Constrained optimization of spatial sampling in skeletal soils using EMI data and continuous simulated annealing. A. Castrignanò1 F. Morari2* C. Fiorentino1 C. Pagliarin2 S. Brenna3 1
CRA – I.S.A., Via Celso Ulpiani, Bari, Italy. DAAPV,University of Padova Viale dell’Università, 16, Legnaro, Italy. 3 ERSAF, via Copernico 38, 20125 Milano, Italy. *
[email protected] 2
Abstract In skeletal soils, surveys are generally time-consuming, labour-intensive and costly. Maps of ECa could usefully be used to direct soil sampling design in order to characterize soil spatial variability. A protocol using a field-scale ECa survey has been applied to three skeletal soils in Lombardy region (northern Italy). Continuous spatial simulated annealing was used as a method to optimize spatial soil sampling schemes. These were optimized at the point-level taking into account sampling constraints, field boundaries and preliminary observations. Two optimization criteria were used. The first (MMSD) optimized even spreading of the point observations over the entire field by minimizing the expectation of the distance between an arbitrarily chosen point and its nearest observation. The second criterion (MWMSD) is a weighted version of the MMSD and used a digital gradient of the gridinterpolated ECa data to calculate optimal sampling density. The spatial simulated annealing procedure has shown great flexibility in adapting to the different environmental conditions within a reasonable calculation time. ECa maps allowed sampling to be optimized, distinguishing between areas with different priority levels. A highly erratic spatial variation of the ECa prevented application of the MWMSD criterion in only one site. Further optimization criteria should be added to the procedure in the future, in particular minimization of cokriging variance, so that more auxiliary variables can be used to direct sampling schemes. Keywords: soil sampling, ECa, continuous simulated annealing, skeletal soil
1. Introduction Soils containing over 35 or 40% of rock fragments (skeletal soils) are widespread in northern Italy, where they identify areas vulnerable to groundwater pollution. Characterizing spatial variation of soil properties in skeletal soils are necessary to improve process-based models aimed at predicting the effects of land use alternatives on groundwater quality and to apply site-specific techniques (e.g. fertilisation) to reduce agricultural pollution. However soil surveys are generally time-consuming, labour-intensive and costly in skeletal soil (Buchter et al., 1994), limiting the possibility of adopting an appropriate sampling intensity to determine variability within the fields. The drive to collect high spatial density soil data to reduce sampling costs led to the adoption of remote and proximal sensing soil sampling methods. Measurement and mapping apparent
electrical
conductivity
(ECa),
using
electromagnetic
induction
(EMI)
instrumentation equipped with GPS, is becoming one of the more widely used proximal sensing technologies for soil mapping (Corwin and Lesch, 2005). Geo-spatial measurements
of ECa may be used as a powerful tool in site-specific management to direct soil sampling. In a geostatistical context optimizing spatial sampling involves estimation of a model for spatial dependence, usually expressed by a variogram, which can be used to optimise interpolation of an environmental geovariable (e.g. McBratney and Webster, 1983). Van Groenigen et al. (1999) proposed an annealing-based algorithm to optimise sampling schemes on a continuous solution space for different quantitative optimization criteria, taking into account physical sampling barriers and earlier measurements. The objective of this paper is to introduce a method of optimising spatial sampling in skeletal soils using auxiliary information on the area in the form of ECa maps. The procedure is an extension of spatial simulated annealing (SSA) for optimization, presented by Van Groenigen et al. (1999), which allows the inclusion of objective weighting factors and use of auxiliary information. The aim of the sampling is to create soil thematic maps at a given level of precision by using a limited number of sampling points. The maps will be used to implement site-specific techniques in three sites of Lombardy Region (northern Italy).
