Massimo Nespolo
email: massimo.nespolo@univ-lorraine.fr

From Bond Strength to Charge Distribution: modern developments of the empirical analysis of crystal structures

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Introduction

A real crystal differs from its idealized model for the presence of static and dynamic defects. Among the formers, we can cite vacancies, interstitials, impurities, precipitates, dislocations and the surface itself.

Among the dynamic defects we have individual and collective atomic vibrations. A classical X-ray diffraction experiment gives a space-and time-average structure which justifies adopting an essentially idealized model of the crystal structure, where the deviations from the ideal model are accounted for in terms of disordered occupation of atomic sites and a finite volume assigned to each atomic position, most often in the shape of an ellipsoid that represents the atomic displacement during the diffraction experiment.

The stability of a crystalline edifice is governed by its thermodynamics. The calculation of the free energy of a crystal is a complex task [START_REF] Jacucci | Free energy calculations for crystals[END_REF][START_REF] Vasileiadis | Calculation of the free energy of crystalline solids[END_REF], not suitable for a routine screening of large samples. A rougher, empirical estimation is in most cases sufficient to rationalize known crystal structures and to predict what type of environment an atom may have in a crystal. This type of considerations constitutes the rationale behind approaches like the crystal field theory [START_REF] Burns | Mineralogical Applications of Crystal Field Theory[END_REF] or the Madelung Part of the Lattice Energy (Hoppe, 1970a;1995). The former is well-known among the readers, whereas the latter has found more limited applications, possibly because of the problems related to the convergence of lattice sums in Madelung's constants [START_REF] Borwein | Convergence of lattice sums and Madelung's constant[END_REF]. Pauling (1929) introduced the well-known empirical rules that today bring his name to estimate the stability of ionic crystals. These rules are based on the description of crystal structures as compact packing of hard spheres bringing a charge that corresponds to the valence of the atoms they represent. The interatomic distance corresponding to a chemical bond is interpreted as the sum of the ionic radii, and the type of coordination, including the coordination number, is governed by the ratio of these radii. The strength of the bond is therefore estimated as the ratio between the charge of the cation centring the coordination polyhedron and its coordination number, i.e. the number of anions occupying the corners of that polyhedron.

Under the very drastic assumption of a uniform coordination polyhedron (i.e. a polyhedron in which all the ________________________________________________________________________________ M. Nespolo. From Bond Strength to Charge Distribution. Page 1/19 cation-anion bond distances have the same length), Pauling's bond strength possesses a significant prediction power, expressed by the second rule, which states that the sum of the bond strengths about a given anion is equal to the formal charge of that anion: q(Aj) = Sisi, si = q(Ci)/Ni (1)

where q is the charge, Aj and Ci are the j-th anion and i-th cation respectively, N i is the coordination number of Ci, si is the bond strength and the summation is performed over the Aj-Ci bonds. Within the conditions used to define Pauling's bond strength, Eq. ( 1) can be used to predict the number of cations bonded to a given anion but also as a very simple rationale behind empirical observations like Loewenstein's rule [START_REF] Loewenstein | The distribution of aluminum in the tetrahedra of silicates and aluminates[END_REF], for which today we dispose of more sophisticated interpretations [START_REF] Larin | The Loewenstein rule: the increase in electron kinetic energy as the reason for instability of Al-O-Al linkage in aluminosilicate zeolites[END_REF].

Whereas the sphere packing model still plays an important role in the description and interpretation of non-molecular structures, at least as the aristotype from which a large number of real structures can be obtained as substitution-distortion hettotypes [START_REF] Umayahara | Derivative structures based on the sphere packing[END_REF], the assumption of ionic bonding is quite restrictive on the type of structures that can be studied. For example, simple structures corresponding to important aristotypes, like cuprite, Cu2O, do not fit Pauling's scheme, because the sum of the ionic radii of Cu + and O 2-(0.96 and 1.40 Å respectively: Pauling radii) is much larger than the actual bond distance (about 1.85 Å), and the ratio, 0.69, suggests a tetragonal antiprism coordination for copper, which instead occurs as linear O-Cu-O coordination. When non-ionic bonds are involved, availability of orbitals, rather than the ratio of cation and anion radii, governs the coordination [START_REF] Brown | What factors determine cation coordination numbers?[END_REF][START_REF] Chiari | Survey on the calcium oxygen coordination[END_REF].

In this context, the terms "cation" and "anion" have to be interpreted as a handy term for "electropositive atom" and "electronegative atoms" respectively.

Even more severe is the assumption of uniform coordination polyhedra. Pauling's bond strength does not include any dependence on the bond length but assigns a unique bond strength to a cation bonded to a given number of anions of the same chemical species ("homoligand polyhedra": [START_REF] Mohri | A new relation between bond valence and bond distance[END_REF]. This assumption becomes less and less realistic the more a coordination polyhedron undergoes deformation. Small polyhedra centred on hard ions 1 like Si-centred tetrahedra in aluminosilicates present a small range of scattering of bond lengths which justifies a description in terms of Pauling's bond strength as a first approximation. In contrast, soft cations like the larger alkali metals usually occur in the cavities of a structure and their interaction with the surrounding anions is spread over a large interval of bond lengths; no net separation appears between a first and a second coordination sphere, so that the coordination number is hardly identified [START_REF] Chiari | On metal-oxygen coordination. A statistical method to determine coordination number[END_REF][START_REF] Balić Žunić | Determination of the centroid or `the best centre' of a coordination polyhedron[END_REF][START_REF] Makovicky | New Measure of Distortion for Coordination Polyhedra[END_REF]. In this type of coordination environment, the bond strength decreases almost asymptotically: not only assigning a unique bond strength gives a completely unrealistic picture; one also needs some criterion to assign the coordination number.

Finally, the possibility of heteroligand polyhedra [START_REF] Mohri | A new relation between bond valence and bond distance[END_REF], i.e. coordination polyhedra in which the atoms at the centre of the polyhedra are bonded to chemically different atoms at the corners, is not even

1 By the terms "hard" and "soft" ions we make reference to the well-known HSAB theory [START_REF] Pearson | Hard and soft acids and bases, HSAB, part 1: Fundamental principles[END_REF] considered in Pauling's approach, which explains the lack of duality in his definition of bond strength. In fact, whereas in ionic solids a cation is often, although not always, bonded to a single chemical species of anion, that anion is usually bonded to different types of cations. Now, heteroligand polyhedra around cations do also occur: a cation can for example be bonded at the same time to oxygen, halogens, sulphur etc. This type of polyhedra, non-uniform by definition, cannot be analysed in terms of Pauling's bond strength.

Different generalisations of Pauling's approach have been introduced, which all have in common the consideration of the dependence of the bond strength from the bond distance. After a very brief critical account of some of them, we concentrate on the Charge Distribution approach, which is the most recent development of Pauling's original ideas.

Bond strength as a function of bond lengths

The dependence of the bond strength from the bond length can be modelled by a function relating the former to the latter and a series of empirical parameters:

si = f(di,p) (2)
where si and di are the bond strength and the bond length of the i-th interatomic distance between two given chemical elements, and p is a vector containing a certain number (usually up to three) of empirical parameters. [START_REF] Baur | Bond length variation and distorted coordination polyhedra in inorganic crystals[END_REF][START_REF] Baur | The prediction of bond length variations in silicon-oxygen bonds[END_REF] showed a linear dependence of the bond length from the bond strength received by each anion (cf. also [START_REF] Ferraris | Generalization of Baur's correlations between bond length and bond strength in inorganic structures[END_REF]. Several functions f have been proposed by various authors, where the bond strength decreases in a more or less exponential way with the increase of the bond length. These functions are often called "R-s curves" [START_REF] Donnay | How to recognize O 2-, OH -, and H2O in crystal structures determined by X-rays[END_REF]Pyatenko, 1973;[START_REF] Brown | Empirical bond-strength-bond-length curves for oxides[END_REF][START_REF] Brown | Empirical parameters for calculating cation-oxygen bond valences[END_REF][START_REF] Allman | Beziehungen zwischen Bindungslängen und Bindungsstärken in Oxidstrukturen[END_REF][START_REF] Trömel | Empirische Beziehungen zu den Bindungslängen in Oxiden. 1. Die Nebengruppenelemente Titan Bis Eisen[END_REF][START_REF] Trömel | Empirische Beziehungen zu den Bindungslängen in Oxiden. 2. Leichtere Hauptgruppenelemente sowie Kobalt, Nickel und Kupfer[END_REF][START_REF] Trömel | Empirische Beziehungen zu den Bindungslängen in Oxiden. 3. Die offenen Koordinationen um Sn, Sb, Te, I und Xe in deren niederen Oxidationsstufen[END_REF][START_REF] Brown | Bond-valence parameters obtained from a systematic analysis of the Inorganic Crystal Structure Database[END_REF][START_REF] Urusov | Semi-empirical groundwork of the bond-valence model[END_REF], where R is the bond length (we use d instead of R for consistency of notation with other approaches). The bond strength computed from Eq. ( 2) is called bond valence, shortened to BV hereafter [START_REF] Donnay | How to recognize O 2-, OH -, and H2O in crystal structures determined by X-rays[END_REF]; the coordination number is obtained as the number of anions after which si is close to zero [START_REF] Brown | Bond valences-a simple structural model for inorganic chemistry[END_REF] but it has been set a priori in the group of structures used for determining the empirical parameter. By replacing Pauling's definition of bond strength with a curve of type

