Modelling spatial variability of saturated hydraulic conductivity ...

Hydrologicaî Sciences -Journal- des Sciences Bydrologiques,39,2t April 1994 143

Modelling spatial variability of saturated hydraulic conductivity using Fourier series analysis*

AJAY KUMAR & R. S. KANWAR Department of Agricultural and Biosystems Engineering, Iowa State University, Ames, Iowa, USA

G. R. HALLBERG University of Iowa Hygienic Laboratory, Iowa City, Iowa, USA

Abstract A study was conducted to develop a computer model to simulate the spatial variability of field-measured saturated hydraulic conductivity (Ks) using Fourier series analysis. Ks measurements, both in situ and in the laboratory, were made at 150 and 300 mm depths at regular intervals of 4.6 m on two perpendicular transects crossing each other at the centre of a no-till field. Three Fourier series models, with a finite number of harmonics, were developed to simulate the variability in Ks values. These models included the regular Fourier series model, the half-range Fourier series model with only cosine terms, and the half-range Fourier series model with only sine terms. Also, a test of significance was conducted with the first five and eight terms of the Fourier series models. This test indicated that with only five or eight terms model performances were not significantly different from each other at the 0.05 level. The overall results of this study indicated that spatial variability in the Ks measurements made with both techniques, field and laboratory, can be represented successfully with half-range Fourier series with only cosine terms. Statistical analysis of predicted and observed Ks values indicated that the two sets were not significantly different from each other at the 0.05 level.

Modélisation de la variabilité spatiale de la conductivité hydraulique à saturation au moyen de l'analyse en series de Fourier Résumé Une étude visant à construite, grâce à l'analyse en séries de Fourier, un modèle de simulation sur ordinateur permettant de simuler la variabilité spatiale de conductivités hydrauliques à saturation au champ (Ks) a été entreprise. Les mesures de Ks, réalisées in situ et au laboratoire, ont été faites sur des échantillons prélevés à 150 et 300 mm de profondeur selon un pas de 4.6 m le long de deux transects perpendiculaires dont l'intersection se situait au centre d'une parcelle non cultivée. Trois modèles en séries de Fourier, comportant un nombre fini d'harmoniques, ont été définis pour simuler la variabilité des valeurs de Ks. Il s'agissait d'un modèle basé sur une série de Fourier complète et de modèles basés respectivement sur un développement ne comportant que les termes en sinus ou en cosinus. Un test de niveau de signification

* Journal Paper No. J-1549S of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa. Project No. 2988. This work was supported in part by the Iowa Department of Natural Resources.

Open for discussion until 1 October 1994

144 Ajay Kumar et al.

a été appliqué à des développements comportant respectivement cinq et huit harmoniques, qui a montré que ces développements n'étaient pas significativement différents au niveau de signification 0.05. Globalement les résultats de l'étude ont montré que la variabilité spatiale des valeurs de Ks, qu'elles soient mesurées in situ ou au laboratoire, pouvait être modélisée de façon satisfaisante par un développement en série de Fourier ne comportant que des termes en cosinus. L'analyse statistique des valeurs estimées et mesurées de Ks a montré que ces valeurs n'étaient pas différentes au niveau de signification 0.05.

INTRODUCTION

Knowledge about the variability of soil properties is probably as old as the soil classification system (Vieira et ah, 1981). Variability in the hydraulic properties of soil units has been studied by several researchers (Biggar & Nielsen, 1976; Freeze, 1980; Mulla, 1988; Mohanty etal., 1991; Gupta et al., 1992). Among all the physical properties of the soil, saturated hydraulic conductivity is one of the most important properties that affect the water and chemical movement within or through the soil. In many of the current models that simulate transport of agricultural chemicals and water to the groundwater, the soil media are assumed to be homogeneous and isotropic in their flow behaviour. However, field soils are seldom homogeneous, and their hydraulic properties have been found to vary in space and time (Nielsen et al., 1973; Peck, 1984; Lee, 1984; Rudra et ah, 1985; Kanwar, 1988; Kanwar et al, 1989, 1991; Mohanty et al., 1991). Field observations show that the hydraulic properties of soils vary significantly with spatial location even within a given soil type (Warrick & Nielsen, 1980; Russo et al., 1981; Sisson et al., 1981). Recent studies indicate that preferential flow paths and spatial variability in the saturated hydraulic conductivity (Ks) of the soil significantly influence the chemical transport from agricultural fields to shallow groundwater (Beven & Germann, 1982; Kanwar et al, 1988, 1990, 1991). Sharma ef al. (1987) have found that subsurface flow should increase with increased Rvalues. Therefore, a more accurate determination of Ks values is required to get reasonable estimates of water and chemical transport to groundwater and/or surface water.