2. Material and Methods 2.1 Spatial Simulated Annealing Spatial Simulated Annealing is an optimisation sampling method which prevents bias estimates by avoiding sample clustering. A central concept in SSA is the fitness function
φ (S ) which has to be minimised. Starting with a sampling scheme S0, let Si and Si+1 represent two solutions with fitness φ ( S i ) and φ ( S i +1 ) , respectively, where Si+1 is derived from the neighbourhood of Si by a random perturbation of one of the variables of Si. This alternative sampling scheme is accepted with probability Pc, using the Metropolis criterion: Pc ( S i → S i +1 ) = 1,
⎛ φ ( S i ) − φ ( S i +1 ) ⎞ Pc ( S i → S i +1 ) = exp⎜ ⎟, c ⎝ ⎠
ifφ ( Si +1 ) ≤ φ ( Si )
(1)
ifφ ( S i +1 ) > φ ( S i )
(2)
where c denotes a positive control parameter. A transition occurs if Si+1 is accepted. The next solution Si+2 is derived from Si+1 and the transition probability Pc is calculated with a similar acceptance criterion. As the optimisation process evolves, c and the maximum perturbation decrease, forcing the sampling scheme to “freeze” in its optimal configuration in a similar way to the physical annealing process of solids. The approach shows a wide
flexibility in defining several optimisation criteria with their corresponding fitness functions because different surveys may have different purposes. The so-called MMSD (Minimization of the Means of the Shortest Distances)-criterion, which minimizes the expectation of the distance of an arbitrary point to its nearest observation point, has frequently been used in the past (Van Groenigen et al., 1999). This condition should be expressed mathematically, for a scheme S of sampling: E[d(x,S)], where d(x,S) is the Euclidean distance between the position x and the nearest neighboured xi ∈ S. When any previous information on the spatial variation
over the area of interest is lacking, the most prudent design is an even sampling by applying MMSD-criterion. However, this criterion cannot be the only best one for agricultural fields, which generally have uneven spatial distribution of soil properties. We therefore modified the MMSD-criterion in order to be able to distinguish between areas with different priority levels. The MWMSD (Minimization of the Weighted Means of the Shortest Distances)-criterion is a weighted version of the MMSD-criterion. A location-dependent weighting function is introduced in the fitness function: r r
r
r
φWMSD ( S ) = ∫ w(x ) x − Vs ( x ) dx
(3)
A
r r r where: x denotes a two-dimensional coordinate vector; w( x ) weighting function; V( x ) r the coordinate vector of the sampling point nearest to x . The symbol ⎪⎪ is used as distance
vector. This function is estimated by:
ne
r r r w( xej ) xej − Vs ( xej )
j =1
ne
φˆWMSD ( S ) = ∑
(4)
r where xe denotes the generic node of a finely meshed grid overposed on the area of
interest. Employing a weighting function offers flexibility in the use of auxiliary high-resolution r data: when EMI data maps of the area are available, w( x ) can be defined as the gradient of ECa, so placing more observations in the area of expected maximum variation. In the absence of a priori information about soil attributes, we assumed that the spatial dependence structure of EMI observations was similar to the spatial dependence structure of those soil properties influencing ECa within the field. We then deemed that a multi-stage sampling design may
lead to more accurate site characterization: a preliminary sampling scheme was obtained by allocating 50% of the total number of samples according to the MMSD-criterion; in this way an even distribution of samples was guaranteed. In the second step the remaining 50% of samples were allocated according to the MWMSD-criterion, so resulting in a coverage of the area that reflected the variation degree of the different priority areas. The sampling procedure used in this work is implemented in the free software SANOS (Van Groenigen et al., 1999).
2.2 Study Sites Description
Three types of skeletal soils were selected in Lombardy Region (northern Italy) according to the nature of the parent materials and dimensions and distribution of the rock fragments along the profile. The first one was a Typic Udorthent sandy skeletal, sub-alkaline soil in a vineyard in Monzambano (south-eastern Lombardy). Rock fragments were coarse gravel (2075 mm), 35-70% in the first 40-cm layer and higher than 70% in the deepest layers. The second soil was an Inceptic Hapludalf, loamy skeletal, located in Ghisalba (central Lombardy). The soil reaction was neutral in the first 50 cm and sub-alkaline in the deepest layers. The rock fragments were a mixture of gravel and cobbles (76-250 mm), more frequent (>35%) in the deepest layers. The last soil was western Lombardy (Boffalora) and was classified as Aquic Udorthent sandy skeletal. The reaction was sub-acid-neutral; the skeletal, in prevalence fine-medium gravel (2-20 mm), was evenly distributed along the profile. The fields in Ghisalba and Boffalora were cultivated with continuous maize.