(2), Eq. (1) becomes much more general and the restrictions to uniform or homoligand coordination polyhedra are removed. Furthermore, bonds between the same type of atom become in principle treatable, provided that a suitable and reliable set of parameters p is available for the pairs of atoms involved in the chemical bond, under the experimental conditions in which the structure was solved and refined. These parameters are obtained by a fit on a large set of reliably refined structures and are frequently redefined and re-refined (Bosi, 2014a,b;[START_REF] Gagné | Comprehensive derivation of bond-valence parameters for ion pairs involving oxygen[END_REF] so that the same crystal structure gives different results when analysed with the same type of curve but with a different set of empirical parameters. [START_REF] Brown | Predicting bond lengths in inorganic crystals[END_REF][START_REF] Brown | Bond valences-a simple structural model for inorganic chemistry[END_REF] developed the theory of bond valence as a method of predicting bond lengths in inorganic crystals. In case of very distorted distributions, however, the results are poor; for the case of V 5+ [START_REF] Gopal | The Crystal Chemistry And Bonding In Vanadates Of Divalent Metal Ions And The Crystal Structure Of Whitlockite[END_REF] reported that "V-O bond lengths can be predicted with some accuracy for systems where the range of bond lengths are not too great (~ 10%)". Several extensions have been proposed with different goals, such as: 1) to treat bonds which occur with a wide range of lengths [START_REF] Brown | Recent developments in the bond valence model of inorganic bonding[END_REF]; 2) to introduce the effect of the lone-pairs on the geometry of the coordination polyhedra (Wang & Liebau, 1996a,b); 3) to take into account zero and negative oxidation states [START_REF] Naskar | New Bond-Valence Sum Model[END_REF]; 4) to correlate bond strength and electron density distribution [START_REF] Gibbs | Power law relationships between bond length, bond strength and electron density distributions[END_REF]; 5) to treat compounds with heteroligand polyhedra [START_REF] Mohri | A new relation between bond valence and bond distance[END_REF]. All these extensions are based on a more or less complex power-law relationship between the bond length and the bond valence and do not have any internal criterion to evaluate whether the method itself is applicable or not to a given structure. Therefore, when a significant deviation from the expected bond valences occurs, the investigator should evaluate independently, and before drawing any conclusion from the bond valence analysis, whether the quality of the refinement of the structure model is reliable, and whether the type of structure (type and pattern of bonding) is suitable for such analysis. A 'blind' application without critical inspection may otherwise result into a discussion in terms of bond valences without solid ground.

Extensions in a different direction were proposed in terms of graph-theoretical approaches. [START_REF] Boisen | Resonance bond numbers: A graph-theoretic study of bond length variations in silicate crystals[END_REF] considered all the Lewis structures that can be associated with a given compound, and defined the resonance bond number (RBN) as the weighted average of the bond order over the Lewis structures. This method ignores the periodicity of the crystalline structures and omits any non-nearest-neighbour interaction:

the calculation requires 10 7 or 10 8 Lewis graphs per atom and the resonance bond numbers obtained in this way are close or identical to the bond valences, but the method has the advantage of being applicable to amorphous as well as to crystalline solids [START_REF] Rutherford | Graph theory applied to bond length variations in silicate minerals[END_REF][START_REF] Rutherford | Theoretical Prediction of Bond-Valence Networks. II. Comparison of the Graph-Matrix and Resonance-Bond Approaches[END_REF]. Hoppe (1970b;[START_REF] Hoppe | Effective coordination numbers (ECoN) and mean fictive ionic radii (MEFIR)[END_REF] criticized the common definitions and applications of the concepts on which bond valence theory is based, namely:

1) the coordination number, which can be hardly identified in those cases in which no net separation appears between a first and a second coordination sphere;

2) the ionic radius, which is far from being uniquely defined, because the interatomic distances cannot always be described as sum of ionic radii, the variation depending partly on geometrical constraints on the crystal structure and especially on the overall connectivity of the atoms in the structure [START_REF] Rutherford | Graph theory applied to bond length variations in silicate minerals[END_REF].

To generalize the concepts of coordination number and ionic radius, [START_REF] Hoppe | Effective coordination numbers (ECoN) and mean fictive ionic radii (MEFIR)[END_REF] introduced the definitions of FIR (Fictive Ionic Radii), MEFIR (MEan Fictive Ionic Radii) and ECoN (Effective Coordination Number), and Hoppe et al. (1989) applied ECoN to the calculation of the distribution of charges in crystalline structures. This "charge" actually has a strict correspondence with the BV, as defined by [START_REF] Brown | Bond valences-a simple structural model for inorganic chemistry[END_REF]. Hoppe et al. (1989) showed that the charge distribution method systematically gives better results than BV when applied to oxide compounds. Despite a large and increasing number of publications where this approach has shown its power, it still remains underestimated with respect to bond valence, possibly because of its more demanding computation requirements. In the following sections we present a systematic analysis of the approach originally developed by Hoppe and a critical comparison with the bond valence methods. The step-by-step developments are available in our series of articles on the subject [START_REF] Nespolo | Charge Distribution as a tool to investigate structural details: meaning and application to pyroxenes[END_REF][START_REF] Nespolo | Charge Distribution as a tool to investigate structural details. II. Extension to hydrogen bonds, distorted and hetero-ligand polyhedra[END_REF][START_REF] Eon | Charge distribution as a tool to investigate structural details. III. Extension to description in term of anion-centred polyhedra[END_REF][START_REF] Nespolo | Charge distribution as a tool to investigate structural details. IV. A new route to heteroligand polyhedra[END_REF].

FIR, MEFIR and ECoN

Atoms in a non-molecular crystal structure can be classified in "polyhedron-centring atoms" (PC-atoms)

and "vertex (corner) atoms" (V-atoms). The assignment of cations ("electropositive atoms") or anions ("electronegative atoms") to each of the two categories can be done interchangeably. The majority of nonmolecular structures are described as corner, edge or, more rarely, face sharing of polyhedra centred on cations, but anion-centred polyhedra represent an alternative, and sometimes preferable description in a number of cases [START_REF] Krivovichev | Structural Crystallography of Inorganic Oxysalts[END_REF][START_REF] Krivovichev | Anion-Centered Tetrahedra in Inorganic Compounds[END_REF].

In the following sections, PC-atoms and V-atoms are indicated as PC(ij) and V(rs) respectively, where i and r identify the atomic site (i.e. the chemical element in absence of isomorphic substitution; otherwise, a statistical mix of chemical elements), j and s the crystallographic type. The latter indices, j and s, are necessary when atoms of the same chemical element are not all symmetrically equivalent but two or more occur in the asymmetric unit. The formal charges (which become weighted average charges in case of isomorphic substitutions) are indicated as q(ij) and q(rs) and depend on j or s through the site occupation factor (sof): for sof = 1, all the PC-atoms with the same i, as well as all the V-atoms with the same r, have the same q. Similarly, the multiplicity of the Wyckoff position is indicated as h(ij) and h(rs). A bond between the atoms PC(ij) and V(rs) is indicated as ijrs, the corresponding distance as d(ijrs). The L-th bond length between PC(ij) and V(rs) is then indicated as d(ijrs)L. For computational purposes, the bond lengths are ordered in increasing length with respect to i, j and r, where d(ijrs)1 is the shortest one and s a sort of dummy index at this stage. Pauling's (1929) statement that in ionic compounds the distance between two bonded ions with charges of opposite sign should be nearly identical to the sum of the ionic radii R(i) and R(r) was generalised by [START_REF] Hoppe | Effective coordination numbers (ECoN) and mean fictive ionic radii (MEFIR)[END_REF], with the introduction of the Fictive Ionic Radii FIR(ij→rs)L concept, defined as:

.

In case of uniform coordination polyhedra, all the d(ij→rs)L are identical and the same holds for

FIR(ij→rs)L; if the crystal is ionic, d(ij→rs) = R(i)+R(r) so that FIR(ijrs) should correspond to R(i).
Deviations from the ionic radius may be read as a measure of the departure from the pure ionic character of the bond. For non-uniform polyhedra, instead, the concept of ionic radius is less well defined; accordingly, a different Fictive Ionic Radius is assigned to each bond, which is then used to compute a weighted average, called the MEan Fictive Ionic Radius (MEFIR). Concretely, an ion at the centre of a uniform polyhedron can be described as a sphere whose dimension, FIR, is directly related to the ionic radius, R( i). For non-uniform polyhedra, instead, the ionic radius shows a dependence on the direction, which leads to a polyhedral shape for the atomic basins (Pendás et al., 1998). The computation of MEFIR corresponds to assigning to the ion a spherical basin whose radius is a weighted average of the ionic radius for each direction. MEFIR is calculated through a convergent iterative process:

(4)

where n is the number of iterations and where the starting value, 0 MEFIR(ij→r), is simply FIR(ij→rs)1. The exponential term is responsible for the asymptotic decrease of the contribution of each FIR with the the increase of the bond distance. MEFIR gives a quantitative meaning to the rule of mutual influence [START_REF] Beck | The co-ordination number rule and the rule of hardness, powerful tools to rationalize inorganic structures[END_REF] of different cations on their seeming size and polarity, according to which the size of an ion depends on the chemical relations with its neighbours. A global n MEFIR(ij) for each PC-atom can also be defined,

because n MEFIR(ij→r) does not change significantly as a function of r:

.

