Mario Caccamo

Hadi Quesneville

Towards the construction of an integrated Wheat Information System Report--June 2012

Introduction

This report was commissioned by the Wheat Initiative scientific board following the proposal adopted by the G20 Agriculture Ministers in June 2011. The aim of this report is to offer recommendations on the best strategies to follow to develop an integrated Wheat Information System (hereafter called WheatIS) and provide the international wheat research community easy access to wheat genetic information, genomic data and bioinformatics tools. This report is based on results obtained from a community--wide consultation and opens the possibility of integrating agronomic data within the WheatIS. This report was built in collaboration with an expert committee composed of wheat scientists and other researchers involved in the development of information systems initiatives (see members names in appendix 1).

I. Community--wide consultation results

A web--based survey questionnaire was implemented to consult the wheat research community on areas of interest to them. The link was circulated to recipients through several mailing lists and relayed by scientists to their collaborators. We also collected feedback from a dedicated workshop that we organised with the support of the Wheat Initiative Scientific Board during the 20th Plant and Animal Genome meeting, held in San Diego in January 2012.

Web--survey

The questionnaire for the consultation was prepared with the input from the expert committee and implemented as web pages to be filled by the survey participants. The survey was open from 12/14/11 to the 03/14/12 and was answered by 282 participants from 30 different countries (Figure 1).

Figure1: Number of respondents per country

The fields of expertise of the surveyed scientists covered the WheatIS targeted scientific domains (Figure 2) Breeding and functional genetics were the most represented. Asked the type of analysis they would like the WheatIS to support, wheat researchers ranked functional and genotype to phenotype analysis ahead of comparative genomics and breeding (Figure 3).

Figure 3: Preferences for analysis types, ranked by responses

To help with develop and populate the WheatIS, with the many types of data that wheat researchers will need to access, we were interested in which data researchers viewed as a priority. Accordingly, we asked which data appeared to be the most important for wheat research in the coming 5 years. When sorted by importance (Figure 4), we observed a smooth progression in the type of data.

Although there is not a clear--cut preference, SNPs, genome assemblies, phenotypes, maps and molecular markers were marginal leaders, which may reflect a current trend in technologies and resources building in the community. However, all data types could be considered important for the future development of wheat research.

Figure 4: Average rank of importance of wheat data types in the coming five years

The WheatIS could operate as a hub and integrate wheat data produced and submitted to the public repositories by the community. In combination with it, data could be transferred and stored on other bioinformatics portals. When asked on which existing platform they would prefer to also store their data, wheat researchers placed the NCBI and GrainGenes portals in first and second position respectively, followed by Gramene and Wheatgenome.info (Figure 5A). However, marked differences were observed between countries (Figure 5B). Considering the 4 countries most represented in the survey, GrainGenes and NCBI were preferred by researchers based in Australia, GrainGenes in the USA, NCBI and Cerealsdb in the UK and URGI in France. This might reflect preferences related to users considerations or specific research interests, and/or the knowledge of local databases by national research communities. When asked what kind of services the WheatIS should provide, wheat researchers placed data browsing, data downloading and genome viewer respectively in first, second and third position (Figure 6). Interestingly, data integration was placed only in fourth position, indicating that this is not considered yet as a top priority. Surprisingly, we also noted that analysis workflows and computing capacities were ranked last. Again this might be due to current technological limitations and lack of access to more advanced bioinformatics tools. Most of the participants supported the data release policy developed by the Bermuda / Fort Lauderdale / Toronto agreements (Nature 461, 168--170, doi:10.1038/461168a), that promotes the early dissemination of whole--genome datasets but preserves the rights for the data generators to lead the analysis and publication of their data in peer--reviewed journals (Figure 7). Nonetheless, a substantial number of responses were also in favour of a 1 to 6--month period embargo.

Plant and Animal Genomes (PAG XX) Conference Workshop

We organized a workshop open to the wheat scientific community during the 20th Plant and Animal Genomes (PAG) meeting held in San Diego in January 2012. We presented preliminary results of the web--survey and proposed a strategy to develop the WheatIS. Around 40 persons attended this event for a round--table discussion that lasted 2 hours.

Some highlights of the audience questions and comments are reported below:

• Data quality with respect to standards (protocol, nomenclature, format) is a key issue for data management. It also facilitates data diffusion to the community. The implementation of strategies for curation and quality checking will be an important aspect that the WheatIS should support. Legacy data are also important to consider as they could be used as gold standards for validation and comparisons. Hence, the added value of the WheatIS will be to provide sets of curated, formatted, and integrated data to other databases (wheat or other related species). • Metadata should include information about the data providers so that their work can be acknowledged, but also to encourage data submission by giving visibility to the scientists who generated them. Literature related to a data set should also be included. Journals' policies to deposit the data in the WheatIS prior to publication should be encouraged by the Wheat Initiative consortium. • The industry partners should be consulted to build a data management policy compatible with their requirements. It is highly desirable that they contribute as data providers, specially for pre--competitive data that could be beneficial for the global research community. • Data integration should be driven by real case studies to avoid effort dispersion.

• Data should be accessible from facilities equipped with computing power and storage (e.g. cloud--based approaches) to facilitate running computing--intensive analyses.

• Multi--lingual information system could be important for breeders not familiar with English (e.g. researchers in emerging economies in Asia and South America).

• The audience pointed that haplotype/map data were not explicitly mentioned in the survey. They should be considered. • The WheatIS should be built on experiences gained by similar initiatives (plant species, animals, and biomedical field). It will have to participate to the data management knowhow exchanges between bioinformatics platforms.

II. A proposal for the WheatIS architecture

This WheatIS proposal takes into account feedback from the wheat scientific community, from experts working on similar initiatives for other species and from the bioinformatics community in general. We built this proposal by taking into account the considerations given below.

First, the WheatIS will have to add value and represent a substantial improvement to what is already available for the community. It should not replace the current wheat information databases, but should offer them new services, for instance to tackle concrete data management issues such as curation, formatting and integration.

Second, several bioinformatics platforms dealing with wheat data already exist and provide useful services to the community. The WheatIS should be built on these platforms as a federated network of existing services, considering that:

i.

Experiences from other species indicate a tendency to build such networks, instead of single information system (e.g. TAIR/IAIC, EBI/Elixir).

ii.

Exiting platforms already offers some services to particular wheat communities. They are specialized on scientific fields, large projects, or countries. They have already served their community for several years, often doing a good job, as indicated by the web--survey. Consequently they are well positioned to continue in a larger framework.

iii.