In quantitative terms, spatial variability could be treated in either a stochastic or a deterministic manner (Rao & Wagenet, 1985). Several studies have been conducted to describe soil hydraulic properties as stochastic variables with respect to space and time (Warren & Price, 1961; Freeze, 1975). Two major approaches have been used to analyze the behaviour of spatial variability in soil hydraulic properties. One approach is to use Monte Carlo techniques for spatial variability studies. This approach involves the sequential generation of data on soil hydraulic properties as an input to the model and subsequent development of the deterministic solution of the flow equation. The Monte Carlo approach was first introduced by Freeze (1975) for the analysis of one-dimensional saturated flow with the assumption of spatially uncorrelated hydraulic conductivities. Smith & Freeze (1979 a,b) later considered a spatial correlation for hydraulic conductivity, using a nearest-neighbour model in the

Modelling spatial variability of saturated hydraulic conductivity 145

analysis of one- and two-dimensional steady saturated flow. Another method is to scale soil water properties according to the concept of similar media as given by Sharma et al. (1980).

Regionalized variable analysis is another approach to deal with the stochastic variability of soil hydraulic properties (Gupta et al., 1992). In this approach, it is assumed that soil properties are stationary in space and that their spatial covariances depend only on the separation of distances between the adjoining measurements and not on the absolute location of the measurement itself. Several researchers have observed that there is a close analogy between the variances of space and time scales and that methods of their analysis are interchangeable (Webster, 1973; Hillel, 1983). Adaptation of time series methods offers promising solutions for the analysis of spatial variability in soil hydraulic properties. Some researchers have explored the possibility of using time series analyses to describe the spatial dependence of soil hydraulic properties in space and time (Peck, 1984; Kachanoski et al., 1985).

Studies have also been conducted to characterize the stochasticity of soil hydraulic properties by auto- or cross-correlograms and variograms (Nielsen et al., 1973; Wierenga et al., 1985; Hillel, 1980). These studies have examined spatial variability patterns by computing the auto-correlation between two measurements separated by a finite distance. Mohanty et al. (1991) used the simplified, split-window, median-polishing technique in conjunction with a robust semivariogram estimator to examine the spatial structure of a glacial till material. Gupta et al. (1992) used Fourier series analysis along with auto-regressive models to model hydraulic conductivity as a stochastic process. Numerous studies have been conducted to study porous media flow by using stochastic approaches (Bakr et al., 1978; Gelhar et al., 1979; Smith et al., 1979; Chung & Austin, 1987; Gupta et al., 1992).

Field experiments have been conducted for this study to collect data on Ks values by using the Guelph permeameter and laboratory methods. Mohanty et al. (1991) found that such data on Ks values contain stochastic components (representing the unexplained proportions of the variances) of only 17% for the laboratory method and 28% for the Guelph. The objectives of the study were to describe the spatial variability in field-measured data on Ks values by using Fourier series models and to study the effects of the number of harmonics of the Fourier series used on the prediction of Ks values. After appropriate scale transformations of measured Ks data (by using a constant head Guelph permeameter), a Fourier series model was developed to predict Ks values for two depths. The stochastic component of these data was small when compared with the periodic component.

FIELD DATA COLLECTION

An experimental plot (115 m x 183 m) was used to collect IÇ data at the Agronomy and Agricultural Engineering Research Center near Ames, Iowa.