(5)

The concepts of FIR and MEFIR have been introduced for ionic bonds but they do apply as well to polar bonds. According to [START_REF] Pauling | The Nature of the Chemical Bond, 3rd edition[END_REF] classification, chemical bonds can be divided into ionic, polar and nonpolar depending on the electronegativity difference of the bonded atoms, this difference being respectively higher than 1.7, between 0.4 and 1.7, and lower than 0.4. For polar bonds, one could simply use covalent radii instead of ionic radii in Eq. ( 3): FIR(ij→rs)1 would then correspond to the covalent radius for bond order 1, or be smaller than it for higher bond orders.

FIR and MEFIR clearly aim at treating atoms as spherical although they are not necessarily spherical. The purpose is to keep the description of a crystal structure as simple as possibl, with nevertheless some consideration of the effect of the environment in which each atom is embedded. In the same line, the classical definition of the coordination number of an atom as "the number of other atoms directly linked to that specified atom" (IUPAC, 1997) becomes less and less satisfactory with the distribution of bond distances over a large interval, from uniform polyhedra to cavities which can hardly be recognized as polyhedra, like in the typical example of alkaline metals showing a very irregular coordination. These considerations led Hoppe (1970) to call the coordination number an "inorganic chameleon" and generalise it to a function of the weighted average of bond distances which he called the Effective Coordination Number, ECoN [START_REF] Hoppe | Effective coordination numbers (ECoN) and mean fictive ionic radii (MEFIR)[END_REF]. ECoN is computed as a normalised sum of bond strengths defined as function of the bond lengths to which a weight is associated that decreases with the increase of the bond length. The approach is therefore somehow similar to that behind Eq. ( 2), with however a fundamental difference: instead of using empirical parameters which depend on the pair PC-V, like in the Bond Valence approach, the function is based on the and by iteration we obtain n d(ij→r

) = n MEFIR(ij→r)•[R(i)+R(r)]/R(i).
The shortest PC-V distance in each polyhedron is used as normalising parameter at the zero-th stage of the iteration, to be replaced by the weighted average until convergence is reached. This exponential term, which acts as a weight applied to each bond length, is precisely the bond weight n w(ij→rs):

. ( 7
)
The sum over the bond weights gives n ECoN(ij→r):

. ( 8)

Because n w(ij→rs) is equivalently defined in terms of FIR/ n MEFIR and of d/ n d (Eq. 7), ECoN too is equivalently defined in terms of MEFIR (as in [START_REF] Hoppe | Effective coordination numbers (ECoN) and mean fictive ionic radii (MEFIR)[END_REF]) and of d(ijr) (as in Hoppe et al., 1989). Eq. ( 6) can therefore be rewritten as:

(6')

MEFIR is the only variable that depends, through FIR, on the ionic radii. BW, ECoN, and the computed 'charges', defined and illustrated in the next section, which are a function of ECoN, do not depend on the ionic radii. For homoligand polyhedra n ECoN(ijr) is actually n ECoN(ij), because r takes only a single value for each coordination polyhedron. Instead, for heteroligand polyhedra, n ECoN(ij) defined simply as the sum of n ECoN(ijr) results from r coordination shells corresponding to each type of anion.

ECoN, as a generalisation of the classical coordination number, is a real number that has to become equal to the integer coordination number for uniform polyhedra. This is the condition imposed to obtain the contraction parameter 6 in the definition of the bond weights. It was in fact obtained by finding the highest value giving an ECoN equal to the number of first neighbours in the structure of simple metals, where a clear separation between a first and a second coordination sphere exists [START_REF] Hoppe | Effective coordination numbers (ECoN) and mean fictive ionic radii (MEFIR)[END_REF]. One exception was later found, for which the contraction parameter had to be modified. In the case of hydrogen bonds, the ratio of two short distances (donor-H and H-acceptor) with a high relative gap makes the weight n w(ij→rs) for the second bond (hydrogen-acceptor bond) negligible so that its presence is not taken into account in the iterative calculation. A revised contraction parameter of 1.6 was then introduced [START_REF] Nespolo | Charge Distribution as a tool to investigate structural details. II. Extension to hydrogen bonds, distorted and hetero-ligand polyhedra[END_REF].

Although Eq. ( 7) can be equally expressed in terms of n d(ij→r) or of n MEFIR(ij→r), the results significantly differ in the case of heteroligand polyhedra. If r takes only one value (homoligand polyhedra), the two expressions give exactly the same result. If instead r takes multiple values, Eq. ( 7) has to be computed separately for each r when using n d(ij→r) so that each heteroligand polyhedron is divided into homoligand subpolyhedra, treated independently, while this is not the case when it is computed in terms of n MEFIR(ij→r). In fact, by using the bond lengths one automatically selects a different normalising parameter (minimal distance) for each type of V-atom: it would make no sense to normalise bond lengths of a V-atom with respect to the shortest bond length of a different V-atom having in general a different size. It would also Table 1 shows the example of a regular octahedron with six PC-V distances of 2.000 Å, undergoing three gradual deformations of bond distances, which however do not affect the arithmetic average, kept constant at 2.000 Å. With increasing distortion, the weighted average distance n d(ij → r) (Eq. 6') decreases with respect to the arithmetic average. This is a normal effect of the exponential decay of the bond weights. When two distances d1 and d2 depart from the value duni corresponding to a uniform polyhedron by the same amount but in the opposite sense (d1 = duni-D; d2 = duni+D), the increase of the bond weight of d1 is larger than the decrease of the bond weight of d2. This leads to a shortening of the weighted average distance and a reduction of ECoN, which is no longer equal to the classical value of the coordination number. Accordingly, the smaller ECoN becomes, the more distorted is the polyhedron. In this example, ECoN decreases slowly for the two first distortions, but for the third distortion is only 4.73, and the distribution of the bond weights

shows that the coordination is closer 4+2 than to 6 (Figure 1). Table 2 shows the example of KNaTiO3, which contains three types of cation (Ti, K, Na; one Table 3 shows the results for a'-Ba 2 TiO 4 . This orthorhombic polymorph of Ba 2 TiO 4 crystallizes in P2 1 nb, which is an unconventional setting of Pna21 (space-group type No. 33). It is a Bieberbach-type of space group [START_REF] Nespolo | Crystallographic shelves: space-group hierarchy explained[END_REF], which does not contain any special Wyckoff position; the multiplicity of the general Wyckoff position is 4. For this compound Z = 12, so that the unit cell contains 24 barium atoms (six sites), 12 titanium atoms (three sites) and 48 oxygen atoms (twelve sites). The six polyhedra centred on barium have eight to ten bond contacts whereas the three titanium atoms are all in tetrahedral coordination.

ECoN from the three titanium atoms is 3.99, showing that the distortion is minimal. On the other hand, for barium ECoN spans a large interval, from 5.54 to 7.69. The minimal value corresponds to Ba1, with eigth Ba-O contacts, of which seven are bonding contacts, whereas the last two contribute for less than 2% to ECoN (0.069 + 0.035 = 0.104, i.e. 1.9 % of 5.54) and can therefore be considered negligible. The discrepancy between ECoN and the number of oxygen atoms bonded to Ba ( 6) is a measure of the degree of distortion of the polyhedron. Ba2 has nine bond contacts, none of which can be considered negligible. The bond weights span however a very large interval, which explains why the nine contacts correspond to ECoN

Charge Distribution Analysis of homoligand polyhedra

ECoN(ij→r) computed through Eq. ( 8) can be distributed among all the bonds around the PC-atom defining a homoligand (sub)polyhedron, obtaining in this way the fractional contribution by each V-atom to ECoN itself.

. (10)

If one sums up DECoN(ij→rs), one gets obviously 1 (cfr. Eq. 8):

Each bond ij→rs connects a PC-atom with formal charge q(ij) and a V-atom with formal charge q(rs). We can read this network of bonds as a graph along which the formal charge q(ij) is transferred ("distributed") from the PC-atom to the V-atoms bonded to it, exactly like Pauling's scheme in Eq. ( 1). Instead of using a single bond strength calculated as q/N, which does not take into account the variation of bond strength with bond lengths, we use DECoN(ij→rs) as a factor to distribute the formal charge of the PC-atom. Because the result is entirely dependent on the geometry of the coordination polyhedron, the distribution scheme is usually read the other way round, saying that the formal charge q(ij) is used as a scaling parameter applied to DECoN, providing the contribution by each PC-V bond to the distributed 'charge':

. (11)

Finally, by summing the Dq(ij→rs) on the PC-atoms around a V-atom, one should obtain the expected 'charge' of the V-atom itself:

. ( 12
)
where q is the input charge (formal oxidation number) and Q is the charge computed as a function of the distribution of ECoN. The ratio of the multiplicities of the respective Wyckoff positions, h(ij)/h(rs), is introduced to avoid counting multiple contributions when a bond connects atoms on Wyckoff positions with different multiplicities. Eq. ( 12) is the extension of Pauling's second rule (Eq. 1) to general coordination polyhedra.

A similar distribution is repeated the other way round and summed up on the V-atoms about a PC-atom.