Providing a single information system which meets everyone's needs may be overambitious and hamper innovation and adaptability. The data are diverse and the number of scientific fields using them important. It might therefore be difficult to build a single platform in one place. We should rely on a world--wide network of expertise. Integration and standardization are important goals. Our aim should be to integrate searches across diverse databases and encourage format sharing.

iv.

No dedicated funding by the Wheat Initiative is expected. Hence, we propose that the WheatIS should be considered as a framework where data and expertise exchanges are facilitated and enhanced. In this framework, platforms will easily exchange their data to provide their users with up--to--date top--quality information. Users will also have access to enriched integrated and curated data from a central repository. With this in mind, we built a proposal consistent with this view from the infrastructure and governance perspectives.

An evolving infrastructure

Instead of proposing a static view of the infrastructure, we present a dynamic model that will be adapted as the system implementation progresses, and where each step builds optimally on the previous infrastructure. The rationale is that bearing in mind the complexity of building a WheatIS, it will be preferable to build it progressively, learning from each phase by the successes and the failures and correcting the strategy accordingly. Users needs will be followed as they will evolve in parallel. This iterative process will have a better chance to succeed, as its evolution will benefit from the community feedback at each step.

We therefore propose to build the WheatIS in three main steps starting from a low--tech, easy--to-achieve infrastructure, towards a more ambitious integrated system.

Step1 : Network building

The first step will be to build a collaborative and interoperable network of platforms, working together to set up the first visible WheatIS service. Initial tasks will be to define standards, formats, and nomenclatures. Data hosted by the WheatIS will follow the defined rules allowing homogeneity, coherence, and reliability between data. Hence, standardized data could be exchanged between users and easily reused in different analysis. Wheat IS partners will define standards collaboratively, taking into account current standardization works of different platforms. Data using this format will be submitted by scientists to a centralized web file repository. The WheatIS platforms will be able to help the community in this task. Success will depend on this being instigated by the Wheat Initiative consortium. In particular, agreements with peer--reviewed journals for proper data deposit in the centralized system will be key for the success of this initial step.

At this stage, the WheatIS will be a web platform allowing the exchange of standardized data files. It will be possible to search the WheatIS metadata using keywords or full text searches. Indexation through Google web and Google scholar search engines will be used to allow scientist to find data through the Google interface without necessarily knowing the WheatIS portal. Solutions are already available to implement such a web platform (e.g. dspace: www.dspace.org, iRODS: https://www.irods.org ). Consequently, we consider this as an achievable and relatively low--tech solution.

In addition, WheatIS partners will be able to download data from the WheatIS repository into their own information system. This will allow users to view the WheatIS data alongside their own data, or to offer a specific service to a dedicated community (e.g. partners of a project). Indeed, several ways of integrating data could be proposed, each resulting in a specific integrated view, depending on the requirements of particular scientific fields (i.e. breeders, population geneticists, functional genomicists) allowing the same data to be viewed differently according to need. The data standardization and the central repository will greatly help the WheatIS partner platforms in this task.

Step2 : Integrated virtual portal

At the end of Step 1, the WheatIS will be seen as a network of platforms, sharing data files on a central web file repository, but each still maintaining their local databases. The second step will be to set up a full text search engine on the WheatIS portal, allowing to dynamically search these local platform databases. The users will be able to connect to the WheatIS portal and type a keyword or a term that will be searched remotely in each database. Results will be provided as a brief summary of the matching data (e.g. Identifier, Name, Short description) with links to access the remotely hosted data.

Step3 : Integrated database

The final step will be to integrate the data in one single, centralised information system. Previous steps will help to provide a broad view of the available data. Being a major task, integration will focus on relevant data sets, chosen with the wheat scientific community. It will not replace the tools built in Steps 1 and 2, but will add a new browsing functionality, allowing users to navigate through data and explore their relationship. It will also answer complex queries involving data that hosted initially in different location (files, databases). This system will also be able to produce integrated, consolidated and consistent information, which could be exported as data files to feed analysis pipelines or other information systems.

Tools and technologies will undoubtedly evolve rapidly in the next coming years. The strategy presented here will be re--evaluated at each step and the WheatIS development strategy adapted accordingly. Agronomic data should be easily integrated in the system at a later stage. This will require dedicated bioinformatics platforms joining the WheatIS network. In addition, we recommend that the WheatIS follows the work of other international bioinformatics initiatives (such as Ontology, IAIC, TransPLANT) by developing synergies and collaborations.

Wheat research community needs: today and in five years.

According to the web--survey results, the main needs for the five coming years are focused on breeding and functional studies requiring information on genome sequence variants (SNP, Indels, CNV), genome assemblies, phenotypes, maps and molecular markers. Therefore, data management efforts should first concern these types of data.

Interactions with users

The web--survey analysis shows a large consensus in favour of a data release policy under the Bermuda / Fort Lauderdale / Toronto agreements. An unrestricted access will be offered to published data and most legacy data. Unpublished data will be distributed upon the signature of an agreement stating the respect of the data producers and the contributors rights to analyse and publish analyses in peer--reviewed publications. Data access could be managed through private accounts in the information system. Access rights will be granted according to the signature of a document. Note that data with restricted access could be accepted as it could also speed up data diffusion allowing submitters and WheatIS staff to check the data as it appears in the system before granting data access to others. However, this embargo duration should not last more than 6 months. Given the popularity of immediate data dissemination, data release under a Bermuda / Fort Lauderdale / Toronto agreement will be encouraged.

Users will have the opportunity to interact with WheatIS management regularly to give feedback and express their needs. Web--surveys will be sent to the community each year to get satisfaction feedback. A users' committee composed of end--user representatives will be set up to discuss the evolution of the system with the WheatIS steering committee. Satellite meetings will be organised around the main wheat conferences to present novelties and to discuss service improvements.

An e--mail help--desk will be available to help users and answer their problems.

