.038 0.380.038 0.38

p variance coefficientsSn

Msafab

(Op/S)1

(^ hf/k)2

(^ 9/k)2

^x Lp/k)2

(WwaP/k)2

0.01441.4488.154

92.68521.9

0.083 0.830.41 4.10.41 4.10.034 0.340,058 0.58

1 For random errors the variances to be summed to obtain total variance are the var-iance for random gamma emission, given for midrange values of d and p in Fig, 1,and the variance coefficients in this table multiplied by the appropriate standard devia-tion squared. At time of determination, errors In the above component parametersusually are random, but in subsequent measurements, using the same parmeters onthe system unchanged except for water content or bulk density, these errors becomeconstant and lead to direct additive bias rather than to random variation. The per-centage error In a component to produce 0,001 or 0,01 g/cm3 bias in 0 or p for mid-range values Is given in the table.

ingly important. For extraordinarily large counts, where vari-ance is reduced to a magnitude comparable to that for columnthickness or attenuation coefficients, the precision limit isabout 0.0035 g/cm3.

Both Fig. 1 and Table 1 deal with errors associated withmidrange values of water content, 0.15 g/cm3, and mineralbulk density, 1.2 g/cm3. Figure 2 shows how the variance inwater content for each component parameter, "^(component)2*changes as water content or bulk density varies away frommidrange values. It may be observed that variances decreaseas e and p decrease and that the greatest change occurs as pchanges.

In many experiments bulk density will not change appre-ciably with time. Therefore, it is possible to use values of pdetermined with the double gamma system and then 137Cscounting by itself to follow changes in water content. Theappropriate equations and error analysis are given by Gardner(1965) and Gardner and Calissendorff (1967).

Source Choice ConsiderationsOverall precision equations involve the attenuation coeffi-

cients which, in turn, are functions of gamma photon energy.The functional relationships for concrete, which resemblessoil, and for water are given by Gardner and Calissendorff(1967). Ideally, we would like to have one source which isattenuated mainly by soil and another source which is attenu-ated mainly by water. With the aid of the relationships be-tween energy and attenuation coefficients, source choice wouldseem to be a simple matter. Other considerations, however,drastically reduce the number of suitable sources available.Some of these considerations are (i) short half-life of isotope;(ii) presence of even small, higher energy peaks in the sourcespectrum or closely spaced peaks; (iii) shielding problems;-(iv) cost of sources; and (v) self absorption by low energysource materials.

Presence of higher energy peaks in the spectrum is impor-tant because of their contribution to the lower energy countthrough Compton scattering. The number of scattered gammaphotons depends on the thickness, water content, and bulkdensity of the sample. The attenuation coefficient is thereforeno longer a simple constant and equations [3] and [4] nolonger are adequate. If the peaks are so closely spaced that thecounting equipment cannot effectively discriminate againstthe lower energy gamma photons without excessive countdrift, then equations [3] and [4] again do not hold.

Some high energy sources require excessive shielding for

0.05 0.10 0.15 0.20 0.25 0.30Water Content - g/cm3

0.35 0.40

Fig. 2 — Variance in water content for each component, o-2e

(comp), as water content or bulk density is varied from stan-dard values of 0 = 0.15 g/cm3 or p = 1.2 g/cm3, S = 10 cm,Nc = 106, for which o-2

9(totan = 0.527 X 10~4.

personnel safety and such long collimators for resolution thattheir use becomes impractical. Self absorption by an isotopeemitting low energy gammas may limit the source strength tosuch low values that the source is not usable.

All factors considered 137Cs emerges as the best availablehigh energy source (0.662 Mev) and 241Am as the low energysource (0.060 Mev). Thermal neutrons (below 0.5 ev) can beused with 241Am with considerably improved precision as isshown in Fig. 1, providing that a sufficiently large neutronsource is available. Neutrons generated at the core of theWashington State University reactor and thermalized withgraphite absorbers have been used for measuring water con-tent (Stewart, 1962; Gee, 1965). The 241Am source is limitedin strength by self absorption and has a small, higher energy(0.103 Mev) peak adjacent to its main peak (Crouthamel,1960). These characteristics cause some problems, but, withappropriate precautions which are discussed later this combi-nation works well for double gamma systems.

