V Romero-García

J V Sánchez-Pérez

L M Garcia-Raffi

Analytical model to predict the effect of a finite impedance surface on the propagation properties of 2D Sonic Crystals

Keywords: numbers: 43.20.-f, 43.50.-Gf, 63.20.D- Sonic Crystals, Finite Impedance, Multiple Scattering, method of images

The use of Sonic Crystals as environmental noise barriers has certain advantages from both the acoustical and the constructive points of view with regard to conventional ones. However the interaction between the Sonic Crystals and the ground has not been studied yet. In this work we are reporting a semi-analytical model, based on the multiple scattering theory and on the method of images, to study this interaction considering the ground as a finite impedance surface. The results obtained here show that this model could be used to design more effective noise barriers based on Sonic Crystals because the excess attenuation of the ground could be modelled in order to improve the attenuation properties of the array of scatterers. The results are compared with experimental data and numerical predictions thus finding a good agreement between them.

Introduction

Periodic arrangements of scatterers embedded in a medium with different physical properties show ranges of frequencies, known as band gaps (BG), where the transmission of waves is forbidden. If these periodic systems are formed by a combination of solids and fluids then they are usually called Sonic Crystals (SC) [START_REF] Martínez-Sala | Sound Attenuation by sculpture[END_REF]. In the last years, an increasing interest has appeared in the potential exploitation of SC as environmental noise barriers [START_REF] Sánchez-Pérez | Acoustic barriers based on periodic arrays of scatterers[END_REF][START_REF] Sánchez-Dehesa | Noise control by sonic crystal barriers made of recycled materials[END_REF][START_REF] Krynkin | Predictions and measurements of sound transmission through a periodic array of elastic shells in air[END_REF].

Some examples of the advantages of using SC instead of conventional screens are the reduction in the size of the foundation or the possibility of designing specific screens for predetermined conditions. However, the acoustical properties of SC depend on several factors showing some particularities in their attenuation properties. For example, the size and position of the band gaps depend on several factors such as the direction of incidence of the wave on the SC and the type of arrangement of the scatterers [5]. As a consequence, the development of the screens based on SC is not a trivial process.

In order to avoid these handicaps several works have been intensively developed in last years. The use of both materials with acoustical properties added or more efficiently distribution of scatterers are two examples. The use of resonators [START_REF] Liu | Locally Resonant Sonic Materials[END_REF] or absorbent materials [START_REF] Tournat | Multiple scattering of acoustic waves and porous absorbing media[END_REF][START_REF] Umnova | Effects of porous covering on sound attenuation by periodic arrays of cylinders[END_REF] in the first case, and the use of Quasi Ordered [START_REF] Romero-García | Band gap creation using quasiordered structures based on sonic crystals[END_REF][START_REF] Romero-García | Hole distribution in phononic crystals: Design and optimization[END_REF] or Quasi Fractal [START_REF] Castiñeira-Ibáñez | Overlapping of acoustic bandgaps using fractal geometries[END_REF] structures in the second case have been studied.

The existence of the ground is one of the factors to be considered in the use of SC as noise barriers. Up to now one of the most appropriate analytical approach to predict the transmission properties of finite SC has been based on the well known Multiple Scattering Theory (MST) [START_REF] Zaviska | The deflection of electro magnetic waves on parallel, infinite long orbital cylinder[END_REF][START_REF] Linton | The interaction of waves with arrays of vertical circular cylinders[END_REF][START_REF] Martin | Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles[END_REF][START_REF] Yu | Theoretical analysis of acoustic stop bands in two-dimensional periodic scattering arrays[END_REF], which is a self-consistent method to calculate the acoustic pressure including all orders of scattering considering the superpositions of the solution for a simple scatterer. MST has been used to predict the acoustical performance of SC in the absence of a ground plane [START_REF] Yu | Theoretical analysis of acoustic stop bands in two-dimensional periodic scattering arrays[END_REF]. On the other hand, some works use MST to study the possibility of modulating the scattering using a ground made of random [START_REF] Boulanger | Reflection of sound from random distributions of semi-cylinders on a hard plane: models and data[END_REF] or periodic inclusions [START_REF] Sainidou | Guided and quasiguided elastic waves in phononic crystal slabs[END_REF]. However the interaction between the Sonic Crystals and the ground has not been studied yet, and in this case the MST is an unrealistic approximation as the presence of the ground should be considered.

In this work, we present a semi-analytical method based on both the MST and the method of images [START_REF] Boulanger | Reflection of sound from random distributions of semi-cylinders on a hard plane: models and data[END_REF][START_REF] Jont | Image method for efficiently simulating small-room acoustics[END_REF] to study the effect of ground planes on the propagation properties of a SC made of rigid scatterers embedded in air. Two types of grounds, acoustically-rigid and with finite impedance, have been considered. In the method of images the system formed by the source, the array of cylinders and the ground is equivalent to a doubled system source-array of cylinders completely symmetric with respect to the axis defined by the ground.

The self-consistent method presented here is completely general, i.e. independent of the impedance model chosen to characterize the ground, so that different ways to determine the ground impedance can be combined with our semi-analytical model. Here we have used a two-parameter impedance method [START_REF] Taherzadeh | Deduction of ground impedance from measurements of excess attenuation spectra[END_REF] due to both the simplicity of the model and the good agreement of the results with the experimental data obtained, although this method is constrained for the low grazing angles.

The paper is organized as follows: in Section 2, we explain the preliminary conditions of our work, including a brief explanation of both the method of images and the impedance model chosen to characterize the ground. In Section 3, we develop the semi-analytical method to model the SC-ground interaction, comparing the results of canonical situations with those obtained with numerical predictions (Finite Elements Method). In Section 4, we present the results of the SC-ground interaction as well as the comparison with the experimental data. Finally, in Section 5, the concluding remarks of our work are shown.

Preliminary conditions: Method of images and ground impedance

The most interesting situation in the SC-ground interaction is likely to involve periodic vertical finite cylinder arrays, but this would require the solution of a 3D problem considering the interaction between finite vertical cylinders and the finite impedance ground. Due to the complexity of the geometry of this general problem, the analytical study could be very complicated and the numerical simulations could require long computational time. Here we have considered the more tractable 2D problem involving a periodic array of cylinders with their axes parallel to the ground (horizontal cylinders). This geometry has recently been used to observe band gaps for water waves propagating over an infinite periodic array of submerged horizontal circular cylinders in deep water [START_REF] Linton | Water waves over arrays of horizontal cylinders: band gaps and Bragg resonance[END_REF].

Defining the problem. Method of images

Consider a line source placed at point O and an array of M circular scatterers placed in the positive half-space which is air, characterized by the sound velocity, c = 344 m/s, and density ρ = 1.23 kg/m 3 . The position of each scatterer C m , m = 1, . . . , M is given by the vector R m . Figure 1A shows the scheme of the problem in the particular case of scatterers arranged in a square lattice which is defined by the lattice constant a. The nearest base of the array of scatterers is placed at a distance H x from the ground, while the nearest vertical base of the array from the source is placed at a distance H y (see Figure 1A)

The geometry used to perform the method of images in our approach is shown in Figure 1B. In this approach one should consider the image of the source as well as of the scatterers. Note that all the waves reflected on the ground can be described as waves coming either from the image source or from the image scatterers, then the images are also interacting with the real space. The image source is placed at point O (from now on, the image vectors will be characterized by a prime), and the image of scatterers C m are placed on the negative half-space. All the vectors measured from the image source are characterized by a prime ( ). In the scattering problem, the image of the source and the scatterers should be modulated by the presence of the ground. Next, an amplitude and a phase should be added to correct the contribution of images to the scattered field. Then, as we will see in the next Section, the complex reflection coefficient should be considered to perform this correction.

Ground effect

In some cases the surface of the ground can be considered perfectly rigid (or totally reflective). However, some cases exist, as for example concrete or soil surfaces with or without vegetation, in which there is absorption of energy of the incident acoustic waves. An accurate prediction of the influence of the ground in the scattering produced in the structure requires the knowledge of the absorptive and reflective properties (the acoustic impedance) of the surface. Motivated by previous works [START_REF] Jont | Image method for efficiently simulating small-room acoustics[END_REF], we characterize in this work the effect of the ground on the scattering problem using the reflection coefficient R( r O , r R ; ν). In general, R( r O , r R ; ν) is a function that mainly depends on the impedance contrast between the two half-spaces separated by the ground surface, on the frequency (ν) and on the positions of both the source ( r O ) and the receiver ( r C ) by means of the angle of incidence on the ground.

The ground surface itself also provides a significant path for transmission of acoustic energy, particularly at low grazing angles and low frequencies. Incident acoustic energy is transformed into vibrational energy and is transmitted along the surface layer. This vibration disturbance can propagate through long distances, before it is dissipated or re-radiated as sound. At these long distances, the transmission of low frequency sound can be dominated by this surface wave mechanism. In this work, we are interested in the interaction between the SC and the ground effect. Therefore, we have studied regions near the source and in the regime of the dispersion frequencies of the array (high frequencies). Thus, this transmission mechanism is neglected in this work.