Required resources

With no specific funding, the WheatIS will rely mainly on existing infrastructures and the contribution from the scientists and bioinformaticians from the community (e.g from wheat platforms that will directly participate in this effort). However, resources will be needed for the implementation of a central repository that should probably be hosted by one of the existing platforms. A key prerequisite will be to guarantee the sustainability of this platform, and therefore the long--term availability of the following resources:

• Hardware infrastructure: petabyte--scale storage must be available with support for backups (such as mirror servers). Note that raw and curated data submitted to NCBI or EBI will be transferred regularly. Servers must be securely kept in a dedicated computer room with fire and intrusion protection. • Personnel: Highly qualified staff will be required. The infrastructure will need database managers, system and network managers, developers and data managers. Around 10 full time engineers/researchers should work for the WheatIS. • Quality control procedures: A management quality system must guarantee the correct functioning of the hosting platform. It will insure effectiveness of the procedures and protocols used the availability of the computer infrastructure, and the quality of the services. • Financial sustainability: Long term service preservation and data accessibility must be guaranteed. Staff salaries, computer licences and maintenance fees, hardware renewal costs, travel and accommodation expenses to coordinating and scientific meetings must be covered.

Governance

The proposed governance is composed of a users committee, an expert committee, and an executive board.

Users committee

This committee brings together around 20 representatives from the different wheat scientific fields and countries. They meet regularly and organize surveys in order to provide one users' feedback per year.

Expert committee (IS, scientists)

The expert committee is composed of information system specialists, PIs or heads of the platforms involved in the WheatIS platforms network, bioinformaticians, and wheat scientists. They will analyse users feedback to make recommendations for the WheatIS on orientation and priorities.

Executive board (2--3 persons)

The executive board will make decisions upon the experts' committee recommendations. They will also manage the WheatIS operational activities. They will follow the activities and report to the wheat initiative scientific board.

Conclusion: Risk analysis of the wheat information system.

The WheatIS proposal can be analyzed as follows to display its strengths and weaknesses.

Strengths

• Promote data sharing, enhance data tracking, secure data in a safe data repository • Federate data that where originally dispersed, establish data links, standardized data.

• The wheat genome is being sequenced, generating a lot of new data that needs to be shared.

• Many recently funded large collaborative projects will provide huge amounts of data (e.g. WISP, Breedwheat, Speed, T--CAP).

• Incremental implementation has a better chance to reach the goal and answer the needs

Figure 2 :

 2 Figure 2: Breakdown of areas of expertise of respondents (number of respondents per field of expertise)

Figure 5 :

 5 Figure 5: Other bioinformatics portals where data would have to be accessible from (X axis: number of positive answers)

Figure 6 :

 6 Figure 6: Preferences for services, ranked by number of responses

Figure 7 :

 7 Figure 7: Preferences for data release policy, ranked by number of responses

Figure 8 :

 8 Figure 8: Average rank of importance of users' needs survey mechanisms

Weaknesses

• No core funding for the WheatIS • Managing a distributed infrastructure is complicated • Lack of established standards to work on such a complex genome • Late deep data integration in the project

Opportunities

• Collaboration between bioinformatics platforms at the international level that will leverage progress in data management. • Definition of missing standards in bioinformatics • New tools development for complex genomes • Stimulate collaboration between scientists (public, private) by providing an infrastructure to share their results. • Accelerate crop improvement by providing wealth of consistent data sets.

Threats

• Different funding priorities from governments that could make resources difficult to find and to coordinate. • Different groups are involved that could have different priorities (data, accessibility).

Priorities could be difficult to determine • Lack of adoption by the community preferring to build their own system to enhance their visibility.

• With no dedicated funding sustainability, resources are more limited.

Appendix 1

Expert committee for the survey and report.


domain: math info

text:
Mark C Kennedy

Anthony O'

Hagan Arnaud Bensadoun

Keywords:

come

Bayesian calibration of computer models.

Residual variablity Specic inputs are supposed to allow the model to predict the value of a real process in a specic condition.

However, if that condition is repeated, the process may not take the same value.

⇒ Variations of the process even when the condition is fully specied. This is residual variability.

Two sources of uncertainty are combined here :

Irreductible

The process is inherently unpredictable and stochastic. The output of a computer code given any particular conguration of inputs is in practice not known until we actually run it with those inputs.

Since it may not be practical to run the code to observe the output for every input conguration of interest, uncertainty about code output needs to be acknowledged. Let f (•) be a function mapping an input x ∈ X into an output y = f (x).

The input space X can be arbitrary but is typically a subset of R q for some q ∈ N . ⇒ We can write x as a vector x = (x1, x2, . . . , xq ).

Mean function m(•), where m(x) = E {f (x)} Covariance function c (•, •) where c (x, x') = cov {f (x), f (x')}
(x) = E {f (x)} Covariance function c (•, •) where c (x, x') = cov {f (x), f (x')} f (•) ∼ N {m(•), c (•, •)} ⇔ f (•)
m(•) m(•) = h(•) T β h(•) = (h1(•), h2(•), . . . , hp (•))
T is a vector of p known functions over X . β = (β1, β2, . . . , βp) is a vector of p unknown coecients.

m(•) m(•) = h(•) T β h(•) = (h1(•), h2(•), . . . , hp (•))
T is a vector of p known functions over X . β = (β1, β2, . . . , βp) is a vector of p unknown coecients. The modelling of c (•, •) is a crucial step. Indeed, it is the correlation between f (x) and f (x') that allow to express : A view that f (x) and f (x') should be similar if x and x' are suciently close in X A belief in smoothness of f (•)

c(•, •) c (•, •) = σ 2 r (x -x')
r (•) r (x -x') = exp  - q X j=1 ω j (x j -x j ) 2 ff
Other possible formulations :

Replaces (x j -x j ) 2 by |x j -x j | α α have a specied value α is another hyperparameter

α j can be dierent in each dimension Set r (d) = exp(-d T Ωd)
where Ω is an unknown symmetric positive denite matrix (in above equation has the form Ω = diag (ω 1 , . . . , ωq))

Calibration inputs

The unknown context-specic inputs we wish to learn about.

They are supposed to take xed but unknown values θ = (θ1, . . . , θq2).

For all the observations that will be used for calibration. For all the instances of the true process that we wish to use the calibrated model to predict.

Variable inputs.

All other model input whose value might change when using the calibrated model.