Other sources, including 144Ce, and e°Co, have been triedin this laboratory but are not recommended because of someof the limitations listed above.

Collimation of Gamma Radiation

Collimation of gamma rays may be accomplished efficientlyusing lead or tungsten. Collimator length required for tung-sten is about % that for lead because of the higher density oftungsten. For maximum resolution, collimating is required onboth source and detector sides and collimating slits or holesshould be parallel and of the same size. The thickness of thesource-side collimator is determined in part by radiation safetyrequirements. Shield thickness to meet US Atomic EnergyCommission laboratory tolerance requirements is given else-

396 SOIL SCI. SOC. AMER, PROC., VOL. 36, 1972

where (Blizard, 1958; Gardner, 1965 corrected in Soil Sci.Soc. Amer. Proc. 32:745-746, 1968). More recent data(Evans, 1968) on the coefficients which are required in theequation lead to a small change in shield thickness from thatshown by Gardner (1965) for a 100 mCi 137Cs source.

Collimator length should be sufficient so that radiation,passing through the source-side collimator walls adjacent tothe soil lamina defined by the collimator slit, does not reachthe detector through the detector-side collimating slit in suffi-cient quantity to affect precision at a desired level. From ananalysis beyond the scope of this paper it has been determinedthat for errors to be less than about 0.001 g/cm3 (computedfor a thickness of 10 cm of soil at a bulk density of 1.2 g/cm3)the source-side collimators should be at least 8 cm long for137Cs and 0.15 cm long for 241Am. Also, the longer the source-side and detector-side collimators the better is the resolutionin terms of low angle scattering of gamma photons which reachsoil adjacent to the soil lamina defined by the collimator slits.Collimator length is limited practically in the case of 241Amby self absorption which limits source strength possible for agiven cross-section and, at least for small collimating slits,necessitates short collimators to keep the count at acceptablelevels. Collimators 1.6 cm long for ^41Am and 9 cm long for137Cs and having a 0.045-cm slit, which were placed on bothsource and detector-sides of a 10-cm soil column showed aresolution of about 0.05 cm for double gamma water contentmeasurements. For narrow slits, at least, the resolution onlyslightly exceeds the thickness of the slit.

EXPERIMENTAL

Various experimental configurations are possible withthe double gamma system, and a number of them havebeen tried in this laboratory. The system described ^herewas set up to determine the water content and bulk densityin the root hair zone of growing plants, and is thereforecollimated to a very narrow beam. Otner experimentswould require other collimator configurations.

Standard equipment was used to detect and record thegamma radiation. The radiation was detected with a thal-lium activated, sodium iodide crystal and photomultipliertube. A 5-cm thick crystal works well for both 137Cs and241Am, although a 2-mm crystal gives a somewhat highercounting efficiency with the 241Am. A single channel ana-lyzer was used to select pulses of the desired energy, andthe pulses were counted using a sealer. A rate meter wasfound helpful in aligning the collimators.

Gamma sources used in this work were 500 mCi 137Csand 500 mCi 241Am. The 137Cs source is in a standard con-figuration not critical to the collimator design and is avail-able from a number of radioisotope suppliers. The 241Amsource was especially designed (Monsanto Research Cor-poration, Dayton, Ohio) to give the highest practical activ-ity per unit area, consistent with the self-absorption of theamericium. Our source has an activity of about 22mCi/mm2 with the active area designed to be a little larger thanthe collimating slit which is 0.045 cm thick and 1.5 cmwide on the source side, increasing to 3 cm on the detectorside.