When airborne sound impinges on the ground, part of the wave is transmitted while another is refracted at right angles onto the surface. For our purposes, we have focused our attention on the reflected waves leaving the surface at the angle of incidence, with its amplitude and phase modified by the impedance of the surface. This reflected wave propagates towards the receiver in addition to the direct wave from the source and, depending on their relative phases and amplitudes, they may constructively add or destructively interfere [START_REF] Piercy | Review of noise propagation in the atmosphere[END_REF][START_REF] Attenborough | Ground parameter information for propagation modeling[END_REF][START_REF] Attenborough | Propagation from a point source over a rough finite impedance boundary[END_REF]. The effect of the ground on the propagating wave is usually called excess attenuation and it can be explained in terms of the existence of two sources: the real one and the image source that models the reflected wave. In this case, the governing equation for pressure p at the receiver, assuming a uniform medium and a line source, in the positive half-space is

p = H 0 (kr) + R( r O , r R ; ν)H 0 (kr ) = H 0 (kr) + R p ( r O , r R , ν)H 0 (kr ) + (1 -R p ( r O , r R , ν))F H 0 (kr ) (1)
where R p is the plane wave coefficient, H 0 is the Hankel function of 0-th order and first kind; parameter F is the boundary-loss factor which is a mathematical function of a variable w called the numerical distance. These functions are [START_REF] Taherzadeh | Deduction of ground impedance from measurements of excess attenuation spectra[END_REF][START_REF] Piercy | Review of noise propagation in the atmosphere[END_REF]

R p ( r O , r R , ν) = cos θ -Z air Z ground cos θ + Z air Z ground
(2)

F = 1 + ı √ πwe -w 2 erfc(-ıw), (3)
where

w = 1 2 ıkr 2 cos θ + Z air Z ground , (4)
Z air and Z ground are the air and ground impedances respectively, r 2 is the distance between the reflection point and the receiver and θ is the reflection angle measured from the normal of the surface. erfc is the complex complementary error function. Usually, fraction β = Z air /Z ground is called the admittance of the homogeneous impedance plane.

The reflected angle can be obtained as

θ = arctan x -x O y + y O (5)
where r O = (x O , y O ) is the position of the source and r R = (x, y) is the position of the receiver point with respect to the origin of coordinates (see Figure 1). The knowledge of the expression of the impedance is necessary in order to account the reflection properties of the surface. In this work we have characterized these type of surfaces (open cell foam layer) using a two parameter impedance model [START_REF] Taherzadeh | Deduction of ground impedance from measurements of excess attenuation spectra[END_REF] with flow resistivity σ e = 4 kPa s/m 2 and porosity at the surface α e = 105 m -1 , being the impedance of the ground

Z ground = ρc 0 0.434 σ e ν (1 + ı) + 9.75ı α e ν , (6)
where ρ and c 0 are the density and the sound velocity of air respectively.

Multiple scattering theory for SC over a finite impedance ground (MSTFIG)

The solution of the appropriate scattering problem satisfies the Helmholtz equation in the half-space that is written in polar coordinates (r, θ) as

∆p( r) + k 2 p( r) = 0, (7)
where

∆ = 1 r ∂ ∂r r ∂ ∂r + 1 r 2
∂ 2 ∂θ 2 , r = r(cos θ, sin θ) is the radius vector, p is acoustic displacement potential, k = ω/c and ω is angular frequency. Equation ( 7) is solved in conjunction with radiation conditions ∂p ∂r

-ikp = o r -1/2 , as r → ∞. (8)
Given that M disjoint cylindrical scatterers located at the positions R m = R 1 , ..., R M all placed above a surface on the symmetry axis (see Figure 1) and a sound source located at point O, one can consider the incident field over the n-th scatterer in the presence of both the other M -1 scatterers and the ground by taking into account the conditions explained in Section 2 about the multiple scattering technique and the method of images.

p n ( r) = p 0 ( r) + M j=1,j =n p j s ( r) + R( R Oj , r R ; ν)p j s ( r) , (9)
where p 0 is the pressure produced by both the real and the image sources and p j s and p j s are the scattered pressure by the j-th cylinder and its image j -th cylinder respectively. Equation ( 9) defines the interaction between the scattering of the array and the ground, therefore the semi-analytical method shown in this work is called multiple scattering theory for finite impedance ground (MSTFIG). The pressure of the sources can be expressed as

p 0 ( r) = H 0 (kr) + R( r O , r R ; ν)H 0 (kr ), (10)
where r = r(cos θ, sin θ) is the vector connecting the real line source and the receiver point and r = r (cos θ , sin θ ) connects the position of the receiver and the image point source. On the other hand, the scattered pressures produced by the cylinders m and m can be represented as

p m s ( r) = ∞ l=-∞ A m l H (1) l (kr m )e ılθm , (11)
p m s ( r) = ∞ l=-∞ A m l R( R Om , r R ; ν)H (1) l (kr m )e ılθ m , (12)
where r m = r m (cos θ m , sin θ m ) is a vector connecting the centre of scatterer C m and the receiver and r m = r m (cos θ m , sin θ m ) connects the receiver with the scatterer image C m . Note that the reflected wave on the cylinder image is produced by the presence of the ground, thus the reflected pressure on the cylinder should be modulated by the reflection coefficient as in the case of the incident wave on the ground.

In order to introduce Equations ( 10), ( 11) and ( 12) in Equation ( 9) all the terms must be expressed in the same origin of coordinates. To do so, the Graf's addition theorems for the Bessel and Hankel functions are necessary [START_REF] Martin | Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles[END_REF][START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF]. Thus, pressures p 0 and p s in the reference system centred at n-th scatterer are,

p 0 ( r) = ∞ l=-∞ (H (1) -l (kR n )e -ılθ Rn + R( r O , r R ; ν)H (1) -l (kR n )e -ılθ R n )J l (kr n )e ılθn , (13)
p j s ( r) = ∞ l=-∞ G jn m + R( R Oj , r R ; ν)G j n m J l (kr n )e ılθn , (14)
G jn m = ∞ s=-∞ A j s H (1) m-s (kR jn )e ı(m-s)θ jn = ∞ s=-∞ A j s G jn ms , (15)
G j n m = ∞ s=-∞ A j s H (1) m-s (kR j n )e ı(m-s)θ j n = ∞ s=-∞ A j s G j n ms , (16)
where vector

R n = R n (cos θ Rn , sin θ Rn ) ( R n = R n (cos θ R n , sin θ R n ))
defines the position of scatterer C n with respect to real (image) line source and vector R jn = R jn (cos θ jn , sin θ jn ) ( R j n = R j n (cos θ j n , sin θ j n )) defines the position of scatterer C j (C j ) with respect to scatterer C n . Finally, due to the geometry of the problem, we can express the total incident wave over the n-th scatterer as

p n ( r) = ∞ s=-∞ B n s J s (kr n )e ısθn . ( 17
)
Introducing Equations ( 13), ( 14) and [START_REF] Sainidou | Guided and quasiguided elastic waves in phononic crystal slabs[END_REF] in Equation ( 9), one can obtain the following system of equations,

B n s = S n s + M j=1 (1 -δ jn )G jn s + R( R Oj , r R ; ν)G j n m , (18)
where

S n s = H (1) -l (kR n )e -ılθ Rn + R( r O , r R ; ν)H (1) -l (kR n )e -ılθ R n . ( 19
)
At this stage B n s , A j s and A j s are unknown coefficients but they can be related using the boundary conditions on the scatterers and the symmetry of the problem. The boundary conditions at the surface of a rigid cylinder relates B j s with A j s and the symmetric condition relates A j s with A j s . In our approach we will consider the general boundary condition, i.e., the continuity of both the pressure and the normal velocity across the interface between the scatterers and the surrounding medium. Note that the asymptotical situation, ρ n → ∞ and c n → ∞, reproduces the case of rigid scatterers (Neumann boundary conditions).

The boundary conditions in the n-th scatterer can be expressed as

p ext | ∂Ωn = p int | ∂Ωn , ( 20
) 1 ρ ∂p ext ∂n | ∂Ωn = 1 ρ n ∂p int ∂n | ∂Ωn , (21)
where ∂Ω n is the boundary of the n-th scatterer, ρ is the density of the surrounding medium and ρ n is the density of the n-th scatterer.

In order to apply the previous boundary conditions, we have considered that the pressure field inside the n-th cylinder can be represented by

P n int = ∞ j=-∞ D n j J j (k 1n r n )e ıjθn , (22)
where k 1n is the wave number inside the n-th cylinder.

Using the boundary conditions and the expressions of both the exterior and interior fields in the n-th scatterer, we can obtain the following relation,

B n j = Γ n j A n j , (23)
where

Γ n j = H j (kb n )J n (kb n /h n ) -g n h n H j (kb n )J j (kb n /h n ) g n h n J j (kb n )J j (kb n /h n ) -J j (kb n )J j (kb n /h n ) . ( 24
)
Here b n is the radius of the n-th cylinder (in this work the radius of the scatterers take the same value for all the cylinders, b n = b), g n = ρ n 1 /ρ is the density ratio, and

h n = k/k n 1 = c n 1 /c
is the sound speed ratio for the i-th cylinder. Note that if the scatterers are acoustically hard, i.e., ρ 1 >> ρ and c 1 >> c, then the coefficients Γ n j coincides with those obtained with the Neumann boundary conditions,

Γ n j = - ∂ r H j (kb n ) ∂ r J j (kb n ) , ( 25
)
where ∂ r is the derivative with respect to polar coordinate r.

The image symmetry can be used to relate A j s with A j s . One has to take into account that r j = r j and that θ j = -θ j , then

p j s ( r) = R( R Oj , r R ; ν) ∞ l=-∞ A j l H (1) l (kr j )e ılθ j = R( R Oj , r R ; ν) ∞ l=-∞ A j l H (1) l (kr j )e -ılθ j = R( R Oj , r R ; ν) ∞ l=-∞ A j -l H (1) -l (kr j )e ılθ j = R( R Oj , r R ; ν) ∞ l=-∞ A j -l (-1) l H (1) l (kr j )e ılθ j , (26)
and

A j l = (-1) l A j -l . (27)
Introducing Equation ( 24) or ( 25) in [START_REF] Attenborough | Propagation from a point source over a rough finite impedance boundary[END_REF] and in [START_REF] Jont | Image method for efficiently simulating small-room acoustics[END_REF], the following infinite system of equations is obtained,

Γ n s A n s = S n s + M j=1 ∞ l=-∞ (1 -δ jn )G jn sl + (-1) l R( R Oj , r R ; ν)G j n s-l A n l . ( 28
)
The methodology previously shown is completely analytical, however the coefficients A n s should be obtained by properly truncating the previous system, i.e., one should use numerical methods to find the solution of the problem. Thus, strictly speaking the methodology is semi-analytical. The subindexes l and s take values from -L to L, therefore the infinite system is truncated to one with 2L + 1 equations. In this work, we have truncated the system using L = 3 which produces an error less than 1% with respect to the values obtained with L = 4.The total acoustic field obtained using the MSTFIG is

P ( r) = H 0 (kr) + R( r O , r R ; ν)H 0 (kr ) + M m=1 ∞ l=-∞ A m l H (1) l (kr m )e ılθm + R( R Oj , r R ; ν)H (1) l (kr m )e -ılθ m . ( 29
)
Note that the methodology shown here is self-consistent and can be applied to any distribution, periodic or random, of scatters. On the other hand, the effect of the finite impedance of the ground only depends on the model to calculate this impedance, i.e., on the calculation of the reflection coefficient, not on the MST procedure. Through this work, we will study the variation of the attenuation properties of a SC with different types of grounds. This variation will be represented using the Insertion Loss (IL) parameter, calculated as

IL = 20 log 10 p 0 P , ( 30
)
where P is calculated using Equation (29).