This time, however, instead of using q(rs) as scaling parameter, in a perfectly symmetrical way, the ratio q(rs)/Q(rs) for the V(rs) bonded to PC(ij) is used: If q(rs)/Q(rs) = 1 for each r and s, i.e. if the distributed charge and formal charge are identical, then the bracket in Eq. ( 13) goes to 1 (it becomes the sum of the fractions of ECoN about each PC-atom) and thus

Q(ij) = q(ij)
. This shows that in a structure correctly solved and perfectly valence-balanced the distribution of ECoN scaled by the formal charges gives back those charges. On the other hand, when a structure has developed some structural strains, its V-atoms may not be perfectly balanced; nevertheless, the distribution of this unbalance, measured by the ratio q(rs)/Q(rs), should give back the expected formal charges on the PC-atoms, otherwise the whole set of coordination polyhedra would be unbalanced and the structure would be unstable. In other words, CHARDI possesses an internal criterion to evaluate the quality of its analysis:

the ratio q(ij)/Q(ij) for the PC-atoms. When this ratio is reasonably close to 1 for all the PC atoms, then the analysis of the connectivity can be approached by the study of the Q(rs), which diverge from q(rs)

proportionally to the structural strains inside the structure. One speaks of over-under-bonding (OUB) effect. Now, because the core of the method is a distribution of ECoN scaled by the formal charges (oxidation numbers), in the trivial case of a structure containing only one type of V-atom, when there is nothing to distribute, Q(rs) is identical to q(rs) and the CHARDI analysis simply does not give any information.

Tables 4 and5 and Figures 2 and3 show the Charge Distribution analysis for KNaTiO3 and a'-Ba2TiO4,

whose ECoN analysis was presented in Tables 2 and3. The same tables provide the Bond Valence Sums (BVS hereafter) as well. The results of the CHARDI analysis are closer to the formal charges (oxidation numbers) than the BVS. The two approaches are fundamentally different: whereas CHARDI, as its name says, distributes the charges and sums back those distributions, BVS computes the charges. With CHARDI it is therefore impossible to get charges that are globally lower (or higher) than the oxidation numbers, as it occurs instead with BVS (i.e. both crystallographic types of oxygen in KNaTiO3 have a BVS lower, as absolute value, than the formal oxidation numbers, whereas with CHARDI the slight excess of one oxygen is compensated by the slight deficit of the other, once the site multiplicity is taken into account). However, because these figures are often used for structure validation, it is evident that the conclusions drawn from BVS are subject to criticism in the spotlight of the better agreement one gets from CHARDI. As a matter of fact, when the method does apply (on the basis of the internal validation criterion q(ij)/Q(ij)), the results are usually better than those obtained with BVS.

Charge Distribution Analysis of heteroligand polyhedra

Eq. ( 5) allows treating all the PC-V distances in a polyhedron at once; this however leads to neglecting the differences between the various V-atoms. In other words, it corresponds to replacing each V-atom with a dimensionless charged point. While this does make sense in the calculation of ECoN(ij), which simply gives a measure of the number of coordinated V-atoms weighted by the respective bond distances, it would prevent any charge distribution analysis. In fact, although every V-atom retains its charge, the relation between the bond distance and the strength of the bond would be lost. There is however one aspect in which the use of FIR and MEFIR instead of the individual and average distances presents a clear advantage.

In a structure built on homoligand polyhedra, every PC(ij) is bonded to V(1s) for some s and the minimal PC-V distance, d(ij→1s)1, obviously corresponds to a chemical bond between the two atoms. Every PC-atom distributes its charge to all the V-atoms bonded to it, in an inverse proportion to the bond length. On the other hand, in a structure containing heteroligand polyhedra not all the PC(ij) are necessarily bonded to every V(rs); this results in two types of problems.

1. d(ij→rs)1 for some r may correspond to a non-bonded contact. If the distances are used directly, then d(ij→rs)1 is scaled to itself in the first step of the calculation of the average distance, d(ij→r), and to avoid considering non-bonded contacts as chemical bonds one needs an external criterion, like the sum of the radii scaled by an expansion factor to allow some tolerance. If FIR and MEFIR are used instead, each polyhedron is treated as homoligand at this stage and the risk of counting non-bonded contacts as chemical bonds is avoided. Concretely, in the analysis of the coordination polyhedra, a first step is performed in terms of FIR and MEFIR, to discriminate between bonding and nonbonding contacts; then, the calculation is repeated by using the distances to treat each coordination sub-polyhedron separately.

2. Each PC-atom distributes a fraction of its charge to each homoligand subpolyhedron. This fraction depends on the relative distances: V-atoms closer to the PC-atom receive a larger fraction than Vatoms farther from it. The problem is how to define this fraction. This is straightforward for structures which are built by homoligand polyhedra in one of the two possible descriptions (cationcentred or anion-centred), while it requires a convergence algorithm when a structure is built by heteroligand polyhedra in both descriptions.

The treatment slightly differs depending on whether the structure under investigation contains only homoligand poyhedra in one description (either cation-centred or anion-centred) or not. Numerous examples have been presented and discussed in Nespolo ( 2016), which we do not repeat here for lack of space.

Structures built by homoligand polyhedra in one description

If a structure contains only one chemical species of cation or of anion, the calculation is straightforward.

The single-type of element (cation or anion) is first assigned to V-atoms (r = 1) and the computation described above is performed. Next, the fraction of Q(ij) coming from each V-atom is calculated by a simple modification of Eq. ( 13):

. ( 14
)
The summation is here over j, i.e. the crystallographic type of PC-atom, instead than r and s. The result is thus the fraction of the formal charge that each V-atom (identified by their crystallographic type s) shares with the i-th chemical species of PC-atoms. The role of PC-atoms and V-atoms is then swapped so that Eq. (14) gives in the next step the fraction that each PC-atom (V-atom in the previous step) shares with each chemical species of V-atom (PC-atom in the previous step):

DQ(1j→r) = -DQ(i→1s). (15)
This corresponds to re-labelling the contribution DQ so that the only chemical species of V atom (r = 1) becomes the only chemical species of PC-atom (i = 1); the index on the crystallographic type is also relabelled from s to j. Finally, the index on the chemical species of PC-atoms (i) is relabelled as index on the chemical species of V-atoms (r). In the case of KNaTiO3 discussed in section 3.1 as cation-centred homoligand cation-centred polyhedra, we had ij = 11: Ti, ij = 21: K, ij = 31: Na, rs = 11: O1 and rs = 12: O2.

In the anion-centred description, the two oxygen-centred polyhedra become heteroligand; ij = 11: O1; ij = 12: O2; rs = 11: Ti; rs = 21: K; rs = 31: Na.

DQ(ij→r) is then used instead of q(ij) in Eq. ( 13), which is replaced by Eq. ( 16):

. ( 16)

Structures built by heteroligand polyhedra in both descriptions

When a structure contains heteroligand polyhedra in both descriptions, we miss the starting point to compute Eq. ( 14). The problem is solved with an iterative procedure in which the starting val ue of DQ(ij→r)

in Eq. ( 15) is assigned as a function of ECoN(ij→r)/ECoN(ij):

(15') Eq. (15') corresponds to Pauling's definition of bond strength, Eq. ( 1) applied to each sub-polyhedron.

From the second step Eq. ( 14) and ( 15) can be used directly and the role of PC and V-atoms is swapped back and forth between cations and anions until the same values of DQ(ij→r) are obtained in each description, meaning that convergence has been obtained.

Comparison of Charge Distribution and Bond Valence approaches

The concept of bond strength has its roots in the XIX century theory of valence and is based on the idea that the total bond valence received by each atom is equal to its atomic valence. Charge Distribution and Bond Valence are both expressions of strengths, but they differ in the way they describe a crystal structure.

The expected outcomes of the two approaches also do not fully overlap, although both methods can be used for structure validation.

Bond Valence assigns a strength (valence) to each bond on the basis of some empirical parameters. These parameters are far from being established once for all and indeed in the literature these parameters are often re-refined or redefined. If a reliable and optimised set of parameters is available, then the Bond Valence approach is potentially able to predict the geometry of polyhedra and molecules in a crystal structure [START_REF] Brown | Bond Valence Theory[END_REF]. However, the empirical parameters depend, in general, on the conditions under which the set of crystal structures used to obtain these parameters have been solved and refined. As a matter of fact, bond lengths do depend on the temperature, pressure, electromagnetic field or other external constraints to which the structure is subjected; empirical parameters that have been optimized for ambient conditions are less adapted to describe structures under conditions in which the bond lengths are significantly shrunk or elongated. Furthermore, bond lengths are also affected by the spin state of the atoms: for example, the Fe-N distance can change as much as 10% following a high-spin to low-spin transition [START_REF] Legrand | On the precision and accuracy of structural analysis of light-induced metastable states[END_REF].

However, the spin state does not normally enter in the definition of the Bond Valence parameters. Attempts have been made to relate the Bond Valence parameters to electron density [START_REF] Adams | Practical Considerations in Determining Bond Valence Parameters[END_REF], although doubts can be reasonably expressed on the ground on which simple empirical parameters can bear information related to quantum mechanics. Finally, Bond Valence lacks an internal criterion to evaluate whether the results of the analysis can be considered correct or not.

Charge Distribution fundamentally differs from Bond Valence in being essentially a geometric analysis of the coordination polyhedra exploiting the observed distances only, rather than comparing them with some 'standard values'. Atoms are replaced by point charges in a Madelung-type scheme and the bond weight is a purely geometric concept which quantitatively tells how strongly the presence of a V-atom at the corner of a polyhedron is felt by the PC-atom at the centre of it. The bond weight is not directly related with the electron density of the bond and the charge is distributed, rather than computed, among each bond. The equivalent of the Bond Valence Sum strictly applies to computed charges of the PC-atoms, not to those of the V-atoms, the two quantities bringing different information: a measure of the OUB effects for latter, the evaluation of the applicability -including structure reliability -for the former.

Despite the Madelung-type approach, CHARDI has proven to be able to treat a wide range of structures with a large chemical heterogeneity, although some types of compounds still remain outside its possibilities.