Soil types at this site were Nicollet and Clarion loam in the Clarion-Nicollet-Webster Soil Association (USDA, 1984). Table 1 summarizes some physical soil properties of the plot area. The plot has gentle slopes of less than 2% to the north and was on a slightly convex rise on the south with low relief. A Guelph permeameter (Reynolds & Elrick, 1987) was used to measure the in situ values of Ks at several sites located on each of two orthogonal transects crossing at the centre of the field, as shown in Fig. 1 with triangulated dotted lines. The transects were oriented in northwest-southeast and northeast-southwest directions along the major and minor axes of the field (Fig. 1). In Fig. 1 the solid line contours divide the soil groups A-F; each plot surrounded by broken dotted lines is drained by a single tile line; parallel solid lines within the site are roads and pavements. This type of sampling is not commonly used

B - Clarion 2-5% C - Clarion 6-9% D - Harps loan 0-2% E - Nicollet t loan 1-3% F - W e b s t e r Silty clay loan 0-2%

Fig. 1 Experimental plot for collecting data on saturated hydraulic conductivity near Ames, Iowa; 66 sampling sites arranged on two transects (northeast-southwest and northwest-southeast) across the field shown by triangulated dotted lines. (See text for other features).


to study the spatial dependence of soil hydraulic conductivities. Ks

measurements with the Guelph permeameter were made at 4.6 m intervals on both transects, at 150 and 300 mm depths. Forty measurements were made on the northwest-southeast transect and 26 on the northeast-southwest. This resulted in 66 in situ measurements of Ks for each depth (as shown in Table 2). All measurements were made in the crop rows to avoid compaction from wheel traffic, which could have affected the Ks values significantly (because of the reduction in total porosity and macroporosity). The Guelph permeameter method described by Reynolds & Elrick (1987) was used to measure the steady rates of recharge at 50 and 100 mm heads, and Ks values were calculated by using the relationship based on Richards' analysis for steady state discharge from a cylindrical well in an unsaturated soil.

Table 1 Physical properties of the soil at the experimental site (Kanwar et al., 1988)

Depth

mm

0-300 300-600 600-900 900-1200 1200-1500

No-till

Particle

sand

200-50

% 38.9 35.5 38.0 53.1 54.8

size(jim)

silt

50-2

% 36.7 37.8 36.0 25.2 31.0

clay

2

% 24.5 26.8 26.0 21.7 14.2

Bulk density

g/cc

1.43 1.33 1.38 1.44 1.65

Conventional tillage

Barticle

sand

200-50

% 39.8 41.3 47.9 55.1 54.8

size(f«n)

silt

50-2

% 38.6 36.5 32.4 29.6 31.0

clay

2

% 21.6 22.2 19.7 15.3 14.2

Bulk density

g/cc

1.46 1.45 1.33 1.45 1.65

In addition to the Guelph data, six undisturbed soil cores were taken from each of the 66 sites, with three replications from each depth. Undisturbed soil cores (76 mm in diameter and 76 mm long) were collected with an Uhland core sampler for hydraulic conductivity measurements in the laboratory. Details on the method of collecting undisturbed soil cores for Ks determination in the laboratory are given by Kanwar et al. (1989) and Mohanty et al. (1991). In the present study, 66 data sets on Ks were used for each depth and each method for developing a computer model to describe the spatial variability in Ks values.

MODEL FORMULATION

Any continuous function may be represented, exactly and at all points, by a Fourier series with an infinite number of sine and cosine terms. A discrete series of data at regular intervals may be fitted exactly by a finite Fourier series with the same number of terms as there are data points, or fitted approximately if a smaller number of terms is used. In the general model formulation, the Fourier series method was used with a finite number of harmonic terms. This model is represented by a Fourier series of the following form:


Ks(l) = M+ E C(M).cos 2irMi +D(M).sm2irM

M=i \_ L L 2 riff T.f.jf. . 2itMll (\\ + D(M).sm w

where: KS(J) = saturated hydraulic conductivity value at location /, / = distance between sampling sites from one end of a transect, (I =

4.6, 9.2 m ). As explained in the text, / is in units of 4.6 m, 1 N

jt£ = mean value of the N Ks values = _ £ K(J) , Ni=i

M = rank number of harmonic, JV = number of sampling points for Ks (N preferably should be odd), L = transect length of Ks(l) (L = NI), €(M) = coefficient of cosine term of Mth rank harmonic, D(M) = coefficient of sine term of Afth rank harmonic, P = the rank number of the highest harmonic terms used.