Acoustically-hard and completely absorbent grounds

For this study we have considered a 5 × 3 square array with lattice constant a =0.3 m and diameter of cylinders D = 0.25 m. The distance from the ground to the centres of the lowest cylinders in the array is half of the lattice constant, H x =0.15 m, so that they are separated from the centre of the cylinders of the image array nearest to the ground by the lattice constant. The distance between the source and the array of scatterers is H y =1 m, and the site of the receiver is (2.5, 0) m. We analyse in this Section, two different types of ground: hard ground (R( r O , r R ; ν) = 1) and completely absorbent ground (R( r O , r R ; ν) = 0). When the real source is placed on the origin of coordinates (O = (0, 0)), then the image source coincides with the real one and depending on the properties of the ground several interesting possibilities can be analysed. With the source on an acoustically hard ground, the predicted IL spectrum of the array should be equivalent to that predicted for an array of double the size (10 × 3) in the free field (without ground), whereas in the presence of an completely absorbent ground, the IL should be the same as the initial array, 5 × 3, in free field. In this Section, we use these canonical situations in order to check our semi-analytical method, specifically using the MSTFIG in the case of a SC on a surface (Section 3) and the MST in the case of a SC in free field. Additional numerical results based on finite element methods (FEM) have also been used to validate the MSTFIG.

The application of FEM to unbounded domains, as for example the case of the scattering problems, involves a domain decomposition by introducing an artificial boundary around the obstacle. This artificial boundary consists of a region in which the discretization can be coupled in various ways to some discrete representation of the analytical solution. In this work, we use a region of perfectly matched layers (PML) [START_REF] Berenguer | A perfectly matched layer for the absorption of electromagnetic waves[END_REF] to numerically approximate the Sommerfeld conditions (see Eq. ( 8)). In this Section the commercial software COMSOL Multiphysics 3.5a is used for the numerical simulations. The PML, originating from electromagnetic computations, is based on simulating an absorbing layer of damping material surrounding the domain of interest, like a thin sponge which absorbs the scattered field radiated on the exterior of this domain. The method was immediately applied to different problems based on the scalar Helmholtz equation as for example acoustics, elasticity, poroelastic media, shallow water waves, etc. Note that MSTFIG gives the pressure field as function of the position, thus one should not solve the entire solution domain to obtain the value at this point. However, FEM should solve the entire solution domain to obtain the value of the field in one point. Thus, FEM takes more computational time than the MSTFIG method to obtain similar convergences.

Figure 2 shows the predicted IL spectra in these two particular cases, acoustically hard and completely absorbent ground. In Figure 2A, the acoustically-rigid ground is analysed. The blue continuous line shows the IL produced by the array placed on an acoustically-hard plane at y = 0 calculated using the MSTFIG. The red open circles show the IL produced by a doubled (i.e. a 10 × 3) lattice in the free field calculated using the MST. The analogous numerical predictions are also shown in Figure 2A with green dots (5 × 3 square array) and black pentagrams (doubled (i.e. a 10 × 3) lattice) respectively. Note the complete agreement between both cases, the red open circles completely coincide with the blue line, and the green dots completely coincide with the black pentagram. The possible small differences could be produced by the precision of the mesh in the numerical discretization. The differences in the predictions at higher frequencies can be reduced by considering more elements on the solution domain. However, the computational time is increased. Figure 2C represents the band structures for a periodic SC with the same periodicity as the finite structure analysed in this Section. These bands have been calculated using the plane wave expansion (PWE) [START_REF] Kushwaha | Theory of acoustic band structure of periodic elastic composites[END_REF]. Grey areas show the pseudogap in ΓX direction (0 • of incidence). The pseudogap covers the range of frequencies (329, 748) Hz. This range is in complete agreement with that obtained using the MSTFIG in the spectra shown in Figure 2A. The differences could be produced by the finite size and the near field effects.

In Figure 2B, the case of the completely absorbent ground (R=0) with the source at (0,0) is analysed. The blue continuous line shows the IL produced by a 5 × 3 square array placed on the absorbing ground using the MSTFIG. The red open circles show the IL produced by a 5 × 3 array in the free field calculated using the MST. The black pentagrams show the FEM predictions. Note the complete agreement between all the cases. Even though the FEM can be used for analysing the situations studied in this paper, its use has been limited to the purpose of validating the MSTFIG methodology. In the rest of this work we will only use the MSTFIG and we will validate our results with experimental data.

Finite impedance surface

When the source is not located on the ground, the semi-analytical solution of the problem has to be obtained using the MSTFIG. In the finite impedance surface case we will use experimental data to check the methodology in Section 4.3. We analyse here the same array as before but varying the impedance of the ground. As we have previously mentioned , the finite impedances considered here are characterized by a two parameter impedance model [START_REF] Taherzadeh | Deduction of ground impedance from measurements of excess attenuation spectra[END_REF] with different values of σ e and α e . We have specifically analysed the cases of a rigid ground, a completely absorbent ground, a covered ground characterized in reference [START_REF] Attenborough | Ground parameter information for propagation modeling[END_REF], the finite impedance ground analysed in reference [START_REF] Taherzadeh | Deduction of ground impedance from measurements of excess attenuation spectra[END_REF] and the sheet of porous material used in this work with σ e = 4000 Pa s/m 2 and α e = 105 m -1 .

Figure 3A shows the IL predictions obtained using the MSTFIG for the five values of the ground impedance. In these cases the source and the receiver are out of the ground, placed at points (0,0.25) m and (2.5, 0.75) m respectively. Figures 3B,3C The orange line with open circles in Figure 3A shows the IL of the array over a rigid ground. At around 1200 Hz, the values of the IL are negative because the excess attenuation shows a clear interaction between the BG of the array and the ground effect. Note that this deep in the attenuation spectrum, due to the excess attenuation, depends on the impedance of the ground, thus it could be modelled depending on the considered impedance. The black horizontally dashed line, the red dots, the green vertically dashed and the blue continuous lines show the reduction or shift of the excess attenuation peaks due to the reduction on the reflection coefficient because of the finite ground impedance. Note that the IL is calculated here considering the pressure level of the system with the ground as a reference. Taking this into account, the negativity of the IL means that, although the sound is attenuated behind the SC, the presence of the SC reduces the effect of the excess attenuation peak. This will be discussed in detail in the following Sections.

Results

In this Section, we analyse the effect of the finite ground impedance on the attenuation properties of a periodic array of scatterers from both semi-analytical and experimental points of view. In both cases we have used an array of 7 × 3 scatterers in square periodicity with lattice constant a = 0.069 m over a ground with a line source placed at point O = (0, 0.235) m. The circular scatterers present a radius r = 0.0275 m. For these simulations, we have considered H y = 0.755 m and H x = 0.0275 m.

Here, we have considered two different grounds: an acoustically-rigid ground (R=1) and a finite impedance ground characterized by a two parameter model (σ e = 4000 Pa s/m 2 and α e = 105 m -1 ).

Symmetry of the acoustic field. Rigid and finite impedance surfaces

In Figure 4, we have analysed the symmetry of the acoustic pressure field with respect to the ground plane depending on the value of the impedance. If the impedance is infinite, i.e., acoustically-hard ground, the acoustic field should be symmetric, which means that the acoustic field in the image space should be symmetrically equivalent to the acoustic field in the real space. However, in the case of a finite impedance ground, the symmetry is broken and the field in the real space and the image space are not identical.

Figure 4A shows the band structure for the considered array calculated using the PWE method. Note that, in this case, the lattice constant is lower than in the previous array, and this produces a shift to higher frequencies of the band gap. Moreover, the filling fraction also affects the width of both the pseudogap and the gap. The range of frequencies between 2478 Hz and 3171 Hz defines the full band gap for this array. Using the MSTFIG, we have predicted the acoustic field at 2000 Hz considering both an acoustically-hard ground and a finite impedance ground. Figures 4B and4C show the pressure maps (Re(P )) at 2000 Hz. One can clearly observe the symmetry of the acoustic pressure field in the hard ground case and how this symmetry is broken in the case of finite impedance.

Band gap-ground plane interaction

For a predetermined source position and the impedance of the ground, the IL defined by Eq. 30 represents the difference between the pressure level measured with and without the array in the presence of the ground. To interpret the IL, one should take into account the effects produced by both the ground and the SC. The ground produces zones with excess attenuation which depends on both the position of the receiver and on the frequency. On the other hand, SC present ranges of attenuated frequencies which depend on both the relative position of the source and the receiver and on the filling fraction. We have calculated the IL spectra using the MSTFIG for the receiver positioned at the interval height y = [0, 0.469] m and for a distance of x = 1.203 m from the source. In all of these cases the IL has been analysed for the two different kinds of grounds defined at the beginning of this Section.

Hard ground

The excess attenuation caused by the ground with the receiver at several heights, due to a rigid ground and without SC, can be observed in Figure 5A. Each horizontal cut, y = y r , of the map in Figure 5 represents the spectrum at point (1.203, y r ) m. The pressure level (P L) in the receiver sites is characterized with the following expression,

P L = 20 log 10 (|H 0 (kr) + H 0 (kr )|) . (31)
The excess attenuation appears in Figure 5A as regions of frequencies with negative values of the P L produced by the destructive interference between the incident wave (from the source) and the reflected wave on the ground. Then, the positive values of the P L mean a positive interference and consequently a reinforcement. Figure 5A shows the dependence of the excess attenuation on the height of the receiver and on the frequency. The higher the height, the lower the frequency of the excess attenuation peak is. Excess attenuation peaks of second order can also be observed for high values of both height and frequencies.