Apart from these rare exceptions, CHARDI can bring important information not always easily obtained from experimental data:

• a structure validation based only on internal criteria, without the reference to empirical parameters;

• the presence of structural anomalies;

• the site-occupancy factor of incompletely occupied sites;

• the most likely oxidation state of atoms with multiple valences;

• the most likely oxidation state of an atomic site in compounds presenting isomorphic substitutions, which gives a clear indication of the fractional occupancy by each type of atom: this information may be difficult to obtain from conventional X-ray diffraction experiments in case of heterovalent substitutions involving atoms of close atomic number or atoms with different valence states;

• the most likely positions of missing light atoms (like hydrogen), who are hidden by the presence of relatively heavy atoms.

CHARDI does not possess the same predictive power as Bond principle be overcome in the seek for missing atoms by computing a CHARDI map reminding the Fourier difference map routinely used in the refinement of crystal structures. Such a development is not available to date and one may reasonably question whether the computational effort would be justified, considering the simplified, Madelung-type description of the structure.

A computationally economical approach could use Bond Valence to predict the bond lengths between an already located atoms and missing atoms. The likely positions could then be estimated from the expected coordination geometry and the chemical good sense. This estimation can then be verified with CHARDI, whose results are normally in better agreement with the expected charges than those provided by the BVS. A few trial-and-error cycles can provide a reasonable estimation of the most likely atomic position. for the structure of KTiNO3. DQ shows the departure from the ideal value of the compute charges (valences).

The higher departure calculated for BVS might suggest a more important over-under bonding effect, whereas this actually comes from the limits of the method itself, as shown by the significantly lower departure calculated for CHARDI.

  ________________________________________________________________________________ M. Nespolo. From Bond Strength to Charge Distribution. Page 3/19

  ________________________________________________________________________________ M. Nespolo. From Bond Strength to Charge Distribution. Page 4/19

  ________________________________________________________________________________ M. Nespolo. From Bond Strength to Charge Distribution. Page 5/19

  ________________________________________________________________________________ M. Nespolo. From Bond Strength to Charge Distribution. Page 6/19 ratio of experimental bond distances in each polyhedron; the bond strength computed in this way is called bond weight, shortened to BW hereafter. The first step in the computation of the bond weight is the calculation of a weighted mean distance n d(ij→r) through a convergent iterative process similar to that used to obtain n MEFIR: (6) with 0 d(ij→r) = d(ij→rs)1. The weighted average d(ij→r) is actually simply MEFIR scaled by the ratio [R(i) +R(r)]/R(i); in fact, solving Eq. (3) for d(ij→rs)L, we rewrite Eq. (6) at step n = 1 as follows:

  ________________________________________________________________________________ M. Nespolo. From Bond Strength to Charge Distribution. Page 7/19

  make no sense to define a unique weighted mean distance n d(ij) by making r in n d(ij→r) variable over all the V-atoms coordinated by the same PC-atom because the corresponding bond distances can be significantly different so that global average would over-estimate bonds with smaller V-atoms and under-estimate bonds with larger V-atoms. Accordingly, n ECoN(ij→r), Eq. (8), is defined unambiguously for each homoligand subpolyhedron, and the resulting n ECoN(ij→r) are then summed up to obtain n ECoN(ij): (9) ignoring thus the effect of the size of the different V-atoms at this stage. ________________________________________________________________________________ M. Nespolo. From Bond Strength to Charge Distribution. Page 8

  crystallographic type each) and one type of anion (O; two crystallographic types: O1 and O2). The index i runs from 1 to 3, and for each of them j takes the only value 1 (ij = 11: Ti; ij = 21: K; ij = 31: Na). The index r takes the only value 1 and the index s runs from 1 to 2 (rs = 11: O1; rs = 12: O2). Na. where K is in contact with twelve oxygen atoms (six pairs of distances), Ti and Na with five. The K-O distances increase in a more or less continuous way, until a jump is observed after the eighth distance. FIR has a parallel behaviour and the bond weight decreases. The contribution of the last four oxygen atoms is almost negligible (lower than 0.01) and the value of ECoN is practically 8. Only the first eight oxygen atoms contribute significantly to the coordination of K. The opposite occurs for Na, where the contribution of each to the five bonds to ECoN is comparable, leading to ECoN practically 5. An intermediate situation occurs for Ti, which makes five bonds with oxygen atoms but ECoN is only 4.56.

  ________________________________________________________________________________ M. Nespolo. From Bond Strength to Charge Distribution. Page 9/19 of only 6.16. Ba3 presents a similar situation, the tenth Ba-O distance having a bond weight negligible. The last three barium polyhedra are less deformed: with eight Ba-O bonds each, their ECoN goes from 7.43 to 7.69.

  ________________________________________________________________________________ M. Nespolo. From Bond Strength to Charge Distribution. Page 11/19

  ________________________________________________________________________________ M. Nespolo. From Bond Strength to Charge Distribution. Page 12/19

  ________________________________________________________________________________ M. Nespolo. From Bond Strength to Charge Distribution. Page 13/19

Figure captions Figure 1 .

 captions1 Figure captions Figure 1. A regular octahedron with six distances d(ij→rs) all equal to 2.000 Å (a) undergoing increasing deformation of bond distances (b) to (d), while preserving the arithmetic average of 2.000 Å. The weighted average distance decreases as does ECoN too. Values in the figures are bond distances in Ångströms. Figure drawn with VESTA (Momma and Izumi, 2011).

Figure 2 .

 2 Figure 2. Graphical representation of the Charge Distribution (CHARDI) and Bond Valence Sum (BVS)

Figure 3 .

 3 Figure 3. Graphical representation of the Charge Distribution (CHARDI) and Bond Valence Sum (BVS) for the structure of a'-Ba2TiO4. Same interpretation as for Figure 1.

Table 1 .

 1 heteroligand.................................................................................................................2 ff., 8, 11 ff., 17 homoligand.....................................................................................................................2 f., 8, 10, 12 f. MEan Fictive Ionic Radius................................................................................................................4 f. MEFIR...............................................................................................................................................4 f. resonance bond number........................................................................................................................4 Resonance bond number.....................................................................................................................15 Simulation of the progressive distortion of a regular octahedron and its effect of the weighted average distance n d(ij → r) and effective coordination number n ECoN(ij → r).

________________________________________________________________________________ M. Nespolo. From Bond Strength to Charge Distribution. Page 19/19

Table 2 .

 2 Calculation of ECoN and the parameters contributing to it for the structure of KNaTiO3 (structural data from[START_REF] Werthmann | Ein neues einfaches Titanat: KNaTiO3[END_REF]. h is the multiplicity of the

			1.000	1.000	1.940	1.000	1.166	1.890	1.000	1.278	1.800	1.000	1.355
	d(ij→rs)2	2.000	1.000	1.000	1.970	1.015	1.075	1.940	1.026	1.125	1.870	1.039	1.133
	d(ij→rs)3	2.000	1.000	1.000	1.990	1.026	1.014	1.970	1.042	1.033	1.920	1.067	0.975
	d(ij→rs)4	2.000	1.000	1.000	2.000	1.031	0.984	2.020	1.069	0.883	1.990	1.106	0.763
	d(ij→rs)5	2.000	1.000	1.000	2.040	1.052	0.866	2.070	1.095	0.739	2.180	1.211	0.302
	d(ij→rs)6	2.000	1.000	1.000	2.060	1.062	0.808	2.110	1.116	0.631	2.240	1.244	0.205
	n d(ij → r)		2.000			1.995			1.918			1.912
	n ECoN(ij → r)			6.00			5.91			5.69			4.73
	Wyckoff position.
				Ti1 4 O1 4	1.7570 0.556	0.596	1.887 1.416	4.56
				O2 8 21.9098 0.604			0.927
				O2 8 22.0034 0.634			0.648
				K1 4 O1 4 22.7924 1.662	1.760	2.959 1.341	8.04
				O1 4 22.9093 1.732			1.101
				O2 8 23.0757 1.831			0.770
				O2 8 23.1244 1.860			0.679
				O2 8 23.6514 2.174			0.079

PC h V h d(ij → rs) FIR n MEFIR n d(ij→r) n w n ECoN(ij → r)

Table 3 .

 3 Calculation of ECoN and the parameters contributing to it for the structure of a'-Ba2TiO4 (structural data from[START_REF] Guenter | Orthorhombic barium orthotitanate, a'-Ba2TiO4[END_REF].PC V d(ij → rs) FIR n MEFIR n d(ij → r) n w n ECoN(ij → r)

	Ba1 4 O3 4	2.5393 1.401	1.509	2.736 1.434	5.54
	O12 4	2.7154 1.498		1.045
	O10 4	2.7240 1.503		1.026
	O9 4	2.7587 1.522		0.950
	O8 4	2.9594 1.633		0.548
	O5 4	3.0263 1.670		0.435
	O12 4	3.3986 1.876		0.069
	O9 4	3.4941 1.928		0.035
	Ba2 4 O7 4	2.5718 1.419	1.564	2.837 1.561	6.16
	O8 4	2.7522 1.519		1.181
	O5 4	2.7872 1.538		1.106
	O12 4	3.0013 1.656		0.669
	O2 4	3.0071 1.659		0.659
	O10 4	3.1208 1.722		0.462
	O1 4	3.3184 1.831		0.210
	O8 4	3.3600 1.854		0.172
	O6 4	3.4019 1.877		0.139

Table 4 .