The collimators were constructed from lead and mountedside by side with a moving platform for positioning thesoil sample precisely in the same position with respect toeach gamma beam. Both source and detector-side collima-tors for 137Cs are 9 cm long and provide about 11 cm ofprotective lead between source and operator. The 241Am

source and detector side collimators are 1.5 and 4 cm long,the latter being longer to block out down-scattered radia-tion from the 137Cs source holder adjacent to, and in tnesame plane as, the 241Am collimator. Lead, considerably inexcess of that required, protects the operator. The 241Amcollimators are not designed for optimum gamma count,which would call for collimators about 0.15 cm long, butare a compromise between high count and desired similar-ity in beam geometry of both gamma sources. The detec-tor crystal for the 241Am source is surrounded by about 4cm of lead to reduce background count. Collimators andsource holders are mounted sturdily so that they can bealigned precisely and the sources are positioned carefullybehind the collimators so as to give the highest possiblecount rate.

In addition to rather stringent collimation requirements,accuracy using the double gamma technique requires thatthe soil column be positioned properly in front of thesources. We accomplished this in the present experimentsby using a rigid frame with a carriage for the soil columns.Vertical positioning was accomplished by turning the screwswhich carried each end of the carriage. The carriage movedup and down as well as across on ball bushings. Stops ateach end of the horizontal rack positioned the columnproperly in front of each source.

Soil containers should be constructed of a material whichhas a sufficient rigidity to maintain a precisely reproduc-able thickness and at the same time attenuate the 241Amradiation as little as possible. Thin aluminum or magnesiumcontainers are best, but lucite containers also work well,and were used for this study. The containers were 10 by 10cm and were constructed from 0.6-cm thick lucite. Cylin-drical containers also work well if narrow collimators areused and care is taken to position the columns properly infront of the sources (R. I. Papendick, personal communica-tion).

Before measurements on soil columns can be made withthe double gamma technique, it is necessary to determinethe corrections and coefficients required in equations [3]and [4]. Dead time corrections for the counting equipment(Fritton, 1969) and nonlinearity corrections associated withthe small 0.103 Mev energy peak above the major 0.060Mev peak for the 241Am source need to be determinedfirst. Attenuation coefficients for soil and water are thendetermined.

The equation for dead time corrections is (Fritton, 1969)

I - R/(l — T K) [10]

where I is the corrected count rate (counts per minute), Ris the observed count rate, and T is the dead time. Since thepossibility of two pulses coming within the instrument deadtime increases with increasing count, this correction needsto be applied only at a high count rate. Once the dead timefor a particular instrument and setting is known, the countrate above which a dead time correction should be appliedis easily found from equation [10], A dead time correctionshould be made if the corrected count differs from the mea-sured count by more than about 0.1 o-(re) or 0.1 X (/)% fora 1-min count.

GARDNER ET AL.: ERRORS IN SOIL BULK DENSITY AND WATER MEASUREMENTS 397

10"10s

Count Rate In Air - cpmFig. 3—Standard deviation in dead-time as a function of count

rate in air for several counting times with a dead-time of 0.05micro minutes.

The dead time will vary somewhat with the instrumentused, the gain setting, and discriminator or single channelanalyzer settings, and must be determined for each instru-ment system used. Dead time may be determined using the137Cs source and two iron absorbers having equal thickness.Count rate is determined for air (R0), for each block sepa-rately and averaged (Rj), and for both blocks together(R2). The dead time is given by

[11](R0 — 2R0R1R2

An error analysis has shown the optimum single absorberthickness to be about 2.4 cm for iron absorbers and 137Csgammas. The optimum thickness does not depend on T orR0. For a given count rate, the standard deviation in T isinversely proportional to the square root of the countingtime. Figure 3 shows the counting time required for a givenprecision in T at different values of R0. The standard devia-tion of ^ is nearly independent of the actual value of T soFig. 5 applies for all values of T around the value shown.

The nonlinearity of the 241Am attenuation curve (Fig.4) due to the small 0.103 Mev peak has been mentionedpreviously. To use 241Am for double gamma measurementsit is necessary to establish a linear approximation of theportion of the curve applicable to overall soil densities

-0.4

-1.2

•z.£ -2.4

-2.8

-3.6

-4.0,' 0 1 2 3 4 5 6

Thickness of Glass Absorber-cm

Fig. 4—Attenuation curve for glass absorbers using 241Amshowing the observed and extrapolated values of In N/Nc.Similar curves have been obtained for water.