In Figure 5B, the IL map produced by the interaction of the rigid ground and the array of scatterers is shown. The vertical dotted line shows the beginning of the pseudogap in ΓX direction whereas the vertical continuous lines show the range of frequencies of the full band gap of the array. This IL is calculated using Equation (30). One can observe that, for all the considered heights, the maximum value of IL is obtained for the frequencies inside the pseudogap in ΓX direction. However, there are heights at which the IL is negative, which represents a reinforcement with respect to the case in which we used a reflecting surface alone. The low values of the pressure at these points and frequencies for the reference taken to calculate the IL (i.e., the pressure only calculated with the presence of the ground) produce that, although the SC attenuates sound because of the BG, the IL was negative. Then, in comparison with the case of only the ground, the negativity of the IL means that, although at these heights a low pressure level exists, the presence of the SC reduces the excess attenuation of the ground.

Soft ground

The excess attenuation caused by a soft ground (σ e = 4000 Pa s/m 2 and α e = 105 m -1 ) at different heights of the receiver and without the array of scatterers can be observed in Figure 6A. In this case, the pressure level (P L) in the receiver sites is characterized using the following expression,

P L = 20 log 10 (|H 0 (kr) + R( r O , r R ; ν)H 0 (kr )|) , (32)
where R( r O , r R ; ν) is calculated using the approach shown in Section 2.2 and is obtained by combining Equations ( 2) and ( 6) in Equation [START_REF] Martínez-Sala | Sound Attenuation by sculpture[END_REF] In Figure 6A, one can observe the pressure level in the case of a sound source with this finite impedance ground. Once again, the excess attenuation also depends on both the frequency and the height of the receiver. However, this dependence changes due to the properties of the ground. The first excess attenuation peak appears at lower frequencies and lower heights than in the case of the acoustically-rigid ground. Excess attenuation peaks of second order can also be observed for lower values of both heights and frequencies than in the case of the rigid ground.

Figure 6B shows the IL maps produced as a result of the interplay between the soft ground and the array of scatterers. The vertical dotted line shows the beginning of the pseudogap in ΓX direction whereas the continuous lines show the range of frequencies of the full band gap of the array. As in the case of the hard ground, the attenuation peaks are once again present due to the influence of the ground. One can also observe that the array of scatterers changes the attenuation properties at the receiver site for frequencies above and below the first peak of excess attenuation adding this effect to that related with the band gap. A 0.03 m thick wooden board, large enough to avoid the diffraction at the edges, was used as a hard surface. The loudspeaker point source was positioned at 0.755 m from the array at the height of the horizontal mid-plane of the array (0.23 m above the ground). The height of the receiver microphone was 0.117 m, 0.235 m or 0.352 m and it was placed in a vertical plane at 0.257 m from the back of the array. The receiver heights were chosen to be below, at, and above, the horizontal mid-plane of the array. In all cases, a constant distance between the microphone and the periodic array has been considered, in such a way that the distance between the source and the receiver was x = 1.203 m. In this study, we have analysed the properties in ΓX direction (0 • ). The attenuation spectra have been obtained from the difference between the sound level recorded at the same point in two different configurations: i) the ground alone and ii) the ground and the periodic array. 4.3.2. Hard ground Figure 7 compares measured and predicted IL spectra for the considered array placed on a hard ground plane with the receiver in three positions with different heights and using the source location described in Section 4.3.1. An R = 1 value has been used for the calculation due to the rigidity of the ground. The agreement between predictions and measurements is fairly good. The discrepancies observed are probably produced by the fact that the experimental ground is not completely rigid. The semi-analytical spectra corresponding to these three heights are also marked in Figure 5B with dotted horizontal lines. One can observe in Figure 7 that the ground effect in the IL due to the excess attenuation peaks for the three heights analysed in this work corresponds with deeps at 4000 Hz, 2000 Hz and 1400 Hz respectively. These frequencies correspond to horizontal dotted lines in Figure 5B In Figure 5C, one can observe that the experimental measurements do not reproduce the second order peak of the excess attenuation at high frequencies (4000 Hz). Once again this discrepancy could be due to the fact that the impedance of the ground used in the laboratory is not infinite, i.e., the reflection coefficient is not R = 1. However, the agreement between the semi-analytical predictions and the laboratory measurements is good. 4.3.3. Soft ground Figure 8 compares the corresponding measured and predicted insertion loss spectra for the considered array on a finite impedance ground for the three receiver heights previously described. For the simulations with the MSTFIG we have used the reflection coefficient obtained from equation 2 with the two parameter impedance model with the values σ e = 4000 Pa s/m 2 and α e = 105 m -1 . Once again, the agreement between the predictions and measurements is fairly good. The horizontal dotted lines in Figure 6B show the corresponding cuts of the IL maps for the three heights analysed in this Section. The semi-analytical predictions of the IL spectra in Figures 8A and8B predict excess attenuation peaks around 1000 Hz in complete disagreement with the experimental results. However, in the case of Figure 8C, the attenuation peak is well defined both semi-analytically and experimentally. This discrepancy could be explained in terms of the interference between the microphone and the finite impedance ground at low grazing angles. The adverse and the additional attenuation influences of the ground effect in the IL spectra are shifted towards lower frequencies due to the finite impedance of the ground. Note that the dependence of the excess attenuation peak on the height of the receiver produces a lower shift in frequency than in the case of the rigid ground, in complete agreement with the results shown in Figure 6B.

Concluding remarks

The effect of both rigid and finite impedance grounds on the attenuation properties of an array of rigid cylindrical scatterers has been booth analytically and experimentally analysed. The Multiple Scattering Theory for a SC over a Finite Impedance Ground (MSTFIG) has been developed as a semi-analytical methodology to study the effect of the ground with different values of the impedance on the propagation properties of an array of rigid scatterers in air. We have also analysed the interaction between the SC and the ground with the receiver at different heights. In order to both compare and validate the model, the MSTFIG has been checked with both numerical predictions (FEM) and experimental results obtaining good agreement between them. The excess attenuation produced by the interplay between the sound source and the ground can be used to tune the attenuation properties of the array of scatterers. Then, the excess attenuation should be taken into account in the design of arrays of scatterers acting as sonic crystal noise barriers.

Figure 1 .

 1 Figure 1. (A) Square lattice array above a perfectly reflecting plane. (B) Schematic of the real and the image sources and the real and the image scatterers.

Figure 2 .

 2 Figure 2. (Colour online) Predicted IL spectra of a square array of rigid cylinders with diameter 0.25 m, lattice constant 0.3 m above ground plane at y = 0 and 'lowest' cylinder centres at H x = 0.15 m. The source and receiver coordinates are (0,0) m and (2.5,0) m respectively. The nearest part of the array is at H y = 1 m from the source. (A) Acoustically-hard ground R( r O , r R , ν) = 1: Blue continuous line (respectively, green dots) shows the IL produced by a 5 × 3 square array of rigid cylinders calculated using the MSTFIG (respectively, finite element method). The open red circles (respectively, black pentagrams) show the IL produced by a doubled (i.e. a 10 × 3) lattice in the free field calculated using the MST (respectively, finite element method). (B) Completely absorbent ground R( r O , r R , ν) = 0: Blue continuous line (respectively, open red circles and black pentagrams) shows the IL produced by a 5 × 3 square array of rigid cylinders calculated using the MSTFIG (respectively, MST and FEM). (C) Band structure in the considered case calculated using plane wave expansion (PWE). The grey area represents the pseudogap in ΓX direction.

Figure 3 .

 3 Figure 3. (Colour online) IL and reflection coefficient predictions obtained using the MSTFIG for five values of ground impedance. (A) IL of the same array as in Figure 2A with the source and the receiver placed at points (0,0.25) m and (2.5, 0.75) m respectively, and H x = 0.15 m and H y = 1 m. The orange continuous dotted line shows the IL predicted in the case of a rigid ground. The green crossed dash line shows the IL predicted in the case of a completely absorbent ground. The black dashed line shows the IL predictions in the case of a covered ground characterized by σ e = 630 kPa s/m 2 and α e = 188 m -1 . Red dots show the IL predicted for the case of a ground characterized by σ e = 272500 Pa s/m 2 and α e = 158 m -1 . The blue continuous line shows the IL predictions in the case of a sheet of porous material characterized by σ e = 4000 Pa s/m 2 and α e = 105 m -1 . The real, imaginary and absolute values of the reflection coefficients of these three grounds are shown in (B), (C) and (D) respectively. The grey area represents the pseudogap in ΓX direction.

  and 3D show the absolute, real and imaginary values of the reflection coefficients calculated at the receiver site for the three considered ground impedances. The calculations of these impedances are based on a two-parameter model, being σ e = 630 kPa s/m 2 and α e = 188 m -1 , σ e = 272500 Pa s/m 2 and α e = 158 m -1 , σ e = 4000 Pa s/m 2 and α e = 105 m -1 respectively.

Figure 4 .

 4 Figure 4. (Colour online) Pressure maps for an array of 7 × 3 scatterers in square array with lattice constant a = 0.069 m considering a line source placed at point O = (0, 0.235) m. The circular scatterers present a radius r = 0.0275 m. For these simulations we have considered H y = 0.755 m and H x = 0.0275 m. (A) Band structure calculated using the PWE method. (B) and (C) pressure maps (Re(P )) (Pa) at 2000 Hz for an acoustically-hard ground and an finite impedance ground (σ e = 4000 Pa s/m 2 and α e = 105 m -1 ) respectively.

Figure 5 .