 4 Charge Distribution (CHARDI) and Bond Valence Sum (BVS) for the structure of KTiNO 3. CHARDI computed with CHARDI2015[START_REF] Nespolo | CHARDI2015: Charge Distribution analysis of non-molecular structures[END_REF]; BVS computed with Kdist[START_REF] Knizek | KDist, version 4.31. part of the Kalvados software suite[END_REF].

		CHARDI BVS
	Ti	4.01 3.92
	K	0.99 0.83
	Na	1.01 1.09
	O1	-2.06 -1.95
	O2	-1.97 -1.94

Table 5 .

 5 Charge Distribution (CHARDI) and Bond Valence Sum (BVS) for the structure of a'-Ba2TiO4. CHARDI computed with CHARDI2015[START_REF] Nespolo | CHARDI2015: Charge Distribution analysis of non-molecular structures[END_REF]; BVS computed with Kdist[START_REF] Knizek | KDist, version 4.31. part of the Kalvados software suite[END_REF].

		CHARDI BVS	CHARDI BVS
	Ba1	2.01 1.78 O1	-2.04 -2.01
	Ba2	1.99 1.56 O2	-1.91 -1.93
	Ba3	1.99 1.59 O3	-1.93 -1.97
	Ba4	2.01 2.04 O4	-2.01 -1.86
	Ba5	2.00 2.17 O5	-2.02 -1.90
	Ba6	2.00 2.05 O6	-2.01 -1.97
	Ti1	4.05 4.14 O7	-2.03 -2.00
	Ti2	3.96 3.98 O8	-2.01 -1.87
	Ti3	3.99 3.97 O9	-1.96 -1.95
		O10	-2.04 -1.96
		O11	-2.00 -1.96
		O12	-2.01 -1.88


Eric Strobl
email: eric.strobl@polytechnique.edu

THE IMPACT OF HURRICANE STRIKES ON LOCAL CROPLAND PRODUCTIVITY 1 : EVIDENCE FROM THE CARIBBEAN

We empirically estimate the impact of hurricane strikes on local crop productivity in the Caribbean region. To this end we first identify local cropland at 1km2 geographical units via Global Land Cover data. We then employ a windfield model combined with a power dissipation equation on hurricane track data to arrive at a scientifically based index of potential local destruction along these 1km2 cropland grid cells for landfalling and passing hurricanes. Cropland productivity at the local level is approximated by annual net primary production values derived from satellite spectral reflectance data. This provides us with a panel of over 150,000 potentially affected cropland areas in the Caribbean over the period 2000-20006. Our econometric results indicate that cropland productivity is substantially reduced after a hurricane strike.

1

Section I: Introduction

Natural disasters are generally associated with considerable economic losses. Particularly alarming in this regard is not only the fact that the last three and a half decades have witnessed an increase in the number of such occurrences, but also that developing countries seem to be those bearing the brunt of these events and ultimately the economic consequences, thus possibly further adding to the perceived gap between the 'rich ' andthe 'poor'. For example, between 1970 and2002 out of a total number of 6436 natural disasters, 77 per cent have taken place in the developing world.

Moreover, the reoccurrence of such extreme events often tends to be concentrated in particular geographic areas, striking certain countries again and again, often with great severity.

For instance, since 1984 Dominica has been struck by 9 different hurricanes, while Hurricane Georges caused losses of around 400 million US$, constituting over 140 per cent of GDP, in the Caribbean islands of St.

Kitts and Nevis in 1998. 1 Importantly, natural disasters such as hurricanes can be particular damaging to agriculture. Since many developing countries tend to be relatively specialized in production, with particular emphasis on agricultural activities, understanding how such events affect the agricultural sector is arguably of upmost importance and of policy 1 See [START_REF] Rasmussen | Macroeconomic Implications of Natural Disasters in the Caribbean[END_REF].

relevance.2 Surprisingly, however, there is to the best of our knowledge specifically trying to quantify the effect of natural disasters on the agricultural sector in developing countries.

In this paper we just specifically set out to rectify this paucity by investigating the effect of hurricane strikes on cropland productivity in the Caribbean. In this regard, arguably focusing on hurricanes' impact on the Caribbean agricultural sector serves as an insightful case study. For one, over the last 50 years over 80 hurricanes made landfall in the region.

Moreover, many of the small Caribbean islands and countries rely heavily on the agricultural sector to generate earnings.

Our approach relies heavily on non-economic data sources that we translate into economic measures in order to achieve the paper's goal.

Firstly, we identify cropland within the Caribbean at a spatially extremely disaggregated level (1km 2 cells) via satellite derived spatial land cover data.

Secondly, we resort to actual historical data tracking the movement of tropical storms across the affected region and employ a wind field model on the hurricane `tracks' that allows us to calculate an approximation of the severity of winds experienced at this detailed geographical level. These local wind estimates are then used in conjunction with a power dissipation index to proxy local potential destructiveness of hurricanes. Finally, we construct measures of cropland productivity at the local level is by using annual net primary production values derived from satellite spectral reflectance data.

All our data sources combined allow us to construct a panel of over 150,000 cropland areas for which we can calculate the annual impact on cropland productivity of hurricane strikes over the 2000-2006 period. Our results show that the effect is rather short-lived, lasting about a year.

Quantitatively, the effects on overall cropland productivity tend to be moderate, although over our sample period the impact was on occasion as large as a 8 per cent reduction in cropland productivity.

The remainder of the paper is as follows. In the next section we briefly describe the basic nature of hurricanes and their potential destructiveness. In Section III we outline the wind field model and power dissipation equation used to derive a local index of local destructiveness.

Section IV describes our data sources. Some destruction estimates using our proxy are given in Section V. We econometrically investigate the impact of hurricanes on cropland productivity in the region in Section VI.

Finally, concluding remarks are provided in the last section.

Section II: Some Basic Facts about Hurricanes and their Destructive Power

A tropical cyclone is a meteorological term for a storm system, characterized by a low pressure system center and thunderstorms that produces strong wind and flooding rain, which forms almost exclusively, and hence its name, in tropical regions of the globe. ( ) (  )

2 1 1 exp 2 sin 1 ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - - = B m B m h m s F R R R V T S V F V (1)
where V m is the maximum sustained wind velocity anywhere in the hurricane , T is the clockwise angle between the forward path of the hurricane and a radial line from the hurricane center to the point of interest, P, V h is the forward velocity of the hurricane, R m is the radius of maximum winds, and R is the radial distance from the center of the hurricane to point P. The relationship between these parameters and P are depicted in Figure 2. Of the remaining ingredients F is the scaling parameter for effects of surface friction, S the scaling parameter for asymmetry due to the forward motion of the storm, and B the scaling parameter controlling the shape of the wind profile curve. The peak wind gust velocity at point P can then be estimated via:

S g GV V = (2)
where G is the gust wind factor.

The next step entails translating these wind field calculations into potential damage estimates. As noted by [START_REF] Emanuel | Increasing Destructiveness of Tropical Cyclones over the past 30 Years[END_REF], both the monetary losses in hurricanes as well as the power dissipation of these storms tend to rise roughly as the cube of the maximum observed wind speed rises. Consequently, he proposes a simplified power dissipation however, no consensus on how such asymmetry should be modeled; see [START_REF] Bao | An Asymmetric Hurricane Wind Model for Storm Surge and Wave Forecasting[END_REF].

index that can serve to measure the potential destructiveness of hurricanes as 9 :

PDI = ∫ τ 0 3 dt V ( 3
)
where V is the maximum sustained wind speed, and τ is the lifetime of the storm as accumulated over time intervals t. Here we modify this index to obtain an index of potential damage of a hurricane at a particular spatial locality. More precisely, the total destruction due to a storm r in country i at locality j in year t is:

HD i,j,r,t = ∑ τ 0 3 , , , s r j i V if V≥119 km/jr (4)
The index in (4) can then be used to calculate annual total destruction in local j by aggregating all its values over a year t.

Section III: Data Sources

Our geographical region of focus are the 25 countries/territories of the Caribbean, where we depict these in Figure 3. To construct our panel of localities we rely on a number of data sources, as described below. 9

This index is a simplified version of the power dissipation equation

rddt V C PD r D t 3 0 0 0 2 ∫ ∫ = ρ π
where the surface drag (CD), surface air density (ρ), and the radius of the storm (r0) are taken as given since these are generally not provided in historical track data. [START_REF] Emanuel | Increasing Destructiveness of Tropical Cyclones over the past 30 Years[END_REF] notes that assuming a fixed radius of a storm is likely to introduce only random errors in the estimation. He similarly argues that surface air density varies over roughly 15%, while the surface drag coefficient levels off at wind speeds in excess of 30m/s, so that assuming that their values are fixed is not unreasonable.

A. Hurricane Data

For data on hurricanes in the Central American and Caribbean region we rely on the North Atlantic Hurricane database (HURDAT), maintained by the National Hurricane Center (NHC). The HURDAT database consists of six-hourly positions and corresponding intensity estimates in terms of maximum wind speed of tropical cyclones in the North Atlantic Basin since 1851 and is the most complete and reliable source of North Atlantic hurricanes. 10 We depict all tropical storm tracks in the region since 1997 in Figure 4, where the segments in red signify the part of tropical storms that reached at least hurricane level of strength.

As can be seen, throughout the region there has been considerable tropical storm activity with ??? tropical storms having navigated the region. However, one may want to note that a large part of this activity has been at a level deemed not (relatively) important in terms of potential damages caused as suggested by speeds of at least hurricane strength.

B. Cropland Productivity

Unsurprisingly, agricultural measures of cropland productivity, such as crop yields, at a fine spatial data over any meaningfully large space and time are essentially non-existent, particularly for developing countries.