2.3 ECa Survey
ECa surveys were conducted using handheld mobile EMI equipment consisting of a singledipole EM38 unit (Geonics, Ontario, Canada), coupled to a DGPS receiver with meter precision and a field computer. The EM38 was used in both horizontal (EMh) and vertical (EMv) dipole orientations collecting data at a 3-s time interval. The horizontal coil configuration concentrates the reading nearer to the soil surface and penetrates to a depth of roughly 1 m, whereas the reading in the vertical configuration penetrates to a depth of 1.5 m. Because of the strong depth-weighted nonlinearity of EM38 measurement (McNeill, 1980), we assumed that the instrument measures to a depth of roughly 0.70-1.0 m in the horizontal configuration and 1.2-1.5 m in the vertical configuration (Corwin and Lesch, 2005). ECa surveys were conducted in July 2006 in Monzalbano and Ghisalba and in October 2006, after
the maize harvest, in Boffalora. The volumetric water content of the soils at the times of the surveys was approximately 10% (Monzalbano) 14% (Ghisalba) and 5.8% (Boffalora).
2.4 The Geostatistical Procedures
Ordinary kriging was applied to interpolate EMI data and reduce spike values due to measurement errors. The variogram study allowed also retrieving important information on the spatial structure of the soil parameter. Before applying univariate analysis, the EM38 data which did not show gaussian distribution were transformed and standardized into Gaussian values Y(x) using an expansion into Hermite polynomials Hk (Wackernagel, 2003). After variogram modelling, ordinary kriging was applied and them the estimates were then backtransformed to the raw variables. In this paper only the horizontal (EMh) results are discussed and used for calculating the weighting function because the horizontal measurements were found more correlated to the properties of the surface layer (Castrignanò et al., 2006), which had a direct significance in applying agricultural site-specific techniques.
3. Results and Discussion
Horizontal ECa variables corresponding to Boffalora and Ghisalba data were submitted to anamorphosis transformation, because they showed marked shifts from the Gaussian distribution. The variogram of the Gaussian variable corresponding to Boffalora data was fitted by a directional nested model including: a nugget effect, a directional Bessel-k model along N50 direction with a range of 49 m (short-range component) and a directional Bessel-k model along N140 direction with a range of 80 m (long-range component). The variogram of the Gaussian variable corresponding to Ghisalba data was fitted by a directional nested model including: a nugget effect, a directional cubic model along E-W direction with a range of 27 m (short-range component) and a directional cubic model along N-S direction with a range of 45 m (long-range component). The Monzalbano data showed a quite normal distribution, therefore the raw data were used in the variography analysis. An isotropic model was fitted to the experimental variogram including three spatial structures: a nugget effect; a spherical model with a range of 5.39 m and a spherical model with a range of 23.97 m. Compared to the previous two datasets, the Monzalbano dataset is characterized by variability occurring within very short distances.
EMh
Figure 1. Map of electrical conductivity in horizontal mode (EMh) in Boffalora
The kriged maps show clear differences between the three fields: the Boffalora field (Fig.1) can be split into two halves along NW-SE direction: the right-hand one with much lower values than the left. There is therefore a transition area characterized by greater variability in a central strip of the field. The Ghisalba field (Fig. 2) shows less variability of the horizontal ECa with lower values than the Boffalora ones.
EMh
Figure 2. Map of electrical conductivity in horizontal mode (EMh) in Ghisalba
However, well defined spatial structures are still evident: a large central area with low values of the horizontal ECa, where the field borders are generally characterized by higher values. The Monzalbano map (Fig. 3) shows a quite different spatial pattern: most variability is erratic without well defined spatial dependence structures. EMh
Figure 3. Map of electrical conductivity in horizontal mode (EMh) in Monzalbano
On the contrary, the horizontal ECa values are the highest compared to the two previous maps. The high readings in some areas of the three fields are quite probably correlated with finer soil texture as previously verified in other experimental sites (Castrignanò et al., 2006). However, this supposition has to be verified in future sampling. It is worth pointing out that the absolute values of conductivity may not necessarily be diagnostic but only the variations (gradient) in conductivity can be used to identify anomalies (Benson et al., 1988).