 5 Figure 5. (Colour online) (A) Pressure level spectra surface produced by the line source in the presence of a rigid ground. The line source is placed at point O = (0, 0.235) m. We ahve calculated the pressure level using Equation 31 at interval height y = [0, 0.469] m for a distance x = 1.203 m from the source. (B) IL map produced by a 7 × 3 array with square periodicity with a = 0.069 m. The circular scatterers present a radius r = 0.0275. For these simulations we have considered H y = 0.755 m and H x = 0.0275 m. We have calculated the IL spectra using the MSTFIG at interval height y = [0, 0.469] m and for a distance of x = 1.203 m from the source. The vertical dashed line marks the beginning of the pseudogap in ΓX direction (0 • ) and the vertical continuous line marks the ranges of frequencies of the full band gap. The horizontal dotted lines show the semi-analytical and experimental cuts shown in Figure 7.

Figure 6 .

 6 Figure 6. (Colour online) (A) Pressure level spectra surface produced by the line source in the presence of a finite impedance ground (σ e = 4000 P a s/m 2 and α e = 105 m -1 ). The line source is placed at point O = (0, 0.235) m. We have calculated the pressure level using Equation 32 at interval height y = [0, 0.469] m and for a distance of x = 1.203 m from the source. (B) IL map produced by a 7 × 3 array with square periodicity with a = 0.069 m above a finite impedance ground. The circular scatterers present a radius r = 0.0275. For these simulations we have considered H y = 0.755 m and H x = 0.0275 m. We have calculated the IL spectra using the MSTFIG at interval height y = [0, 0.469] m and for a distance of x = 1.203 m from the source. The vertical dashed line marks the beginning of the pseudogap in ΓX direction (0 • ) and the vertical continuous line marks the ranges of frequencies of the full band gap. The horizontal dotted lines show the semi-analytical and experimental cuts shown in Figure 8.

4. 3 .

 3 Comparisons between data and predictions 4.3.1. Laboratory experiment 2 m long PVC cylinders with outer diameters of 55 mm have been used to construct the 7 × 3 square periodic array with lattice constant a = 0.069 m. The sound source was a Bruel & Kjaer point source loudspeaker controlled by a Maximum-Length Sequence System Analyzer (MLSSA) enabling the determination of impulse responses. Measurements of the insertion loss (IL) spectra for arrays of cylinders placed near a ground surface in an anechoic chamber have been obtained.

Figure 7 .

 7 Figure 7. Measured (open circle with dotted line) and predicted (continuous line) insertion loss spectra for a source at coordinates (0,0.235) m and 0.755 m from a 7 × 3 array of rigid cylinders of a 0.055 m diameter over acoustically-hard ground with receiver coordinates (A) (1.203,0.117) m, (B) (1.203,0.235) m and (C) (1.203,0.352) m. Arrays of cylinders placed near the ground surface.

Figure 8 .

 8 Figure 8. Measured (solid line) and predicted (broken line) insertion loss spectra for a source at coordinates (0,0.235) m and 0.755 m from a 7 × 3 array of rigid cylinders of a 0.055 m diameter over a finite impedance ground with receiver coordinates (A) (1.203,0.117) m, (B) (1.203,0.235) m and (C) (1.203,0.352) m. Arrays of cylinders placed near the ground surface.

Acknowledgments

The authors would like to acknowledge The Open University (UK) for the use of their facilities. This work was supported by MEC (Spanish Government) and FEDER funds, under Grant No. MAT2009-09438.


Mahmoud Rezagholi

Erik Svend

Mathiassen

Svend Erik Mathiassen

Erik

Cost-Efficient Design of Occupational Exposure Assessment Strategies-A Review

Keywords: error, information, performance, cost, optimization error, information, performance, cost, optimization

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INTRODUCTION

Acquiring sufficient information on occupational exposures is a persistent challenge in occupational epidemiologic studies. Imprecise and biased estimations of exposure averages may result in vague or even false conclusions regarding the exposure status of a group, and may thus even jeopardize groups or conditions, as in epidemiology and intervention research.

Using biomechanical exposure as an example, guidance on measurement designs has therefore been developed during recent years to improve the information provided by occupational exposure assessments. This guidance includes considerations to the appropriateness of different measurement instruments, acknowledging possible systematic errors in using them [START_REF] Burdorf | Exposure assessment strategies for work-related risk factors for musculoskeletal disorders[END_REF][START_REF] David | Ergonomic methods for assessing exposure to risk factors for work-related musculoskeletal disorders[END_REF][START_REF] Van Der Beek | Assessment of mechanical exposure in ergonomic epidemiology[END_REF][START_REF] Winkel | Assessment of physical work load in epidemiologic studies: concepts, issues and operational considerations[END_REF][START_REF] Burdorf | Design of strategies to assess lumbar posture during work[END_REF][START_REF] Stanton | Handbook of Human Factors and Ergonomics Methods[END_REF][START_REF] Takala | Systematic evaluation of observational methods assessing biomechanical exposures at work[END_REF], and it includes discussions on sampling strategies focussing on how to deal with random sources of error [START_REF] Burdorf | Reducing random measurement error in assessing postural load on the back in epidemiologic surveys[END_REF][START_REF] Mathiassen | Statistical power and measurement allocation in ergonomic intervention studies assessing upper trapezius EMG amplitude. A case study of assembly work[END_REF]Mathiassen et al., 2003;[START_REF] Nordander | Precision of measurements of physical workload during standardised manual handling. Part I: Surface electromyography of m. trapezius, m. infraspinatus and the forearm extensors[END_REF]. The literature also includes an extensive discussion on which method is more accurate and most applicable when measuring different biomechanical exposures in working life (e.g. [START_REF] Van Der Beek | Assessment of mechanical exposure in ergonomic epidemiology[END_REF][START_REF] Winkel | Assessment of physical work load in epidemiologic studies: concepts, issues and operational considerations[END_REF]. The appropriateness and statistical efficiency of the employed measurement design, however, are not the only criteria determining whether or not a specific design will or should be employed. A measurement design should also be cost-efficient as pointed out and addressed decades ago by, e.g. [START_REF] Cochran | Sampling techniques[END_REF]. While statistical efficiency concerns minimizing the variance of an unbiased estimator of the target exposure variable, the economic term of cost efficiency refers to either technical efficiency, i.e. generating the maximum possible "output" for the given "inputs" (measurement efforts), or productive efficiency, i.e. producing one unit of "output" at the lowest possible cost. In the context of exposure measurement, statistical efficiency is an appropriate measure of "output", since it measures the amount of information generated by the measurement design. With this definition of output, cost efficiency is an appropriate comprehensive concept to use, because it emphasizes both economic and statistical performance. Notably, the concept of cost efficiency 3 when measuring exposures differs from the term "cost-effectiveness analysis" as frequently used in health economics. While cost efficiency analysis as discussed in the present paper addresses an economic evaluation of measurement designs that produce information on exposure variables, cost-effectiveness analysis is a methodology for economic evaluation of, in particular, treatment programs. Consequently, in cost efficiency, the term "efficiency" refers to a statistical concept focussing the quantity of information, while the term "effectiveness" in cost-effectiveness analysis refers to the outcomes of a treatment intervention on patients, for instance as measured by improved quality of life.

The measurement method and the number of sampling units both affect the total cost of the measurement as well as the statistical efficiency of the estimated variable. A reduction of errors in exposure assessment, i.e. an improvement of statistical efficiency, can be achieved by using more suitable or advanced technical instruments and/or by increasing the number of sampling units. Such improvements, on the other hand, most often increase the total cost of the measurement. Thus, a measurement design that generates low error (high statistical efficiency) is not necessarily cost-efficient. A cost-efficient measurement design balances the cost and the amount of information produced according to either technical or productive efficiency criteria by, in principle, manipulating the number of sampling units at different stages as well as by considering which measurement methods to employ in the statistical production.

Analysis of the cost efficiency of producing information on exposure variables is an important field of research, yet so far with a limited body of literature. As an example, the literature on occupational biomechanical exposure assessment lacks studies devoted to the issue of how to supply "sufficient" information on exposures at a "low" cost. None of the measurement design studies cited above have considered cost efficiency criteria when assessing occupational exposures; all have focused on appropriateness and/or statistical performance.

An interesting study, however, by [START_REF] Trask | Measuring low back injury risk factors in challenging work environments: An evaluation of cost and feasibility[END_REF] was devoted to classification and estimation of all important cost components associated with different methods of measuring low back injury risk factors. While this study clearly demonstrated that costs differ between exposure assessment technologies, it did not discuss statistical performance.

While our personal interests concern mainly biomechanical exposures in working life, the present review addresses all available studies devoted to the trade-off between "cost" and "efficiency" in assessment of any exposure variable. The purpose of the review is to analyse statistical and economical models suggested and/or applied in the literature for analysing cost efficiency in exposure assessment, so as to set a stage for future research. In order to aid the health economics and medical statistics. The relevance of the studies to the present review was judged using three criteria: 1) the analysis of cost efficiency should address assessment of exposure variables, 2) both "cost" and "efficiency" (or terms with equivalent meanings) must be considered, 3) the analysis of cost efficiency should be based on at least one mathematical model. First, the titles of all 211 papers were inspected. If the title suggested the paper to be relevant according to the inclusion criteria, the abstract was examined. Thus, the abstracts of 183 publications were examined. On the basis of this examination, nine papers were identified as relevant that fulfilled the inclusion criteria. Several other publications were considered relevant to the general topic of cost efficiency analysis in data collection. These papers are addressed when appropriate in the discussion below, but they were not included in the core review. Reviewing the reference lists of the nine included papers did not lead to the identification of additional relevant papers. Also, a citation report was retrieved for all nine papers using Google Scholar, but this did not reveal any additional studies either. Thus, a total of nine papers were accepted for a thorough review.