In order to obtain a proxy of cropland productivity we instead resort to the concept of 'net primary production' (NPP). 'Production' in this regard 10 Elsner and Jagger (2004).

refers to the creation of new organic matter. For example, when a crop of wheat grows, new organic matter is created by the process of photosynthesis, where light energy is converted to energy stored in plants, in turn spurning plant growth. 'Gross primary production' (GPP) thus refers to the rate at which an ecosystem's producers convert solar energy into chemical energy as biomass. Since plants use some of their energy for respiration, the amount of energy available for energy consumption by consumers is just gross primary production minus respirations costs, i.e., NPP, usually measured in terms of kcal/m 2 /year. In essence NPP quantifies the conversion of atmospheric CO 2 into plant biomass. As noted by Hicke et al. ( 2004), NPP can change in response to shifts in different crops, changes in crop management practices (ex: fertilization, irrigation, pest management etc.), and climate (ex: precipitation, temperature, solar radiation). One may also want to note that one of the advantages for using NPP to proxy cropland productivity over large areas and over time is that it, unlike economic data, provides a common metric among different crop types, thereby facilitating comparisons and aggregation over all crop types. As a matter of fact, there numerous studies have used yield data to derive NPP estimates in order to assess cropland productivity across areas; see, for instance, Monfreda et al (2008) and Veron (2002) to name a few.

The most natural starting point to try and estimate NPP for a particular spatial location would theoretically of course seem to be through actual ground level measurement. However, even part from questions of cost and organizational feasibility, using ground data to measure local NPP is exceedingly difficult over large geographical areas because of the spatial variability of environmental conditions and the limitations in direct measurement techniques; see Goetz and Prince (1999). An attractive alternative in this regard is to instead use NPP measurements derived from satellite data on spectral reflectance. More daily mosaics ("S1" format). We first overlaid the data into the grid format of the MOD17A2 via krigging. We then used the land cover categories (i) cropland (upland cropland or inundated/flooded crops), (ii) mosaic of cropland / shrub or herbaceous cover, and (iii) mosaic of cropland / tree cover / other natural vegetation to identify those 1 km cells that are used for cropland. We provide a graphical depiction of these croplands in Figure 5.

One may want to note that our identification of cropland will necessarily suffer from two weaknesses. Firstly, we cannot take account of any changes in land cover over our sample period since the data is time invariant. Secondly, our identification crucially depends on the accuracy of the classification system of the GLC 2000. In this regard, one would suspect that small scale farming may not be captured in our analysis.

C. Other Data

As noted above, one of the factors that can affect NPP is the local climate, such as rainfall and temperature, that a region is exposed to. To control for such factors we climatic data from the Willmott, Matsuura and Collaborators' Global Climate database at the University of Delaware.

More specifically, this provides monthly time series over the 1900-2006 period on precipitation and temperature at 0.5 times 0.5 degree cells of the global land area. Since the grid classification is more aggregate than our 1km 2 schemata used for our measures cropland productivity we interpolate values via distance weighting to arrive at measures at a similar spatial aggregation as our NPP data.

Overall the collection of our data sources above results in annual values of hurricane destruction, cropland productivity, and climatic variables for 157,116 local cropland regions at a 1km 2 size over the 2000-2006 period. Summary statistics of these are provided in Table 1.

Section IV: Hurricane Destruction Estimates

To calculate local and aggregate wind speed damage estimates due to hurricanes, we first need to estimate local wind speeds experienced by relevant localities. One should note that of all the parameters necessary to estimate (1) and ( 2) some are given by the hurricane best track data, while for others values need to be assumed as in [START_REF] Boose | Landscape and Regional Impacts of Hurricanes in Puerto Rico[END_REF]. In particular, the raw hurricane data set provides values for maximum sustained wind velocity, V m , at particular locations at particular time intervals and from these one can then estimate V h , the forward velocity, and, relative to the point of interest P, the clockwise angle between the forward path of the hurricane T, and, R, the radial line from the hurricane center.

The scaling parameters, F, S, B, and G in ( 1) and ( 2) control for surface friction, forward motion of the hurricane, the shape of the hurricane, and the gust factor, respectively. Here we use the figures as suggested by Boone et al (2005). In particular, F is assumed to take on values of 1.0 and 0.8 for points on water and land respectively, while G uses respective values of 1.2 and 1.5 for these surface types. S and B are assumed to be 1.0 and 1.3, respectively. Finally, one should note that while the radius of maximum winds, R m , i.e., the distance between the center of the cyclone and its band of strongest winds, is considered to be an important parameter in tropical cyclone forecasting, historical hurricane best track data generally do not provide estimates of this parameter. 13 We thus assume this to take on the value of 50 (km), which corresponds to its average value found for hurricanes with central pressures falling between 909 and 993 hPa. 14With these parameter inputs in hand the wind field model in ( 1)-( 2)

enables us to estimate the wind intensity experienced by any location relative to the position and maximum wind speed of a hurricane (as given by the best track data). However, one may want to note that while the raw cyclone data provides six hourly positions of tropical cyclones, these storms may travel considerable distance within six hours. Thus in order to ensure that we do not neglect areas that may be affected but do not fall within any significant distance (in the sense of experiencing severe winds) in our six hour windows, we linearly interpolated the positions P and wind speeds between the six hourly data to obtain three hourly track data. 15 In choosing all possible positions for which to calculate wind speeds experienced, we compiled the location of the center of each grid cell used for the population data within our region of interest.

In terms of applying our wind field model to obtain local wind intensity estimates for the Caribbean region, we then followed each tropical cyclone over each point of the interpolated track and calculated the wind intensity relative to the center of each grid cells in the schemata provided by the population data as long as these fell within 500 km of the hurricane's location. 16 This provided us with a complete set of estimates of wind fields experienced by all spatially relevant localities relative to each position of each tropical cyclone. We were then able to calculate local destruction according to our index of (4). 15 One should note that interpolating the track data to obtain more frequent observations of the tropical cyclone is standard in the literature; see, for instance, [START_REF] Jagger | Climatology Models for Extreme Hurricane Winds near the United States[END_REF]. 16 Hurricanes have been observed to reach up to a maximum of size of 1000km in diameter.

We first depict all hurricane tracks that according to our wind damage index were associated with at least some damage in one of the countries in the CAC region in Figure 6, where the red portions of the tracks indicate when these reached hurricane strength. Accordingly, only ??? storms, i.e., ?? per cent of all tropical storms that occurred since 1999 in the North Atlantic and Eastern North Pacific, came within close enough distance and reached high enough strength to affect local areas of the countries in the Caribbean region according to our HD index.

As a demonstration of how our HD index translates into estimates of local destruction for individual hurricane occurrences we next calculated and plotted its value over all affected localities for Hurricanes David in As can be seen from Figure 8, Hurricane Dennis only made landfall at hurricane strength in Cuba, causing damages throughout the island.

Noteworthy in this regard is that the extent of damages differed widely, where being close to the actual traveled track does not necessarily mean large destruction in terms of national importance because of a non-even spread of population densities.17 One may also want to take note that while no other islands were directly struck in terms of landfall, Hurricane Dennis' winds were strong enough to affect Haiti, Jamaica and small parts of the Bahamas.

In Figure 9 we plot the average degree of destruction suffered by individual localities in the region over our sample period -where the scale increases as colors change from yellow to red. As can be seen, the potential damage to croplands even with countries is not evenly distributed. For instance, north eastern Cube suffered to a much greater extent than other parts of the countries.

Averaging the values calculated from the HD proxy over all hurricanes r can also serve to compare the destructiveness of hurricanes relative to each other in terms of the damages done across economies.

We show a ranking of these in terms of average destruction in Table 3. As can be seen, croplands in the Cayman Islands were potentially the most affected, followed by Jamaica, Anguilla, and the Virgin Islands. In contrast, over our sample period Costa Rica experienced no potential hurricane destruction to its croplands. Comparing the actual damage index figures across countries, one may want to note the large dispersion, with the top affected having values multiple times those at the lower half of the table.

Section V: Econometric Analysis

Our main econometric task is to investigate the macroeconomic impact of hurricane strikes in the Central American and Caribbean region using our index of destruction. To do so we take our panel of countries for which we also have macroeconomic data and specify a simple growth equation:

log(NPP) i,j,t = α + β 1 RAIN i,j,t + β 2 TEMP i,j,t + β 3 HD i,j,t + ε i,t ( 5
)
where NPP is net primary production, RAIN precipitation, TEMP temperature, HD our index from (4) summed over a year, and ε is an error term. One worry with estimating is that we do not control for the different types of crops and that some crops may be more affected by hurricane strikes than others. If the choice of crops depends at least to some extent on the probability that a hurricane strikes the area, then this could bias our estimate of the coefficient on HD. We assume here that this effect is time invariant and control for it by running a fixed effects estimator, which purges all time invariant unobservables from the equation. Another concern is that, particular due to our use of spatially very disaggregated data, that there may be spatial dependence across localities, causing spatial correlation among the error terms. To take account of this we employ the nonparametric covariance matrix estimator proposed by Driscoll and Kray (1998), which produces heteroskedasticity consistent standard errors that are robust to very general forms of spatial and temporal dependence.

Our results of estimating (4) are given in Table 3, where year specific time dummies are included but not reported. In the first column we only include our climatic variables. As can be seen, while rain has a positive impact on NPP, the is no such effect in terms of temperature. This may be because of the interpolation procedure underlying fitting the climatic data onto our 1km 2 grid cells. However, more likely this is in large part due to the fact that temperature tends, in contrast to rainfall, vary much less locally, and thus that any effect of temperature is captured by our time dummies.