The Fourier coefficients of the cosine and sine terms in equation (1) may be computed as:

cm = ll and

Ni"i

N D(M) = 1%

iv/=i

Ks(l).cos^Mi

v m • 2-wMl Ks{T).nn—-— L

M = 1,2 (iV-l)/2 (2)

M = l,2,...,(JY-l)/2 (3)

In practice, one may restrict the number of harmonics in the above equations with small loss of accuracy in the fitting of the data. The appropriate number may be identified after making statistical tests (Kottegoda, 1980). Therefore, the number of harmonics that contribute significantly needs to be selected. The rank number of harmonics to be included in the series was selected on the basis of the contributions made towards the periodic component of the data. The contribution of the Mth rank harmonics (VM) was computed as follows:

F = iCm2+D(M)2f5 (4) M j

There are two ways to select the number of significant harmonics: (1) by using an analysis of variance (Kottegoda, 1980); and (2) by using a subjective analysis of periodograms. In this study, neither method was applied. The successive values of VM obtained were neither convergent nor divergent. The sequence fluctuated with one VM value being either lower or higher than its neighbours. The rank number of significant harmonic terms was therefore selected based on some past research work in this field (Gupta et al., 1992). Further, the total contribution of the first five harmonics was more than 80%. Gupta et al. (1992) selected five as the significant rank number of harmonics on the basis of a subjective analysis of the periodograms.

The formulated model can be expressed as:


Ks(l) = /* + E

where JÏ = rank*

C(M). cos +D(M).sin v-̂

number of significant harmonics. The other two models of Fourier series (Franklin, 1958) were formulated

with only cosine and sine terms in the series and are given in equations (6) and (7), respectively:

K,(Q = M+ E C(M)xos^€ (6) M=l JL

Kfi) = n+lD(M).sm^Mi (7) M=l A*

The coefficients, C(M) and D(M), were computed by using equations (2) and (3), respectively. A computer program was written to calculate the contribution of each rank of harmonics to estimate the periodic Ks values. Each of the three models given by equations (5), (6) and (7) was tested, first with five harmonics in the series and then with eight harmonics in the series. The basis for the selection of the significant number of harmonics was as explained earlier and on statistical interpretation. Essentially, these models were evaluated on the basis of their statistical significance by determining whether a greater number of harmonic terms makes any significant improvement on fitting the Ks

observed values.

DISCUSSION OF RESULTS

À summary of the experimental data on the Ks values collected by using Guelph permeameter and laboratory methods is presented in Table 2. Three Fourier series models, given by equations (5), (6), and (7), were used to describe the spatial variability in the Ks data. The following sections discuss the results of this study of the Ks values as a function of depth and of the method of Ks determination.

Table 2 Summary ofKs data sets based on 66 sampling sites arranged on two transects across the field, perpendicular to each other (Mohanty et al., 1991)

Saturated hydraulic conductivity,J^(cm s"1)

*Laboiatory method, Ks at depth Guelph permeameter, Ks at depth

Items 150 mm 300 mm 150 mm 300 mm

Number of observations 185 „ 188 . 66 , 66 Average Jt (em sr) , 6.29 X HT? 8.74xl0"f 5.53x10"? 3.92X10-" Maximum Ks (cm s".1) 2.66x10"' 2.16x10"' 2.06x10"? 1.81x10"' Minimum K.(cm s"1) 5.24x10"* 3.04X10"5, 2.03x10"* 1.82x10"* Standari deviation 5.66xl0~i 6.30x10"? 4.41x10"* 3.43X10"4

Variance 3.21 X10"7 3.97X10"7 1.95X10"7 1.18X10"7

Coefficient of variation (%) 90 72.07 81.35 87.46 * These statistical parameters are based on average K, (i.e. of three replicates)


KB values at 150 and 300 mm depth using the Guelph permeameter

Figures 2(a) and 3(a) show the observed Ks values and the values predicted by the three models containing the first five harmonics of the series. Figures 2(b) and 3(b) show Ks values predicted by the same models but containing the first eight harmonics of the series. It is evident from Figs 2(a) and 2(b) that the best fit of the observed Ks data is described by the Fourier series model containing only cosine terms. Table 3 shows the statistical parameters for the various models. When a significance test was conducted on the Ks values predicted by the three models, the Ks values predicted by model 2 (i.e. Fourier series containing only cosine terms) were not significantly different at the 0.05 level from the observed Ks values. However, it is clear from Table 3 that the Ks

values predicted by models 1 and 3 were significantly different from the observed Ks values (with the Guelph permeameter) at 300 mm depth. It was also observed from the Duncan Groupings that the Ks values predicted by model 2 with either five harmonics or eight harmonics fall in the same group. Similar results were obtained when the Ks data at 150 mm depth (using the Guelph permeameter method) were used to fit the models.