In examining these nine articles, we focused on three issues: 1) the indicator of statistical efficiency and the operational model applied to assess it, 2) methods used to assess "cost", and 3) the general conclusion of the cost efficiency analysis. Some studies reported empirical data to illustrate their approach, but we considered these quantitative data not to be generalizable, and thus have not commented them in this review. Early in the review process, we found that the identified papers employ concepts and tools that are specifically adapted either to evaluation/comparison of measurement designs or to their optimization. Since these two approaches are fundamentally different, conceptually as well as with respect to the applied methods, we chose to structure the review accordingly, and sort the nine identified papers into either category. Tables 1.a and 1.b summarize the nine studies by reporting their indicators of "efficiency", the purpose in their cost efficiency analysis, and the underlying assumptions in the applied statistical and economical models. Studies comparing measurement methods/strategies

Comparison of alternative sampling strategies

Based on variance components estimated by a random effect model, [START_REF] Lemasters | Balancing cost and precision in exposure assessment studies[END_REF] compared 32 possible alternative sampling strategies with various combinations of periods, weeks, and days that might be adopted in assessments of coal-dust exposure for an individual miner. Each of these alternatives is compared to a "pilot" strategy in terms of cost and precision. Gains or losses in "relative accuracy" are assessed using the expression ˆ( ) 100 ˆ( )

X X choice pilot σ σ ×
, where ˆX σ denotes the standard error of the mean. The assessed "relative accuracy" is evaluated together with the "cost saving", which is set equal to the percentage reduction in the total number of measurements per person. The paper then identifies "the 10 best choices" among the 32 alternatives, for which the trade-off between cost savings and changed precision is most favourable. Thus, among "the 10 best choices", the reader can identify the design that provides the highest precision for a particular budget, or the design that offers an acceptable precision at the lowest cost.

In a similar yet more advanced study, [START_REF] Shukla | Sampling over time: developing a cost effective and precise exposure assessment program[END_REF] also compared alternative sampling strategies in terms of cost and precision of the estimated mean, using the same structure as Lemasters et al. in assessing environmental exposure for periods, weeks, and days. In the paper by Shukla et al., the cost efficiency analysis is based on three concepts: "cost", "precision" and "design efficiency". Like Lemasters, the authors define "cost" as being directly proportional to the total number of measurements, while "precision" is quantified as the inverse of the standard error of the mean exposure estimate resulting from the sampling strategy. The "design efficiency" of a specific strategy is defined as its precision relative to the pilot strategy, divided by its relative cost. The alternative sampling strategies are then ranked according to their design efficiencies.

Unlike Shukla's study, the Lemasters paper confuses the terms "precision" (reproducibility)

and "accuracy" (unbiasedness) when it describes "relative accuracy" by standard errors in alternative designs. The two studies both use statistical models for estimating the precision of the mean that are based on the assumption of an additive random effect, while not discussing possible violations of this assumption. Neither of the studies considers fixed costs associated with the compared designs. Further, they equalize the cost and the number of measurements when estimating the cost of each design. This very simple approach of assessing costs has the advantage that it is easily employed for economic evaluation of different strategies. However,

F o r R e v i e w O n l y
it does not account for the fact that the cost of measurements is only part of the total cost of a strategy, and that it may vary over time. Thus, it seems improbable that the reduction of the total number of measurements per worker is equivalent to the "cost saving". The Lemasters study identifies "the 10 best choices" among several investigated strategies, but the cost efficiency analysis does not introduce a general method for ranking alternative designs. Thus, Lemasters' approach and results are not generally applicable in comparing measurement designs. The study by Shukla et al. solves this by introducing the concept of "design efficiency".

Comparison of two alternative measurement methods

In two similar studies, [START_REF] Armstrong | Study design for exposure assessment in epidemiological studies[END_REF][START_REF] Armstrong | Optimizing power in allocating resources to exposure assessment in an epidemiologic study[END_REF] formulated principles for comparing the cost efficiency of two exposure assessment methods. His approach is based on the concepts asymptotic relative efficiency (ARE), validity coefficient ( XZ ρ ) and reliability ( 2 XZ ρ ). In the context of Armstrong's studies, asymptotic relative efficiency is given an extended definition as the ratio of the total costs of two measurement designs that achieve equal statistical power, i.e. that yield the same precision of an estimated mean exposure. The validity coefficient of an approximate exposure measure Z is its correlation with the "true" exposure X, and the reliability of Z as a measure of X is the square of the validity coefficient. In comparing an "approximate" and a "perfect" exposure measurement, Armstrong suggests the following model:

[ ] 2 ( ) / ( ) Z XZ I X I Z X ARE C C C C ρ = + + [1]
In this equation, c I denotes the basic cost of including a subject, while c X and c Z refer to the cost of getting a perfect and an approximate measurement, respectively, from that subject.

Armstrong states that Z and X are equally efficient when 1 ARE = , and that X is preferable

if 1 ARE < , which occurs when 2 [( ) / ( )] XZ I Z I X C C C C ρ < + + .
For comparing an approximate exposure measure Z 1 with a more accurate one Z 2 , he suggests another model:

1 2 2 2 1 2 1 2 ( / )[( ) / ( )] XZ XZ I Z I Z ARE C C C C ρ ρ = + + [2]
The more accurate measure Z 2 is preferable if

2 2 2 2 1 1 ( / ) [( ) / ( )] XZ I Z I Z XZ C C C C ρ ρ > + + .
Armstrong (1996) concludes, therefore, that "investment in increased precision is worthwhile up to the point at which the proportional increase in the total cost per subject exceeds the proportional gain in 2 XZ ρ , the square of the validity coefficient (reliability)". The terms "power" (the probability of rejecting a null hypothesis that is, in fact, false), "accuracy" and "precision" as indicators of statistical efficiency are interchangeably employed by Armstrong, which is confusing, given the sources of error actually addressed by the study. Armstrong's suggested models for comparing the cost efficiency of alternative measurement methods are based on the concept "asymptotic relative efficiency", which is theoretically advantageous. However, it presents difficulties in practice, since it involves the terms "validity coefficient" and "reliability", which are principally unknown whenever the "true exposure" is inaccessible. Thus, in principle, Armstrong's comparison approach cannot clearly identify the most cost-efficient measurement method when "the true exposure" is not available. Despite the development of advanced instruments during recent years, perfect measurements are not possible in most situations. Realizing this, [START_REF] Armstrong | Optimizing power in allocating resources to exposure assessment in an epidemiologic study[END_REF] relaxes the requirement for a "perfect measurement" to the "most accurate" when exemplifying his approach in an assessment of exposure to nitrogen dioxide. Armstrong's model for calculating total cost for each measurement method includes a fixed cost, which is important when comparing alternative designs. However, as Lemasters et al. and Shukla et al., Armstrong assumes that the cost associated with using either method is linearly related to the number of measurements.

Studies optimizing resource allocation

Optimal allocation of resources between two measurement methods [START_REF] Duan | Combination of direct and indirect approaches for exposure assessment[END_REF] have addressed the approach of combining direct and indirect methods for assessment of pollution exposure. The direct measurement method is superior in precision but expensive, while the indirect exposure estimation is cheaper but associated with a larger error. The combined approach is a way to balance the drawbacks of either method. The ( )

* 2 2 ( ) ( ) 1 ( ) D T T f c I c D ρ ρ = - [3]
The formula states, as could be expected, that if an indirect measurement is a good predictor of the direct measurement, i.e. if 2 T ρ is large, and if the cost of the indirect measurement is low compared to that of the direct measurement, then the optimal sample allocation will call for a small number of expensive direct measurements.

Duan and Mage's study constitutes a step forward as compared to the results by Armstrong in introducing analytical tools to optimize the fraction of direct measurements. The confusion between "precision" and "accuracy" found in the Lemasters and Armstrong studies is also seen here, since the authors measure the "precision" of the mean exposure estimate by its mean square error, which also includes the bias of an estimator. Duan and Mage's statistical modeling of "approximate mean square error" is thus potentially advantageous, provided that the sub-sample selected for direct measurements is representative to the entire population.

Surprisingly, the optimal sample allocation fraction is allowed to exceed one without imposing the additional constraint * ( )

1 D f ≤ .

Optimal allocation of resources between main and validation studies

In two similar studies of cost-efficient designs for assessing exposure and response variables in epidemiologic research, [START_REF] Spiegelman | Cost-efficient study design for binary response data with Gaussian covariate measurement error[END_REF], and [START_REF] Spiegelman | Cost-efficient study designs for relative risk modeling with covariate measurement error[END_REF]

L U L U z V n n V n n α β β π -   -+ -Φ ≥       and 1 / 2 1 2 1 2 ( , ) ( , ) U L U L z V n n V n n α β β π -   - -+ Φ ≥       , F o r R e v i e w O n l y
where D r , X r , and x r are the unit costs for assessing the (binary) response variable, the imperfect exposure measurement, and the superior exposure measurement, respectively. Spiegelman's optimization approach thus attempts to provide a cost-minimized, yet powerful epidemiologic study, where both superior and imperfect measurement methods are used to assess a continuous exposure variable, similar to the dual sampling idea also addressed by [START_REF] Duan | Combination of direct and indirect approaches for exposure assessment[END_REF]. As expected, the optimal resource allocation between the main study and the external validation study depends on many parameters, such as the desired confidence level, unit cost components, distance between two alternative values of relative risk, magnitude of hypothesized values of parameters, and the specified non-linear model of measurement error.

The choice of "discriminatory power" as an indicator of efficiency is highly relevant when assessing relationships between response and exposure variables using approximate and more accurate methods for measuring exposures. However, the constraint inequalities in the Spiegelman papers results in the optimization problem having no closed-form mathematical solution for n 1 and n 2 . This is unfortunate, since closed-from solutions provide elementary demand functions for the number of sampling units, which are discrete and offer the potential for sensitivity analysis. An alternative approach in epidemiologic studies is to assume a fixed or maximized precision of regression coefficients instead of inequalities, which can lead to closed-form solutions [START_REF] Reilly | Optimal sampling strategies for two-stage studies[END_REF]. Like in the studies by Armstrong and by Duan and Mage, Spiegelman's cost model is based on the assumption of the measurement costs for each method being linearly related to the number of measurements; that is, the cost does not vary with replicate measurements or between subjects.