They are assumed to have known values for each of the observations that will be used for calibration. ζ(x) be the true value of the real process when variable inputs takes values x z = (z1, . . . , zn ) be the calibration data, where z i is an observation of ζ(x i ) for known variable inputs x i , but subject to error. y = (y1, . . . , y T N ) the output from N runs of the computer code where y j = η(x * j , t j ) ζ(x) be the true value of the real process when variable inputs takes values x z = (z1, . . . , zn ) be the calibration data, where z i is an observation of ζ(x i ) for known variable inputs x i , but subject to error. y = (y1, . . . , y T N ) the output from N runs of the computer code where y j = η(x * j , t j )

z i = ζ(x i ) + e i = ρ η(x i , θ) + δ(x i ) + e i
Where e i is the observation error for the i th observation ρ is an unknown regression parameter δ(•) is the model inadequacy function that is independant of the code output η(•, •)

η(•, •) ∼ N ˘m1(•, •), c1{(•, •), (•, •)} δ(•) ∼ N {m2(•), c2(•, •)}
In each case, the mean and variance functions are modelled hierarchically At the rst step, mean functions are modelled using linear model form :

m1(x, t) = h1(x, t) T β 1 m2(x) = h2(x) T
β = (β T 1 , β T
2 ) ⇒ No specication of particular forms for covariance functions, but supposition that they are expressed in terms of some further hyperparameters denoted by ψ ⇒ φ denote (ρ, λ, ψ)

The complete set of parameters therefore comprises

The calibration parameters θ

The location parameters β

The hyperparameters φ It is reasonable to suppose that the prior information about θ is independant of the others, and with the prior expression of β, we can write :

Remaining uncertainty

The model now take explicitly into account all the sources of uncertainty previously identied.

However, the compromise proposed here is to not account fully for all these sources.

By xing λ at an estimated value, we do not account fully for observation

Data and model Results

Gaussian plume model

The dispersion of radionuclides is a highly complex process involving various chemical and environmental process which are not directly observable.

Introduction Taxonomy of uncertainties in computer models

Bayesian inference for functions Bayesian calibration Example Discussion

Data and model Results

Gaussian plume model

The dispersion of radionuclides is a highly complex process involving various chemical and environmental process which are not directly observable.

Many simplifying assumptions are made in the Gaussian plume model.

For instance, the speed and direction of the wind are assumed to remain constant during the travel time of the particles released.

Introduction Taxonomy of uncertainties in computer models

Bayesian inference for functions Bayesian calibration Example Discussion

Data and model Results

Gaussian plume model

The dispersion of radionuclides is a highly complex process involving various chemical and environmental process which are not directly observable.

Many simplifying assumptions are made in the Gaussian plume model.

For instance, the speed and direction of the wind are assumed to remain constant during the travel time of the particles released.

Introduction Taxonomy of uncertainties in computer models

Bayesian inference for functions Bayesian calibration Example Discussion

Data and model Results

Gaussian plume model

The dispersion of radionuclides is a highly complex process involving various chemical and environmental process which are not directly observable.

Many simplifying assumptions are made in the Gaussian plume model.

For instance, the speed and direction of the wind are assumed to remain constant during the travel time of the particles released.

Data and model Results

Data

Data used here comes from an accident at the Tomsk-7 chemical plant in 1993 (detail in Shershakov et al., 1995).

We consider the deposition of ruthenium 106 ( 106 Ru).

Data were obtained from an aerial survey which started to the source and continued to about 40 km downwind in such a way that consecutive measurements are very close.

A total of 695 measurements were made.

Introduction Taxonomy of uncertainties in computer models

Bayesian inference for functions Bayesian calibration Example Discussion

Data and model Results

Data

Data used here comes from an accident at the Tomsk-7 chemical plant in 1993 (detail in Shershakov et al., 1995).

We consider the deposition of ruthenium 106 ( 106 Ru).

Data were obtained from an aerial survey which started to the source and continued to about 40 km downwind in such a way that consecutive measurements are very close.

A total of 695 measurements were made.

A subset of n = 10 was choosen to represent a small sample of observed data similar to that which might be collected shortly after an accident.

Additional points were taken, giving subset of size 10, 15, 20 and 25 points.

Introduction Taxonomy of uncertainties in computer models

Bayesian inference for functions Bayesian calibration Example Discussion

Data and model Results

Assumptions

The log-transformation is used to approximate better the assumptions of normality.

The true log-deposition for variable inputs x is ζ(x).

The z i s are the logarithm of observed depositions.

Introduction Taxonomy of uncertainties in computer models inference for functions Bayesian calibration Example Discussion

Data and model Results

Assumptions

The log-transformation is used to approximate better the assumptions of normality.

The true log-deposition for variable inputs x is ζ(x). The choices made here, especially for the correlation function, will not be appropriate for all applications.

A brief examination of plausible alternative formulations has been carried out (further details in Kennedy and O'Hagan, 2000b) and suggested that Any improvement due integrating with respect to the hyperparameters, as opposed to maximising, is likely to be small. The eect of using alternative covariance structures is small. This indicates some degree of robustness, but also a need to consider models that allows for more localized structure.

The method thus need to be explored with more and varied applications.

  Parametric variablity z = η(X , θ) + δ(X ) + ε (X , θ) + δ(X ) + ε

  h(•) describes a class of shapes.The model for m(•) expresses a belief f (•) may be approximated by a

  error and residual variability/uncertainty. By xing ρ and the hyperparameters ψ2 of c2(•, •) at estimated values we do not account fully for model inadequacy. By xing the hyperparameter ψ1 of c1{(•, •), (•, •)} at estimated values we do not account fully for code uncertainty. plume model (Clarke, 1979) is used to predict the dispersion and subsequent deposition of radioactive material following an accidental release. Code inputs can be divided into two types : Atmospheric conditions at the time of the

  The z i s are the logarithm of observed depositions.Critical unkown inputs are the source term and deposition velocity ⇒ The logarithms of these are the calibration parameters θ A normal distribution is used as prior for θ prior means are given by authorities prior variances are set to represent vague prior knowledge prior covariances are assumed to be data sets, posterior means and variance of z (x) were calculated for all the 670 unobserved points, and accuracy was assessed on the basis of the true values at these points. data sets, posterior means and variance of z (x) were calculated for all the 670 unobserved points, and accuracy was assessed on the basis of the true values at these points.The following strategies were compared Startegy 1 : Using a Gaussian process interpolation of the physical observation alone, taking into account measurement errors, but making no use of the Gaussian plume model. data sets, posterior means and variance of z (x) were calculated for all the 670 unobserved points, and accuracy was assessed on the basis of the true values at these points. The following strategies were compared Startegy 1 : Using a Gaussian process interpolation of the physical observation alone, taking into account measurement errors, but making no use of the Gaussian plume model. Startegy 2 : Using Bayesian calibration and model inadequacy correction as described here. data sets, posterior means and variance of z (x) were calculated for all the 670 unobserved points, and accuracy was assessed on the basis of the true values at these points. The following strategies were compared Startegy 1 : Using a Gaussian process interpolation of the physical observation alone, taking into account measurement errors, but making no use of the Gaussian plume model. Startegy 2 : Using Bayesian calibration and model inadequacy correction as described here. Startegy 3 : Using the Gaussian plume model with plug-in input parameters. The physical data are not interpolated in any way. Instead, selection of the input parameters is done by minimizing the sum of squared dierences between the model and the data.

  is a measure of a physical system ζ and is subject to error

	Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion
	Plan Notations
	1 Introduction
	2 Taxonomy of uncertainties in computer models
	Bayesian calibration of computer models. 3 Bayesian inference for functions
	Gaussian Process
	Modelling issues
	4 Bayesian calibration
	Calibration and variable inputs
	Model
	Posterior distribution
	Estimating hyperparameters

z z = ζ(X ) + ε

  has a Gaussian Process distribution with mean function m(•) and covariance function c (•, •).

	Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Discussion Example	Gaussian Process Modelling issues Gaussian Process Modelling issues Gaussian Process Modelling issues
	Necessary conditions Modelling m(•) and c(•, •) Modelling
	m(•) may be any function on X We can separate the mean and covariance structures by writing
	f (x) = m(x) + e (x)
	c (•, •) must have the property that for every n = 1, 2, . . . the
	variance-covariance matrix of f (x1), f (x2), . . . , f (xn) (comprising elements where e(x) is a zero-mean GP with covariance function c(•, •).

  = (t1, . . . , tq2) be the vector of calibration inputs. η(x, t) the corresponding code output.Note that the unknown value θ of the unknown calibration inputs is distinct from the (known) value t that is set as inputs when running the model.

	Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion	Calibration and variable inputs Model Posterior distribution Estimating hyperparameters Calibration and variable inputs Model Posterior distribution Estimating hyperparameters Calibration and variable inputs Model Posterior distribution Estimating hyperparameters
	Inputs Inputs Data and Real process
	Let
	Let Let

x = (x1, . . . , xq1) be the vector of variable inputs. t = (t1, . . . , tq2) be the vector of calibration inputs. η(x, t) the corresponding code output.

x = (x1, . . . , xq1) be the vector of variable inputs. t

  i includes any residual variation as well as observation error. i includes any residual variation as well as observation error. Since it's not possible to have replication of observations in circumstances where variable inputs AND all unrecognized conditions were the same, it's not possible to separate the two sources of uncertainty. i includes any residual variation as well as observation error. Since it's not possible to have replication of observations in circumstances where variable inputs AND all unrecognized conditions were the same, it's not possible to separate the two sources of uncertainty.

	Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Observation error Strictly speaking, Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Observation error Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Observation error iid ∼ N (0, λ) Strictly speaking, Proposition : e i Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Prior information about unknown functions Calibration and variable inputs Model Posterior distribution Estimating hyperparameters Calibration and variable inputs Model Posterior distribution Estimating hyperparameters Calibration and variable inputs Model Posterior distribution Estimating hyperparameters Calibration and variable inputs Model Posterior distribution Estimating hyperparameters Prior information about both unknown functions η(•, •) and δ(•) are represented Strictly speaking, Introduction by Gaussian Processes
	Remark : Assumption of normality may require a transformation of the raw data

e e e

  to integrate β out to give p(θ, φ|d) However p(θ, φ|d) is a highly intractable function of φ Authors propose to derive plausible estimates of the components of φ and act as if these were xed. Use only the code output y to estimate the hyperparameter ψ1 of c1{(•, •), (•, •)} ⇒ Use z to estimate ρ, λ and the hyperparameters ψ2 of c2(•, •), having now xed ψ1.

	Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion	Calibration and variable inputs Model Posterior distribution Estimating hyperparameters Calibration and variable inputs Model Posterior distribution Estimating hyperparameters Calibration and variable inputs Model Posterior distribution Estimating hyperparameters Calibration and variable inputs Model Posterior distribution Estimating hyperparameters Calibration and variable inputs Model Posterior distribution Estimating hyperparameters Calibration and variable inputs Model Posterior distribution Estimating hyperparameters Calibration and variable inputs Model Posterior distribution Estimating hyperparameters
	Posterior Analysis Posterior Analysis Posterior Analysis Hyperparameters Remaining uncertainty
	Notations Expectations Expectations
	Set of points at which the code output y are available : Let H1(D1) denote the matrix with rows h1(x * Hence E (d|θ, β, φ) = m d (θ) = H(θ)β H(θ) = " H1(D1) 0 ρ H1{D2(θ)} H2(D2) ⇒ Second stage : where The model now take explicitly into account all the sources of uncertainty « previously identied. However, the compromise proposed here is to not account fully for all these It is possible Fisrt stage : sources.
	Arnaud Bensadoun	Bayesian calibration of computer models

D1 = {(x * 1 , t1), . . . , (x * N , t N )}

Set of points for the observations z of the real process : D2 = {x1, . . . , xn } Augmenting each points of D2 by the calibration parameter θ, we dene : D2(θ) = {(x1, θ), . . . , (xn, θ)}

  Generally, developers of computer model will have given concrete physical meaning to the calibration inputs, but the true value of these physical quantities do not necessarly equate to θ It may be that the physically true value gives a worse t and less accurate future prediction than another value.⇒ It is dangerous to interpret the estimates of θ (obtained by calibration) as estimates of the true physical values.

	Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion	Data and model Results
	Results Discussion Discussion
	Sensitivity to modelling assumptions
	True parameter values
		RMSEs for the following values of n
	Strategy	n = 10 n = 15 n = 20	n = 25
	1	0.75	0.76	0.86	0.79
	2	0.42	0.41	0.37	0.36
	3	0.82	0.79	0.76	0.66
	For comparison, RMSE=0.84 is obtained using the code with input parameters
	xed at their prior mean
		Arnaud Bensadoun	Bayesian calibration of computer models

  Computer models with multivariate outputs have not been considered here.However, eective calibration should ideally use all available code outputs and corresponding physical measurement.The use of multiple code outputs is an ongoing topic of research (insight can be found in Higdon et al.2004, 2008)

	Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion Introduction Taxonomy of uncertainties in computer models Bayesian inference for functions Bayesian calibration Example Discussion
	Discussion Merci pour votre attention
	Des questions ?
	Multivariate output
	La vie c'est comme une chaîne de Markov, on ne sait jamais
	sur quoi on va tomber !
	Arnaud Bensadoun	Bayesian calibration of computer models

Gaussian Process f (•) is seen as an unknown function.Arnaud BensadounBayesian calibration of computer models

f (•) is a random function. GP represent prior knowledge about f (•).

f (•) is a random function. GP represent prior knowledge about f (•).non-Bayesian framework f (•) is drawn randomly from some population of functions.GP represent the distribution of functions in that population.Arnaud BensadounBayesian calibration of computer models

f (•) has a Gaussian Process distribution if for every n = 1, 2, 3, . . . the joint distribution of f (x1), . . . , f (xn) is multivariate normal for all x1, . . . , xn ∈ X

Arnaud BensadounBayesian calibration of computer models

p(θ, β, φ) = p(θ)p(φ)

Data and model

Results

Discussion

Data and model

Results

Discussion

Data

Data used here comes from an accident at the Tomsk-7 chemical plant in 1993 (detail in Shershakov et al., 1995).