Table 3 Statistical parameters significance test on Ks values

and results of analysis of

Statistical parameters related to Ks values Models Mean Standard «--•-__

(XlCT'cm

Methods (soil depth) deviation

l s - ' ) ( x K r * c cm s

Variance - 1 ) x l ( T 6

Duncan Grouping

Guelph permeameter (300 mm)

Guelph permeameter (150 mm)

Laboratory (300 mm)

Laboratory (150 mm)

Observed Ml Ml M3 M4 MS M6 Observed Ml Ml M3 M4 MS M6 Observed Ml Ml M3 M4 MS M6 Observed Ml Ml Ml M4 MS M6

0.39132 1.67154 0.39680 4.21505 1.67152 0.39703 4.21480 0.54138 1.53869 0.55113 3.44895 1.54094 0.55183 3.45049

0.87826 2.24449 0.87436 4.93509 2.24069 0.87313 4.93251

0.63317 2.17980 0.64356 5.19683 2.18208 0.64512 5.19754

0.34826 0.74536 0.10799 2.19273 0.74902 0.11834 2.19207 0.44863 0.61736 0.11940 1.67088 0.62090 0.12506 1.67484

0.63821 0.79440 0.05809 2.32818 0.80305 0.08276 2.32577

0.57421 0.93550 0.21383 2.61436 0.94489 0.23263 2.62026

0.1195 0.5475 0.0115 4.7338 0.5525 0.0138 4.7349 0.1984 0.3744 0.0141 2.7509 0.3795 0.0154 2.7633

0.4011 0.6222 0.0033 5.3389 0.6348 0.0068 5.3296

0.3247 0.8623 0.0450 6.7324 0.8803 0.0533 6.7624

A B A C B A C A B A C B A C

A B A C B A C

A B A C B A C

Models with same Duncan Grouping letters are not significantly different at the 0.05 level.

Ml = sine cosine series with 5 terms; Ml — cosine series with 5 terms Mi = sine series with 5 terms; Af4 = sine cosine series with 8 terms MS = cosine series with 8 terms; M6 = sine series with 8 terms


0 90 180 270 360 450 540 630 720 810 900 Distance (ft)

Fig. 2 Predicted and observed values of saturated hydraulic conductivity at 300 mm depth (using Guelph permeameter) as a function of the separation distance: (a) when only five harmonics were used in model predictions; and (b) when only eight harmonics were used in model predictions.

Ks values at 150 and 300 mm depth using laboratory method

The observed Ks values using the laboratory method and the Ks values predicted by the three models are shown in Figs 4(a), 4(b), 5(a), and 5(b). Figures 4(a) and 5(a) show the Ks values predicted by model 2 with the first five harmonics. Figures 4(b) and 5(b) show the Ks values predicted by model 2 with the first eight harmonics of the Fourier series. It is evident from Figs 4(a) and 4(b) that the Fourier series model containing only cosine terms best describes the variability in the Ks values. Table 3 shows the statistical


(a) o.c

(b)

0.006

0.005

0.004

0.003

a 0.002

0.001

0.007

0.006

0.005

0.004

0.003

1? 0.002

o 0.001

" OBSERVED

+ PRED.SINCOS

o PRED.COS

A PRED.Sihi

0 90 180 270 360 450 540 630 720 810 900

Distance (ft)

OBSERVED

PRED.SINCOS . model 1

- model 2 /v * f

r

: i I S ! M i_l i i_i _ c i i 0 90 180 270 360 450 540 630 720 810 900

Distance (ft)

Fig. 3 Predicted and observed values of saturated hydraulic conductivity at 150 mm depth (using Guelph permeameter) as a Junction of the separation distance: (a) when only Jive harmonics were used in model predictions; and (b) when only eight harmonics were used.