Figure 4. Evenly distributed (MMSD-criterion) sampling points in Monzalbano
We allocated 40 sampled points in each studied area: the number was defined by financial and labour constraints typical of skeletal soils. For Monzalbano (Fig. 4) all 40 points, whereas for Boffalora (Fig. 5) and Ghisalba (Fig. 6) only the first 20 points were allocated starting
from a random spatial distribution and using the MMSD criterion to minimize the mean distance between the generic point and the nearest observation, in order to obtain a uniform distribution of the points in the studied area and prevent biased estimates. To optimize the sampling scheme and obtain the same precision level in estimation all over the field, we allocated the other 20 points, treating the 20 previous samples as fixed, by assigning a weight to each node of the grid. In the light of the above considerations, we used the gradient of the horizontal ECa estimates as the associated weights to increase the number of sampling points on the sub-regions with expected higher variability in soil properties. To be sure the absolute minimum in annealing simulation had been reached, we repeated the procedure 3 times with a different initial seed, verifying that the point distributions on the maps actually remained the same. It is quite clear that the weights in fig 5 and 6 cause the number of samples to increase in the sub-regions with higher priority (greater gradient).
Figure 5. ECa gradient map and sampling points in Boffalora (red dots = MMSD-criterion and yellow dots = MWMSD-criterion).
As regards the Monzalbano data we preferred to define the sampling scheme in one step by applying only the MMSD criterion, because of the highly erratic spatial variation. Of course the validity of this procedure in sampling planning is based on the assumed positive relationship of horizontal ECa with most soil parameters, in particular particle size variables.
Figure 6. ECa gradient map and sampling points in Ghisalba (red dots = MMSD-criterion and yellow dots = MWMSD-criterion).
4. Conclusions
In this paper, we have described the use of SSA in optimizing spatial sampling in soils with a high proportion of gravel and stones. The proposed MWMSD -criterion can use any spatial weight function to focus sampling in those areas with by high variability and reduce sampling in the areas with expected low heterogeneity. One of the crucial issues in the proposed approach is the definition of the weighting function. We have preferred to avoid semi-quantitative priority values based on expert-knowledge and therefore, to a certain extent, arbitrary. However, we think that more research should be dedicated to the best setting of the weighting factors because the possibility of attaching priorities to a certain area may also prove to be a valuable tool also in decision making.
Acknowledgement
Research supported by ERSAF, Project GAZOSA.
5. References
Benson R.R., Glaccum A, and Noel M.R. (1988) Geophysical techniques for sensing buried wastes and waste migration. National Water Well Association, Dublin, OH, Buchter, B., Hinz, C. and Flühler, H., 1994. Sample size determination of coarse fragment content in a stony soil. Geoderma, 63: 265-275
Castrignanò, A., Morari, F., and Morelli, G., 2006. Assessment of spatial relationship between some soil properties and electromagnetic induction scans. Agricultural Engineering For A Better World. XVI CIGR World Congress, Bonn, Germany. Corwin,, D.L., and Lesch, S.M., 2005. Apparent soil electrical conductivity measurements in agriculture. Computers and Electronics in Agriculture, 46: 11-43. McBratney A.B. and Webster, R., 1983. Optimal interpolation and isarhythmic mapping of soil properties V. Coregionalisation and multiple sampling strategy. Journal of Soil Science, 34: 137-162. McNeill, J.D., 1980. Survey Interpretation Techniques: EM38. Tech. Note TN-6, Geonics Pry. Ltd, Ontario, Canada. Van Groenigen, J.W., Siderius, W. and Stein, A., 1999. Constrained optimisation of soil sampling for minimisation of the kriging variance. Geoderma, 87: 239-259. Wackernagel, H., 2003. Multivariate Geostatistics. An Introduction with Applications. Springer Verlag, Berlin.