Optimizing a two-stage sampling strategy

For the purpose of sampling diet records in a nutrition study, [START_REF] Stram | Cost-efficient design of a diet validation study[END_REF] σ and 2 ε σ are the variabilities in nutrient consumption within and between subjects, respectively, S is the cost of recruiting a new subject and obtaining the first 1-day record, and R is the cost of making an additional measurement on a subject who has already been measured once. The optimal number of recording days when S R > depends on the ratios of variances and costs as follows:

2 2 1 S m R ε σ σ   = -     [4]
The unusual specification of the cost model in Stram's study leads to an optimal allocation that has a global solution only in the case of S R > . If S R < or S R = , for instance if the cost of recruiting a subject is negligible, the variance has a local minimum when m takes its smallest possible value, i.e. 1 m = . Stram et al. do not state whether any economical assumptions or theories can explain that they separate the cost of the first-day recording from the cost of additional recordings, and instead add it to the cost of recruiting a subject. When optimizing a two-stage sampling strategy, the costs of primary and secondary sampling units usually appears as separate parts of the total sampling cost [START_REF] Cochran | Sampling techniques[END_REF]. In that case, the cost model could take the form Other studies besides the one by Stram et al. have addressed optimization of a resource allocation between two sampling stages, either when assessing a continuous response variable [START_REF] Shoukri | Issues of cost and efficiency in the design of reliability studies[END_REF] or when addressing two different treatment conditions [START_REF] Allison | Power and Money: Designing statistically powerful studies while minimizing financial cost[END_REF]. As in Stram's study, these papers optimize a two-stage sampling strategy by manipulating the number of subjects and the number of replicates per subject. However, rather than addressing the precision of a mean value, they express statistical performance through the size of the intraclass correlation coefficient (Shoukri et al.) or through the correlation between replicate measurements (Allison et al.) The estimated total cost in the studies consists of costs associated with recruitment and observations (Shoukri et al.) or treatments (Allison et al.) [START_REF] Eliasziw | A cost-function approach to the design of reliability studies[END_REF]. However, the results of this study are not directly applicable to sampling strategies for mean exposure assessment.

Optimizing a three-stage sampling strategy

Using a three-stage sampling procedure (county, n 1 , census block, n 2 , household, n 3 ) for assessment of human environmental exposure to metals and volatile organic compounds, [START_REF] Whitmore | Cost/variance optimization for human exposure assessment studies[END_REF] aimed at determining the optimal numbers of sampling units to achieve a given level of precision at a minimum cost. The analytical tools applied in sampling optimization consisted of a linear cost model ρ and 2 ρ . The authors then provide the optimal sample sizes for the second and third stages, 2 n and 3 n , as functions of variance components (or intraclass correlations) and unit costs:

2 1 1 2 1 2 2, 2 2 1 2 1 (1 ) opt C C n C C ρ ρ σ ρ σ - = = 2 2 3 2 2 3, 2 3 2 3 2 (1 ) opt C C n C C σ ρ ρ σ - = = [5]
The authors conclude that greater variability and/or lower costs at a certain sampling stage calls for a larger sample size at that stage. themselves". Obviously, optimization based on intraclass correlations will give the same result as optimization using raw variance components, and the authors do not present empirical evidence to support that optimization will be easier or more correct if using intraclass correlations, to compensate for the increased difficulties in translating results into practical sampling advice.

Addressing continuous response variables, [START_REF] Foster | Improved allocation of costs through analysis of variation in data: planning of laboratory studies[END_REF] also developed a methodology for optimizing a three-stage sampling strategy based on the same cost and variance models as in Whitmore et al. The authors optimized sampling strategies both in terms of minimizing cost at a constant precision, and in terms of maximizing precision under a budget constraint. The cost and precision of the optimal strategies are compared with some non-optimal sampling strategies. The principles for optimization are useful also for exposure studies.

GENERAL DISCUSSION

Strikingly, most of the cost efficiency studies identified in this review were published more than ten years ago. This shows that while the interest in identifying exposure measurement designs worth their price is obvious in both research and practice, little effort has been devoted to developing this issue. However, the few identified and reviewed studies are very valuable as a basis for further research in that they offer the initial necessary theories and methods. They clearly demonstrate the need of developing cost-efficient designs of exposure assessment strategies by analysing their economic and statistical performance in a systematic fashion. This requires data on cost components and sources of error, and also well-behaved economical and statistical models. The review also demonstrated that additional theories and experiences within microeconomics and sampling statistics, including empirical data, will be necessary to further develop cost efficiency analysis.

As noted previously, cost efficiency analyses, regardless of their specific approach and application, strive to achieve a balance between the cost and statistical performance of a measurement design. However, the specific concepts and methods chosen for the cost efficiency analysis depends on whether the aim is to evaluate and compare alternative designs, as pursued by some of the reviewed studies, or to determine an optimal allocation of resources between different measurement efforts as attempted by others. Since the reviewed papers are so few and disparate, it is not justified to derive any more specific guidelines for

F o r R e v i e w O n l y
what might be the optimal exposure measurement design in a particular occupational setting.

The concept of "efficiency" in the reviewed studies was expressed by indicators such as "precision", "accuracy" or "statistical power". Some studies suffered from confusion in the definition and operationalization of these indicators of "efficiency". Such confusion can be observed in the work of [START_REF] Lemasters | Balancing cost and precision in exposure assessment studies[END_REF], [START_REF] Armstrong | Study design for exposure assessment in epidemiological studies[END_REF][START_REF] Armstrong | Optimizing power in allocating resources to exposure assessment in an epidemiologic study[END_REF], and [START_REF] Duan | Combination of direct and indirect approaches for exposure assessment[END_REF], as explained in the specific reviews of these studies. Which indicator of "efficiency"

to select in a particular study depends on the specific purpose and limitations of the study. For optimization of a resource allocation, the most useful indicator of efficiency is "precision", as employed by [START_REF] Whitmore | Cost/variance optimization for human exposure assessment studies[END_REF] and [START_REF] Stram | Cost-efficient design of a diet validation study[END_REF], since a possible bias of a particular exposure assessment method does not influence how to optimally allocate samples.

The concept of "statistical power" can only be employed in an optimization approach with some difficulty, as in the studies by [START_REF] Spiegelman | Cost-efficient study design for binary response data with Gaussian covariate measurement error[END_REF] and [START_REF] Spiegelman | Cost-efficient study designs for relative risk modeling with covariate measurement error[END_REF]. For

evaluating and comparing different measurement methods, on the other hand, it is appropriate to use indicators that include both systematic and random sources of error or, as a minimum, to assume that exposure assessment is non-biased only after assuring that the possible bias is negligible compared to the uncertainty associated with data. Likewise, including fixed costs associated with exposure measurement is important when evaluating or comparing measurement designs. In their comparisons, [START_REF] Armstrong | Study design for exposure assessment in epidemiological studies[END_REF][START_REF] Armstrong | Optimizing power in allocating resources to exposure assessment in an epidemiologic study[END_REF] did consider fixed costs, while [START_REF] Lemasters | Balancing cost and precision in exposure assessment studies[END_REF] and [START_REF] Shukla | Sampling over time: developing a cost effective and precise exposure assessment program[END_REF] did not. In studies addressing optimal resource allocation, on the other hand, fixed costs do not need to be included in the proposed cost model, as all fixed costs will disappear in the differentiation contained in the optimization procedure and thus do not influence optimal allocation.

Modern mixed-model statistics has, in the past decade, gained increasing intention as a tool for identifying exposure determinants with a satisfying predictive ability, mainly for the purpose of exposure control and intervention [START_REF] Burdorf | Identification of determinants of exposure: consequences for measurement and control strategies[END_REF]. However, determinants that are "cheap" to obtain and, at the same time, correlate well with "true" exposures also represent a potentially cost efficient avenue for exposure assessment. Besides the studies addressing cost efficient allocation of subjects to indirect/imperfect and direct/superior exposure measurements [START_REF] Duan | Combination of direct and indirect approaches for exposure assessment[END_REF][START_REF] Spiegelman | Cost-efficient study designs for relative risk modeling with covariate measurement error[END_REF][START_REF] Spiegelman | Cost-efficient study design for binary response data with Gaussian covariate measurement error[END_REF], none of the studies in the present review discuss this possibility of modeling "expensive" exposure variables by "cheap" determinants. Various ideas for exposure modeling have been analyzed

with regards to statistical performance (e.g. [START_REF] Chen | Using exposure prediction rules for exposure assessment: an example on wholebody vibration in taxi drivers[END_REF][START_REF] Mathiassen | Task-based estimation of mechanical job exposure in occupational groups[END_REF][START_REF] Preller | Modeling long-term average exposure in occupational exposure-response analysis[END_REF], while the possible and probable economic benefits of exposure modeling remains an issue for future research. 16 All together, the reviewed studies showed two typical and common characteristics:

Simplification

All of the reviewed studies assume that the total cost of data collection is linearly related to the number of subjects and measurements per subject. In the studies by [START_REF] Lemasters | Balancing cost and precision in exposure assessment studies[END_REF] and [START_REF] Shukla | Sampling over time: developing a cost effective and precise exposure assessment program[END_REF], this linearity is particularly simplified in assuming a straightforward equivalence between the number and the cost of measurements. However, the recruitment costs may well differ between subjects, for instance because they live at different distances from researcher's laboratory and thus cause different travelling cost, or because different efforts need to be invested in convincing them to participate. The cost of measurements for a particular subject can also vary, for instance as a result of subjects getting more familiar with measurement procedures, and thus saving working time. Costs can also vary over time not only due to the effect of economical factors as inflation rate but also due to changes of exposure patterns. An appropriate economic model should therefore allow for cost variations. A linear development of input costs exhibits a constant return to scale for exposure measurements irrespective of the applied measurement technique, because the marginal cost, i.e. the cost of one additional unit of measurement, is equal to the average cost at each level of output. This assumption is a simplified view of statistical production. When adopting a linear cost function, the optimized sampling strategy will be the same irrespective of whether marginal costs of sampling units or their average costs are addressed in the cost model [START_REF] Whitmore | Cost/variance optimization for human exposure assessment studies[END_REF]. However, calculation of the total cost of a study would be incorrect if the statistical production does not exhibit constant return to scale.