We consider the deposition of ruthenium 106 ( 106 Ru).

Variances

Dene V1(D1) to be the matrix with (j, j ) element c1{(x * j , t j ), (x * j , t j )} Dene V1{D2(θ)} and V2(D2) similarly Let C1{D1, D2(θ)} be the matrix with (j, i )

Full joint postrior distribution

We can now obtain the full joint postrior distribution

Note that we have explicitly shown dependance on θ but m d (θ) also depends on β and ρ whereas V d (θ)


domain: shs sdv

text:
Keywords: wastewater management, septic system, sewer system, residential water pollution, land use, commuting, polycentrism Q2, R1, R2, R3

We study the interplay between residential location choice, sprawl and water quality. We propose an urban economics model of a, first, monocentric, then, polycentric city with two different residential areas : sewer-serviced suburbia, with small residential lot size, and exurbia where wastewater management is individual and on-site and residential lots are larger to accomodate sanitary requirements. Sewer and septic are also characterized by different abatement efficiencies. Within this framework, where development is assumed contiguous, we analyse how wastewater management and commuting costs impact on residential location choice and consequently on sprawl and water quality. According to the abatement efficiency gap between sewer and septic technologies, improving water quality may be achieved at the expense of higher or lower sprawl. The extension to the polycentric setting allows introducing heterogeneities in wastewater and commuting costs that illustrate how independant policy makers may impact the sprawl and water quality of the entire metropolis.

Introduction

An increasing share of the population lives in cities and their close hinterland. It is then crucial to understand how urban development, and the form this development takes, impact on the environment. Indeed, urban growth is associated with numerous damage to the environment : the transport of people and goods contributes to local air pollution and greenhouse gaz emissions [START_REF] Kahn | Green cities : urban growth and the environment[END_REF][START_REF] Glaeser | The greenness of cities: Carbon dioxide emissions and urban development[END_REF]Hensher, 2002); the increase in impervious surfaces alters water ecosystems functioning [START_REF] Lohse | Forecasting relative impacts of land use on anadromous fish habitat to guide conservation planning[END_REF]; land fragmentation is detrimental to biodiversity [START_REF] Merenlender | Exurban development influences woodland bird composition[END_REF][START_REF] Tannier | Assessing the capacity of different urban forms to preserve the connectivity of ecological habitats[END_REF]; firms contribute to air pollution [START_REF] Hwang | The Effects of Pollution Taxes on Urban Areas with an Endogenous Plant Location[END_REF]; etc. Furthermore, urban development can take different forms that don't impact these issues in the same way. Polycentric urban structures are becoming a prominent feature of the landscape worldwide, and the impacts of the decentralization of jobs and people within metropolises on the patterns of sprawl and greenhouse gaz emissions are not straightforward [START_REF] Gaigné | Are compact cities environmentally friendly?[END_REF][START_REF] Legras | Urban form and sustainable development[END_REF]. Beyond the organisation of productive activities, the nature of residential areas also matters in assessing the environmental impacts of urban growth. [START_REF] Heimlich | Development at the urban fringe and beyond : impacts on agriculture and rural land[END_REF] state that recent land development in the US takes two major forms : the continuous accretion of urban growth at the fringe of urban areas, in suburbia, and the multiplication of larget-lot housing beyond the urban fringe and in nonmetropolitan counties, exurbia. Suburban and exurban development represent two fundamentally different types of growth : the former relies on access to sewer and small residential lots (< 1 acre), the latter is not bound to collective sewer and hence relies on septic systems and larger lot size (5 -40 acres per unit) [START_REF] Heimlich | Development at the urban fringe and beyond : impacts on agriculture and rural land[END_REF]Newburn and Berck, 2011a). In a recent paper, Newburn and Berck (2011a) show the crucial role that wastewater technology choice plays on the type of development occuring at the urban fringe : additional costs entailed by municipal sewer extension may significantly reduce the reservation price of agricultural land in suburban use, while exurban development can occur immediately and entails only the onsite conversion costs. The bid rent of households with a preference for large lots may then exceed agricultural landowner's reservation price for future suburban use in what they coin an exurban "feasible zone" where development leapfrogs. This theoretical work establishes a strong link between wastewater technology choice and sprawl.

This link was indeed invoked by Maryland Governor O'Malley when advocating a "septic bill", adopted last year. This piece of legislation aims at limiting the development of new residential lots based on septic systems, especially in rural areas. Its justification is twofold : to reduce wastewater-related polluting inputs to the Chesapeake Bay and to curb urban sprawl. Relying on wastewater management as an indirect candidate to impact on sprawl appears appealing, given the seemingly lack of efficiency of direct instruments to limit sprawl (Newburn and Berck, 2011b;[START_REF] Harrison | Smart growth and the septic tank : wastewater treatment and growth management in the Baltimore region[END_REF]). An empirical examination by Newburn and Berck (2011b) shows that policies aimed at limiting urban growth are more effective to manage suburban rather than exurban development, with the potential adverse effect of encouraging development in exurbia, hence higher reliance on septic systems and a greater hold on land ressources. In the same vein, [START_REF] Harrison | Smart growth and the septic tank : wastewater treatment and growth management in the Baltimore region[END_REF] show that septicbased development has increased significantly in the Baltimore region since the passage of Maryland's Priority Funding Area, a legislation aiming at focusing new development where there is existing infrastructure.

In this paper we analyse how wastewater management and commuting costs affect urban spread1 and water quality to identify to what extent strategies undertaken to reduce one issue (e.g. altering the wastewater bill) may impact on the other one. We show that, depending on the relative abatement efficiencies of septic and sewer systems, strategies that improve water quality may either increase or decrease urban spread.