Water Content-g/g.10 .15 .20 .25

40L OFig. 5—Water content and bulk density distribution with depth

for Ritzville soil into which root hairs were growing at thesurface. The parameter is the number of days following uni-form wetting of the soil column (Campbell, 1968).

involved in measurements. The value of Nc is determinedby extrapolating this line to zero thickness. To obtain Ncthe thicknesses of several pieces of a plate glass absorberwere accurately measured with a micrometer. Absorbersused were such that counts ranging from N/NC — 0.1 toN/NC = 0.01 were obtained. Ten increments of thickness,Slt S.2 . . . Sw, in this range were used. The absorbers were

398 SOIL SCI. SOC. AMER. PROC., VOL. 36, 1972

Table 2—Representative values of soil and water massattenuation coefficients.

Palouse silt loamRitzville fine sandy loamSalkum silty clay loamEverett gravelly loamAsotin B2 clayWater

'"Cs

0, 07600.07570.07550. 07600.07550.0858

M1Am

0.3050.2980.3110.2900.3230. 2036

Thermal neutrons

0.413

2.632

placed in the soil container and with appropriate countingequipment gain, threshold, and window settings 10-mincounts were made through the 10 absorber combinations.The best least squares fit of a straight line plot of 5 vs. InN was then obtained and Nc was taken as the extrapolationof In N to S = 0. A 60-min air count was obtained imme-diately after the count was made in glass and a correctionfactor (C) for conversion of Nair to Nc was obtained. Forall subsequent experiments, Nc was obtained from CNair.

Once the appropriate correction factors had been ob-tained, attenuation coefficients were determined for waterand soil. From Table 1 it is apparent that ps must be knownwith considerable accuracy to obtain reasonable accuracyin 6 and p. To obtain fj,w the soil container was filled withwater and counts are taken through it with both 137Cs and24IAm. The counts were averaged and corrections weremade to get Na, Nca, Nb and Ncb. These were substitutedinto equations [1] and [2] with the measured S, 6 = 1,and p = O (S'/a.cpc = O because Nc rather than Ng is beingused) to compute M,TO and ̂ ^

Soil attenuation coefficients were obtained by cutting oneof the soil containers into upper, middle, and lower sections,reassembling it with masking tape, and packing it as uni-formly as possible with air-dry soil. Counts were takenthrough the middle section at several locations with both137Cs and 241Am. The top section was carefully removedand the soil scraped off with a straight edge and discarded.The middle section was removed and the soil scraped off,weighed, dried, and reweighed. From these weights andthe measured volume of the container section, ps and 6(g/cm3) were calculated. These values along with the cor-rected counts and the appropriate p.w were substituted intoequations [1] and [2] to get p,s. Table 2 shows attenuationcoefficients obtained for water and for several soils. Thesenumbers vary somewhat with collimator and sample geom-etry and with counting equipment characteristics and set-tings, so it is necessary to determine them under the condi-tions which will be used in the experiment if accuratemeasurements are to be obtained.

Results of a preliminary test of the double gamma systemon a simulated soil (glass plates and water-filled glass cells)are given by Gardner and Calissendorff (1967). These re-sults indicate that, with the previously discussed precau-tions, errors in 6 and p measurements were about the sizepredicted by the error analysis.

Water content and bulk density measurements were alsomade in soil (Campbell, 1968). Plants were grown on afine screen which allowed root hairs to penetrate but keptroots above the screen. The screen was placed on a 10-cmsquare by 5-cm deep column of a Ritzville fine sandy loam.Thus water was removed at the upper soil boundary by the

plants. The gamma beam was used to determine water con-tent and bulk density distributions in the soil column as thesoil dried. The results are shown in Fig. 5. Bulk densitywas computed using equation [4] and the average of sevendeterminations is presented. These bulk densities were usedto calculate water content from the 137Cs counts. Gravi-metric samples which were taken on runs similar to theone shown in Fig. 5 gave water content values whichwere within 0.01 g/g of the corresponding double gammameasurements.