parameters for the various models. The significance tests (F test) conducted on the Ks values predicted by the three models also indicated that the Ks values predicted by model 2 (i.e. Fourier series model containing only cosine terms) were not significantly different at the 0.05 level from the observed Ks values. However, it is evident from Table 3 that the Ks values predicted by model 1 and model 3 were significantly different from the observed Ks values. Similar results were obtained when the Ks data at the 150 mm depth (using the


(a)

y 3

(b)

o 3

180 270 360 450 540 630 720 810 900

Distança (ft)

Fig. 4 Predicted and observed values of saturated hydraulic conductivity at 300 mm depth (using lab method) as a function of the separation distance: (a) when only five harmonics were used in model predictions; and (b) when only eight harmonics were used.

laboratory method) were used to fit the models. It was also observed that the Duncan Groupings for the various Fourier series models for the laboratory method were similar to those of the Guelph permeameter method.

Effect of number of harmonics on Kt predictions

Statistical analyses were conducted on the predicted Ks values by each model with five and eight harmonics and the observed Ks values. The test parameters


(a)

(b)

• OBSERVED

t- PRED.SINCOS . mode! 1

o PRED.COS -model 2

M KP

& PRED.Si^ - model 3 J > / ^

,x : JS , - \tf ^A - *$S*A / * . .. •_;•** >*».

* " ,%»•, '"'V»» ,• * • > "* m

• a

;****•% y*^ ^ " i m " " * ' • " '

P^*4' B

0 90 180 270 360 460 540 630 720 810 900

distance (ft)

0.01

0.009

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0

0.011

_ 0.01 o | 0.009 o >. 0.008

> § 0.007 ~o § 0.006 o | 0.005

-c 0,004

ra 0.003

J5 0.002

0.001

0 0 90 180 270 360 450 540 630 720 810 900

distance (ft)

Fig. S Predicted and observed values of saturated hydraulic conductivity at 150 mm depth (using lab method) as a function of the separation distance: (a) when only five harmonics were used in model predictions; and (b) when only eight harmonics were used.

• OBSERVED

+ PRED.SINCOS

o PRED.COS

a PRED.SiN

. 1

- model 1 tS/^

- model 2 , ^ V

- model 3 ;\f i

/J

are presented in Table 3. For J^ data obtained by using the Guelph permeameter at 150 and 300 mm depths, the critical F values at the 0.05 level of significance were 118.4 and 233.3, respectively. Similar F values for the laboratory measured Ks data at the two depths were 121.23 and 122.1, respectively. Other statistical parameters on the overall mean, standard deviation and variance are also presented in Table 3. Duncan Groupings from the significance test (Table 3) show that the Ks values predicted by a model with five harmonics and the Ks values predicted by the model with eight harmonics are not significantly different at the 0.05 level, and the models fall


in the same group. These results indicate that Ks values predicted by models with five harmonics are not significantly different from the Ks values predicted by the models with eight harmonics. This clearly shows that inclusion of more harmonics does not improve the prediction of Ks values, and these results are consistent with the results obtained by other researchers (Franklin, 1958; Kottegoda, 1980).

SUMMARY AND CONCLUSIONS

Fourier series analysis was conducted on Ks data collected from two soil depths by using two Ks measurement techniques. For each depth and each method, three models were developed, and predictions on Ks values were made. The results of this study indicate that half-range series (Franklin, 1958) with only cosine terms gives the best fit for the Ks data at each depth for each method. Even though statistically predicted and observed values were not significantly different from each other at the 0.05 level, from the practical point of view the observed difference (Figs 2 and 3) appears to be significant. However, it is clear from the discussion that the variation in Ks values can be explained by a Fourier series of cosine terms only (equation (6)). Therefore, it was concluded that a Fourier series with only cosine terms can accurately predict the spatial variability in Ks values for the Nicollet loam soil. The soil depth and Ks

measurement method do not seem to affect the description of the spatial variability in the Ks values.

Another interesting result of this study was that the effect of the number of harmonics on the prediction of Ks values was negligible. The Ks values predicted by models with five harmonics were not significantly different from Ks values predicted by using eight harmonics. The inclusion of more harmonics did not improve the predictions of Ks values for this data set.

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Received 23 August 1993; accepted 24 January 1994

Modelling spatial variability of saturated hydraulic conductivity ...

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