Key to the proposition of a joint management of water quality and sprawl is the assumption that septic systems are associated with a higher contribution to pollution load than collective sewer ones. Indeed, a number of papers have raised awareness on the detrimental impacts of poorly designed, sitted or maintained septic systems on water quality, especially groundwater [START_REF] Arnade | Seasonal correlation of well contamination and septic tank distance[END_REF][START_REF] Borchardt | Septic system density and infectious diarrhea in a defined population of children[END_REF][START_REF] Moore | Lake eutrophication at the urban fringe, USA[END_REF]. Sewer systems also contribute to water pollution, and another range of papers assess the environmental performance of different types of collective sewer systems [START_REF] Lassaux | Life cycle assessment of water from the pumping station to the wastewater treatment plant[END_REF].

There lacks studies that compare the impacts of sewer and septic systems on water quality; indeed they are not homogeneous in this respect : the former mainly impacts surface water while the latter has a more direct effect on groundwater; the effluents they generate are not homogeneous in type and quantity; septic failure is diffuse and develops over time without being noticed while sewer-related issues are more ponctual and quickly dealt with. Consequently, no a priori assumption is made regarding the efficiency gap in this paper, since it drives most of the results. Assessing which conditions on abatement apply to which context is an empirical matter outside the scope of this paper.

The paper is organised as follows. Section 2 presents the model of a monocentric city with sewer-serviced suburbia and septic-dependant exurbia. In section 3 we derive the spatial equilibrium, and analyse how the costs of wastewater management and commuting affect the population distribution over suburbia and exurbia. In section 4, we assess how these costs impact on two environmental variables associated with urban development : urban spread and water quality. Section 5 extends the analysis to a polycentric context and introduces heterogeneous commuting and wastewater costs. Section 6 concludes.

The Model

We first set our analysis within the standard framework of a monocentric city. It comprises a central business district (CBD), where all firms are located and do not consume land, and two types of residential rings : suburbia, where access to a sewer system is available and residential lot size h s is fixed and small, and exurbia, where there is no access to sewer : residential lot size h e is larger to accomodate the sanitary regulation on individual septic systems. Space has one dimension, x ∈ R and is symmetric to x = 0 : without loss of generality, we will focus on the right hand side of the city where x ≥ 0. The borders of the various areas are determined endogenously.

The city comprises a continuum of N individuals, exogeneously determined since we consider a closed city. The agents consume a level q(x) of an aggregate consumption commodity and an amount h s or h e of living space, depending if they reside in suburbia or exurbia, with h s < h e . They commute from their place of residence to the CBD at a unitary cost t > 0 per distance travelled. They generate water pollution e(x) as a by-product of consumption. Wastewater infrastructure abate pollution to a different extent according to the system in use, sewer or septic. The amount of pollution emitted by a household located at x is as follows :

e(x) = γ i q(x) i ∈ {s, e} (1)
Abatement efficiencies are fixed but different according to the system in use. Wastewater systems are also characterized by different installation and monitoring costs. Septic only involves fixed costs C e ≥ 0 while sewer entails fixed costs and a variable component :

C s (x) = a + bq(x), a, b ≥ 0.
Wastewater infrastructure is assumed to be produced, installed and monitored by firms established outside the metropolis : refer to [START_REF] Quaas | Pollution-reducing infrastructure and urban environmental policy[END_REF] for a model with endogeneous infrastructure production.

On the production side, we assume that the aggregate composite commodity is produced in totality by the firms established in the CBD, so that a firm profit function is given by :

Π = pQ -φw (2)
with p the price of the commodity, Q the total production, w the wage paid to workers and φ the number of labor units necessary to produce Q.

Hence, we consider identical agents which sort between suburbia and exurbia. Refer to (Newburn and Berck, 2011a) for the detailed analysis of how agricultural land is converted into either suburban use in the contiguity of the existing city limit, or exurban use in a "feasibility exurban zone" where households that highly value space induce leapfrog development for residential use with a septic system and a large lot size required for sanitary purposes. Here, we assume that exurban development is contiguous but keep the assumption that exurbia is associated with larger lot size and recourse to septic wastewater management, while in suburbia connection to sewer is guaranted and lot size is smaller.

This allows us to quantify the environmental impacts of urban development, according to households' choice of residence in suburbia or exurbia, analytically.

Equilibrium monocentric structure

To analyse the environmental impacts of wastewater management, we derive the equilibrium city structure in the case of a closed monocentric city. N households maximise their utility, given their budget constraint, by chosing their composite commodity consumption level and their residential location; firms maximise profits. In equilibrium, all markets clear, and all households enjoy the same utility level.

We use a quasi-linear utility function : U(q(x), h(x)) = q(x) + v ln h(x) with v > 0. An agent's budget constraint depends on his location in suburbia (subscript s) or exurbia (subscript e) :

BC s : w = tx + R s (x)h s + pq s (x) + a + bq s (x)
BC e : w = tx + R e (x)h e + pq e (x) + C e where R s and R e are the bid rents functions. As agents settle continuously along the city line, we can express the values of the borders between suburbia and exurbia, x s , and between exurbia and the agricultural hinterland, x e , as follows, where β, 0 < β < 1, is the share of population in suburbia :

x s = Nβh s and x e = x s + N(1 -β)h e

To derive the equilibrium repartition of population, we equate the equilibrium utilities of agents settled in suburbia and of those settled in exurbia. To do so, we derive the bid rent functions in each area by recognizing that, due to the fixed lot size assumption, there is equality of urban costs (commuting plus rent costs) within each area. Absentee landowners allocate land to the highest bidder. The opportunity rent is the agricultural rent R a ; it is set equal to zero for simplicity. The rent function is :

R(x) = max{R s (x), R e (x), R a }
where the bid rent functions are given by the following expressions :

R s (x) = t h s (x s -x) + t h e (x e -x s ) and R e (x) = t h e (x e -x)
We plug these bid rent functions in the budget constraints to obtain the equilibrium consumption values in each zone 2 . Using these equilibrium values and the definitions of x s

and x e , we equate the equilibrium utility fonctions in exurbia and suburbia to derive the value of β * , the equilibrium share of population in suburbia :

β * = 1 - p t( xe -xs ) v ln h e h s - C e -(a + bq * e ( xs ) p + b (3)
where xe = x e (β = 0) = Nh e is the eastern limit of a totally exurban city and xs = x e (β = 1) = Nh s is the eastern limit of a totally suburban city. It can be shown that 0 < β * < 1 under conditions (C1) and (C2) described in Appendix 1.