Generally, the reviewed studies do not discuss whether the underlying assumptions of the statistical models used for assessing precision of the mean exposure estimates were met; and all studies employ a standard additive random effect model. It should be investigated more thoroughly whether errors at different stages of sampling are, indeed, additive, and which consequences a violation might have for the cost efficiency analysis. Further, variabilities at the different stages of sampling are assumed to be homogeneous in the exposure studies as well in the response and intervention studies, which may also in many cases be a critical assumption.

In studies aiming at sampling optimization, the selection of simple cost and error models may be the result of a strive to obtain easy mathematical solutions to the optimization problem.

With complex cost and error models, the mathematical form for optimizing sample sizes is also more complicated, so that operations may not lead to closed-form optimal solutions without the use of additional assumptions. In many cases, numerical analysis, such as "nonlinear multiparameter constrained optimization subroutine" in the studies by [START_REF] Spiegelman | Cost-efficient study design for binary response data with Gaussian covariate measurement error[END_REF][START_REF] Spiegelman | Cost-efficient study designs for relative risk modeling with covariate measurement error[END_REF], may then be the most viable alternative.

Incompleteness

The reviewed studies have given much less attention to the cost model and economic analyses than to error models and statistical interpretations. It is often not clear which cost curves (unit cost, average cost or marginal cost) or which input costs (capital cost, labor cost, energy cost or material cost) are considered and estimated in the cost models. The social costs of biased and imprecise estimates of exposures, i.e. the opportunity costs of not using the most productive measurement design, have never been modelled, since the outcomes have been limited to statistical performance of the exposure measurement design. Further, systematic methods for estimating the actual costs have not been discussed in the studies, and hence not either the reliability of assessing various cost components, even though mathematical techniques are available for this purpose. These techniques include approaches based on Bayesian statistics [START_REF] Lambert | Estimating the costeffectiveness of an intervention in a clinical trial when partial cost information is available: a Bayesian approach[END_REF] or regression analysis [START_REF] Lin | Regression analysis on incomplete medical cost data[END_REF], by which costs can be estimated reliably in spite of uncertainty and incomplete data. Also, non-parametric estimation methods are available to address the distributional form of measurement cost data [START_REF] Cooper | Use of Bayesian Markov Chain Monte Carlo Methods to Model Cost-of-Illness Data[END_REF].

No studies have given attention to the cost and productivity associated especially with the labour inputs or to further analysis of associations between (minimized) total cost and statistical performance. A fundamental issue to consider when making economic decisions for exposure assessment, which has not been addressed by any of the reviewed studies, is the resulting from one percent change in its cost. Both these type of elasticities could be useful as a basis for decision-making when designing exposure measurement.

Also, methodologies developed for economic evaluation of competing treatment interventions in the field of health economics [START_REF] Tompa | Economic evaluation of interventions for occupational health and safety: developing good practice[END_REF][START_REF] Goossens | Principles of economic evaluation for interventions of chronic musculoskeletal pain[END_REF]) could be useful when providing cost-efficient exposure measurement designs. The basic purpose of an economic evaluation in this case is to ensure that the opportunity cost of implemented interventions is compensated by their (positive) effects on health. In contrast to cost efficiency analysis of exposure assessment, economic evaluation of alternative health care interventions has a long scientific history. Since the generic issue of how to optimize economic resource consumption with respect to a desirable output is shared by cost efficiency analysis and health economics, research in exposure assessment could benefit from consulting the health economics literature; for instance on approaches for modelling cost data, comparing the cost efficiency of alternative measurement designs where the "perfect measurement" does not exist, and/or optimizing resource allocation if the cost of measurement inputs, but not the variabilities, are taken into consideration. Adopting approaches from health economics into exposure science will present challenges in terms of replacing various dimensions of health outcomes by appropriate indicators of statistical performance, while it will, at the same time, offer opportunities of expressing the success of measurement designs in wider terms than mere statistical performance.

Distinct and well defined concepts, well-specified models for costs and error, complete cost analysis, and estimation of elasticities should all be emphasized when attempting to econometrically model the production of information on different exposure variables. Our suggestion for future research devoted to cost efficiency analysis of exposure assessment is to first formulate appropriate indices for measuring the quantity of information produced on the exposure variable according to the specific need of a study, and then to construct a statistical production function and its corresponding cost function, i.e. its dual cost function in economics terms, so as to provide the basis for a comprehensive economic analysis. The measure of the quantity of information produced should account for all sources of error and not assume an "unbiased estimation" or the existence of a "perfect measurement". Methods should also be developed for evaluating the output of statistical production, e.g. statistical performance, in monetary terms, so as to make it directly comparable to the total cost of data collection and processing. The production and cost functions should give all information allocation to the principles and algorithms used for optimization is not known so far. This includes research into the practical need of using well-specified, though complex, models for estimating costs and errors, and the loss of performance when designing exposure assessment strategies that deviate from the optimal choice.

  review even considers some studies devoted to cost-efficient assessments of response variables, and also points to usable methodologies from the literature on health economics. The review considers cost efficiency analysis in the two applications occurring in the literature: evaluation/comparison of alternative measurement designs and optimization of resource allocation in statistical production.METHODSA systematic search in the medical database PubMed and the cross-disciplinary scientific database ScienceDirect was conducted in February 2009, using two search algorithms:1.[cost (precision accuracy power) assessment exposure] cost-efficient validation study) ∩ . These search algorithms identified 99 papers (78+21) in PubMed and 112 (22+90) papers in ScienceDirect, primarily in the areas of epidemiology,

  sample of subject is principally separated into two sub-samples: a dual sample and an indirect-only sample. Both measurement methods are used simultaneously in the dual sample, while only the indirect method is used in the indirect-only sub-sample. The dual sample is used to estimate the relationship between the direct measurements and the indirect estimates.The estimated relationship is then applied to the indirect-only sample to calibrate the indirect estimates, thus predicting the direct measurements that were never carried out. Finally, the the dual sample and the predicted direct measurements in the indirectonly sample are combined into an estimate of mean exposure in the entire target population.Duan and Mage provide an algorithm that can be used to determine the optimal allocation of subjects between the dual and the indirect-only samples, in the sense that it minimizes the measurement error subject to a linear budget constraint. The optimal fraction of direct measurements, * ( ) D f , depends on the costs of collecting the indirect data and the direct data for each individual in the dual sample, c(I) and c(D), and the correlation between the direct measurements and indirect estimates, 2 T ρ , as follows:

  are the variances of the estimated regression coefficient β between exposure and response under the two alternative values L β and U β , respectively. In a simple explanation, the power constraints reflect an epidemiologic study designed to discriminate between two values of relative risk obtained by two different measurement methods.

  denotes the average cost of recruiting a subject and R stands for the average cost of one measurement from a subject. The demand function for m would then be has a global solution also in the case ofS R< and S R = .

  the three stages; C 1 , C 2 , C 3 are the variable costs of adding one more county, census, and household, and C 0 is a fixed cost, which, while not specified in Whitmore's work, could include equipment costs or the salaries of the key executives that run the research. The authors transform the variance formula above to an error model based on intraclass correlations for the first and second sampling stages, 1

  various elasticities associated with statistical production. Elasticities measure the dynamic behaviour of cost and efficiency by a ratio of the percent change in one variable to the associated percent change in another variable. Elasticities include, for instance, cost elasticity of output, c y E , and price elasticity of demand, p D E , measuring the responsiveness of an optimized function to changes in the required output and input cost, respectively. c y E would, for instance, show the percentage change in the (minimized) cost as a result of a one percent change in the required precision of the mean exposure estimate, and p D E would measure the percentage change in demand of a measurement input, e.g. the optimal number of subjects,

  effects of inputs on statistical production in order to facilitate decisions about how to invest an increased budget or implement a saving. Another area for future cost efficiency related research concerns the influence of the specifications of the statistical and economical models applied on the results from comparison or optimization of measurement designs. The results and principles developed by the reviewed studies might have been different if other models had been applied, but the sensitivity of, for instance, an "optimal"

Table 1 .a Basic characteristics of the identified studies on cost-efficient exposure assessment in comparison approach

 1

	Reference	Indicator of statistical	Purpose	Statistical assumption	Economical assumption
		efficiency
	Lemasters et al.	Precision	Comparison of 32	Additive random error,	Total cost directly
	(1996)		alternative sampling	three-stage sampling	proportional to the
			strategies		number of measurements
	Shukla et al.	Precision	Comparison of	Additive random error,	Total cost directly
	(2005)		alternative sampling	three-stage sampling	proportional to the
			strategies		number of measurements
	Armstrong	Accuracy	Comparison of two	Existence of perfect	One linear cost
	(1995, 1996)		measurement methods	measurement, single-	component differing
				stage sampling	between measurement
					methods, same fixed cost
					for both measurement
					methods

Table 1 .b Basic characteristics of the identified studies on cost-efficient exposure assessment in optimization approach

 1

	Reference	Indicator of statistical	Purpose	Statistical assumption	Economical assumption
		efficiency
	Duan and Mage	Accuracy	Optimization of resource	Correlation between	One linear cost
	(1997)		allocation between two	indirect and direct	component differing
			measurement methods	measurements, single-	between direct and
				stage sampling	indirect measurement
					methods
	Spiegelman and Gray

  in two-stage sampling procedures, but does not apply to sampling at three stages, for instance subjects, days and units within days. The study byAllison et al. identifies solutions to a number of additional basic optimization problems with applications in health economics, besides the two-stage allocation issue. Another similar study

	assessing continuous variables is available, addressing how to optimize a general two-stage sampling strategy for conducting
	a reliability study
	F o r
	R
	e v
	i e
	w
	O n l
	y
	13

. The optimization model in Shoukri's study is appropriate for

Annals of Occupational Hygiene

Acknowledgements

Professor Eva Vingård, Occupational and Environmental Medicine, Uppsala University, and Dr. Apostolos Bantekas, Group of Economics, University of Gävle, are gratefully acknowledged for their support in writing this review. Parts of the study were supported by a grant from the Swedish Council for Working Life and Social Research (FAS DNr. 2005-0183).