In the second column we include our HD index, which turns out to be negative but insignificant. In contrast, the t-1 lag of the index, as shown in the third column, is statistically significant. We suspect this to be due firstly because most hurricanes take place in the third and fourth quarter of any calendar year and hence their effect may only be picked up in the NPP measures of the following calendar year. Additionally, it could also be that calendar years and growing seasons of crops are unlikely to perfectly overlap.

As can be seen including further lags (up to t-3) produces no additional long term effect of a hurricane strike on cropland productivity.

As a robustness check we re-ran (5) only including our significant variables, namely HD at t-1 and RAIN at t, but the coefficient on HD remains significant and of similar size. One possibility may also be that RAIN is picking up some of the damage due to hurricanes since these tend to be heavily correlated with strong rainfall particularly in the outer wind bands of a hurricane. However, as shown in the last column, excluding RAIN changes little in terms of the coefficient on HD.

We can also use our results to gain some insight into the economic significance of hurricane strikes on cropland productivity. More specifically, our estimated coefficient on the HD at t-1 suggests that the average hurricane reduces cropland productivity by about 0.7 percentage points. In contrast, the largest annual exposure could reduce productivity of croplands up to 6.1 percentage points.

Section VI: Concluding Remarks

In this paper we investigated the impact of hurricane strikes on local cropland productivity in the Caribbean region using non-economic data and methods to construct economic measures. Our results show that the impact has been unevenly distributed across the regions, with countries/territories like Cayman Islands and Jamaica having been potentially most affected over our sample period (2000)(2001)(2002)(2003)(2004)(2005)(2006). Our econometric analysis demonstrates that hurricane strikes have indeed had a statistically significant impact on cropland productivity. More specifically, our estimates suggest that the average hurricane reduces cropland productivity by about 0.7 percentage points, but that local cropland may experience a loss of up to over 6 percentage points. Notes: The degree of destruction increases as the color scheme changes from yellow to red.

  precisely, in contrast to NPP measures based on interpolations from widely separated ground point observations, remotely sensed data are spatially contiguous and relatively frequent (generally daily). The physical basis for the observed correlation between spectral reflectance and NPP is the existence of a relationship between spectral reflectance and the absorption of solar radiation by vegetation canopies, and in turn the link between the amount of absorbed photosynthetically active radiation and its utilization for NPP.11 To then actually deriving NPP from APAR one needs an estimate of the production efficiency, ε. ε can, however, potentially vary widely across different plant types and biomes for two reasons since respiration costs appear to increase with plant size and suboptimal climatic conditions can also attribute to its variability. The main challenge to then actually deriving quantitative measures of NPP from satellite reflectance data is estimating this ε.12 In this paper we use the MOD17A2 NPP measures derived from images on spectral reflectances of terrestrial vegetation using the Moderate Resolution Imaging Spectroradiometer (MODIS) on the NASA Earth Observing System (ES) Terra satellite. More precisely, the Earth Observing System Data Information System (EOSDIS) computes calibrated and atmospherically corrected reflectances from each spectral channel of the MODIS sensor for each cloud free pixel. Daily GPP computation at the local level then rests additionally on the biome type, which is recomputed annually, the fraction of photosynthetically absorbed radiation, which can change weekly, and the daily surface climate conditions, which change diurnally. GPP and NPP and are then calculated using these input variables and the complex ecosystem model BIOME-BGC. Global data at the 1 km spatial level is calculated for GPP daily and summed every 8 days, while NPP measures are provided at an annuals basis at the end of the year since plants grow during different parts of the year. While NPP estimates in the MOD17A2 data set are provided for all land areas, we are for the purposes of the current paper only interested in 12 It is of course of interest to know how well satellite derived measures of NPP are able to approximate their from ground data computed alternatives. In this regard, Lobell et al. (2002), for instance, have shown that there is good agreement between yield and satellite derived estimates of NPP for the US. those pertaining to cropland areas. To identify these we resort to the 2000 Global Land Cover (GLC 2000) data set. The GLC 2000 data classifies land across the globe into 22 distinct land cover categories based on 14 months (1 Nov. 1999 -31 Dec. 200) of daily 1-km resolution satellite data acquired over the whole globe by the VEGETATION instrument on-board the SPOT 4 satellite and delivered as multi-channel

Figure 7 ,

 7 Figure7, where shading moving from yellow to red indicates the rising

Figure 1 :Figure 2 :

 12 Figure 1: The Typical Structure of a Hurricane

Figure 3 :Figure 4 :

 34 Figure 3: Caribbean and Central American (CAC) Region

Figure 5 :

 5 Figure 5: Cropland Areas in the Caribbean Region

Figure 6 :

 6 Figure 6: Relevant Hurricanes

Figure 7 :

 7 Figure 7: Hurricane Dennis (2005)

Figure 8 :

 8 Figure 8: Hurricane Dennis' (2005) Destruction Path

Figure 9 :

 9 Figure 9: Average Local Degree of Destruction, 2000-2006

Section III: Hurricane Wind Damage Index

  3 Depending on their location and strength, tropical cyclones are referred to by various other names, such as hurricane, typhoon, tropical storm, cyclonic storm, and tropical depression. Tropical storms in the North Atlantic and the North East Pacific region, as we study here, are generally termed hurricanes if they are of sufficient strength.4 In terms of its structure, a hurricane will typically harbor an area of sinking air at the center of circulation, known as the 'eye, where weather in the eye is normally calm and free of clouds, though the sea may be extremely violent.

	high winds pushing on the ocean's surface cause the water near the
	coast to pile up higher than the ordinary sea level combined with the low
	pressure at the center of the weather system and the bathymetry of the
	body of water results in storm surges. Storm surges can cause severe salt
	contamination of agricultural areas, where flooding on the coast may
	occur 3-5 hours before the arrival of the center of the hurricane. 6
	While the extent of potential damages caused by hurricanes may
	depend on many factors, such as slope of the continental shelf and the
	shape of the coastline in the landfall region in the case of storm surges, it is
	typically measured in terms of wind speed, and we similarly follow this
	approach. More specifically, our hurricane wind damage index is based
	on being able to estimate local wind speeds at any particular locality
	end of May and last where a hurricane strength tropical storm passes over or nearby. To do so
	until the end of November. we rely on the meteorological wind field model developed by Boose et al
	Hurricane damages in terms of agriculture typically take a number (2004). 7 , which provides estimates of wind field velocity of any point
	of forms. Firstly, the strong winds associated with hurricanes may cause relative to the 'eye' of the hurricane. This model is based on Holland's
	considerable structural damage to crops. Secondly, strong rainfall can well known equation for cyclostrophic wind 8 and sustained wind velocity
	result in extensive flooding and, in sloped areas, landslides. Finally, the at any point P is estimated as:

5

Outside of the eye curved bands of clouds and thunderstorms move away from the eye wall in a spiral fashion, where these bands are capable of producing heavy bursts of rain, wind, and tornadoes. The typical structure of a hurricane is depicted in Figure

1

. Hurricane strength tropical cyclones are typically about 483 km wide, although they can vary considerably. The season for hurricanes in the two regions can start as early as the

Table 2 : Mean Hurricane Destruction experienced in Croplands by Country/Territory

 2

	Country/Territory	Wind Destruction
	CAYMAN ISLANDS	347.0138
	JAMAICA	213.4461
	ANGUILLA	178.8457
	VIRGIN ISLANDS	161.7454
	SAINT KITTS AND NEVIS	145.892
	BELIZE	119.5626
	TURKS AND CAICOS ISLANDS	108.6281
	CUBA	99.56162
	ANTIGUA AND BARBUDA	79.38261
	ARUBA	75.60234
	BAHAMAS	74.66311
	MONTSERRAT	69.56258
	NETHERLANDS ANTILLES	69.23227
	PUERTO RICO	66.95869
	GUADELOUPE	42.64012
	DOMINICA	20.62569
	SAINT VINCENT AND THE GRENADINES	19.85767
	TRINIDAD AND TOBAGO	19.23702
	HAITI	13.73098
	DOMINICAN REPUBLIC	11.39064
	BARBADOS	11.02453
	SAINT LUCIA	6.029252
	MARTINIQUE	0.189819
	COSTA RICA	0.000000

See Albala-Bertrand, J.M.(1993).

The term "cyclone" derives from cyclonic nature of such storms, with counterclockwise rotation in the Northern Hemisphere and clockwise rotation in the Southern Hemisphere.

Generally at least 119 km/hr.

National Weather Service (October 19, 2005). Tropical Cyclone Structure. JetStream -An Online School for Weather. National Oceanic & Atmospheric Administration.

[START_REF] Yang | Coping with Disaster: The Impact of Hurricanes on International Financial Flows, 1970-2002[END_REF].

This wind field model was, for instance, verified by the authors on data for Puerto Rico.

See Holland (1980). One may want to note that Holland's model is an axisymmetric model in that the true asymmetric nature of a hurricane cannot be represented. There is,

This relationship was first noted byMonteith (1972).

This parameter is traditionally measured by reconnaissance aircraft in the Atlantic basin, so that there is no information in this regard for older hurricanes.

See Hsu and Yana (1998). This roughly corresponds to the central pressures of tropical storms of hurricane strength, where central pressure is inversely related to strength.

Most obviously, some areas, despite being very close to the actual track, were estimated to have zero damages because the local population was zero.