Increasing the wage and the preference for larger lots, or reducing the transport cost tend to reduce the share of population settled in suburbia : by relaxing the budget constraint or increasing the utility from the consumption of a larger lot, it allows more agents to settle far from the CBD in the exurban area.

Wastewater management costs also have an impact on residential location choice (see Appendix 2) : any increase of the cost relative to septic tanks has the effect of increasing the share of population in suburbia, hence of decreasing the population in exurbia. This alters the borders of the city, moving x s to the east and x e to the west (since h s < h e ). On the contrary, raising the costs of sewer increases settlement in exurbia. 2 The equilibrium consumption values are : q * s = w-a-tx s -t(x e -x s )h s /h e p+b and q * e = w-C e -tx e p .

Due to their effect of the equilibrium repartition of the fixed population between exurbia and suburbia, wastewater management costs ought to impact on the environmental performance of the metropolis, that we grasp through two variables : the total width of the city to assess the amount of natural or agricultural land converted to urban development and the aggregate residential pollution load.

Land conversion and water quality in a monocentric city

This section is devoted to the analysis of the impact of wastewater and commuting costs on the two environmental variables that we focus on in this paper : city width and residential pollution load. We chose to analyze these two costs since they are both policy bases of choice to manage the environmental impacts of urban development3 .

The total width of the city is the location of the eastern border x e :

S = x e = Nβ * h s + N(1 -β * )h e (4)
Residential pollution load is expressed as the sum over all agents of their pollution emissions, proportional to their consumption levels :

E = x s x 0 γ s h s q(x)dx + x e
x s γ e h e q(x)dx (5)

Proposition 1. Higher sewer costs, lower septic costs or higher commuting costs increase the spread of the city.

Proof. See Appendix 2.

Proposition 1 follows from the clear-cut impact of the cost parameters on β * : cheaper septic systems induce more agents in exurbia, where lot sizes are larger than in suburbiaconsequently more land is consumed by an agent choosing to live in exurbia compared to an agent settling in exurbia. Also, by tightening the budget constraints of all households, increasing commuting costs induce the city to shrink with more people settling in suburbia where the lot size is lower than in exurbia, and from where distances to work are smaller.

The impacts of wastewater-related and commuting costs on S only channel through a change in the repartition of the population between suburbia and exurbia. In the case of E, changing the costs affects the repartition of population, which in turn affects the equilibrium consumption level, from where the pollution emission is generated. Consequently, for each parameter a, b, C e and t there are three effects at play, so that the aggregate impact is not as clear-cut.

Proposition 2. Raising wastewater and commuting costs affects aggregate water pollution in three ways :

-a consumption effect that is always negative : increasing any type of cost decreases the level of consumption, hence the amount of pollution emitted, whatever the residential location;

-an exurbia effect stemming from the change of location of x s and x e : it is positive for a and b, and negative for C e and t;

-a suburbia effect coming from the change of location of x s : it is of opposite sign as the exurbia effect.

The aggregate impact depends on the abatement efficiency gap. When sewer is highly efficient or septic poorly efficient, any alteration of the costs that induce more agents to settle in suburbia result in a lower aggregate pollution load. When the opposite applies, reducing pollution entails inducing agents to settle in exurbia. The efficiency gap threshold depends on which cost is analyzed.

Proof. The equilibrium aggregate pollution load is given by the following expression :

E * = γ s h s q * s x * s + γ e h e q * e (x * e -x * s ) (6)
where the equilibrium consumption levels and area borders depend on the cost parameters.

Consequently, the impact of a cost parameter j ∈ {a, b, C e , t} can be broken down into the following terms :

∂E ∂ j = γ s x * s h s ∂q * s ∂ j + γ e (x * e -x * s ) h e ∂q * e ∂ j + γ s q * s h s ∂x * s ∂ j + γ e q * e h e ∂x * e ∂ j - ∂x * s ∂ j (7)
The first two terms are negative for all cost parameters : it is the consumption effect that captures a decrease in consumption, hence in pollution emitted, when the budget constraint is tightened by the increase of a cost.

The third and fourth terms capture the impacts due to a change of repartition of the population between suburbia and exurbia. They are of opposite sign for each parameter :

increasing the cost of any type of wastewater management induces more agents to settle in the area where the other system prevails; while increasing commuting costs unambiguously induces more agents to settle in suburbia.

The aggregate impacts can be signed after rearranging the terms (see Appendix 2). Raising the costs of sewer increases pollution load if the following applies :

γ s γ e ≤ γ a = 1 - pv ln h e h s + tN(h e -h s ) -pv ln h e h s + tN(h e -h s ) + pq * s ( xs ) + p p+b (C e -a -bq * e ( xs )) (C3)
In other words, if the ratio is low, with a very efficient sewer and poorly efficient septic, increasing the cost of sewer has the detrimental impact of increasing aggregate pollution load : while both consumption levels decrease, the transfert of population from suburbia to exurbia induce more use of a poorly efficient system, leading to a decrease in water quality. For a high ratio, increasing the cost of sewer induces less pollution emitted.

Raising the cost of septic increases the aggregate level of water pollution if the following applies :

γ s γ e ≥ γ C e = 1 - v ln h e h s q * s ( xs ) (C4)
Under an efficient septic technology, and a less efficient sewer one, increasing septic costs may induce a higher equilibrium level of residential pollution. There needs to be a sufficient gap in depolluting efficiencies between the two technologies to ensure that water quality is improved when the cost of septic increases, reducing both consumption levels and the share of population settled in exurbia.

Raising commuting costs increases the aggregate level of water pollution if the following applies :

γ s γ e ≥ γ t = 1+ t 2 N 2 h 2 s (h e -h s ) + (w -a)[(p + b)(vp ln h e h s -C e ) + wb + pa] t 2 N 2 h s (p(h e -h s ) + bh e ) -vp ln h e h s [(p + b)(vp ln h e h s -C e ) + wb + pa] (C5)
As for septic costs, since increasing transport costs induce more agents to settle in suburbia, it also leads to more pollution emitted if the abatement efficiency gap that septic is highly efficient as compared with sewer.

From Propositions 1 and 2 it appears that according to the abatement efficiency gap between sewer and septic technologies, the environmental objectives of reducing sprawl and residential pollution load, pursued by a policy maker, may appear conflicting.

Corollary 1. Implementing a transport tax/subsidy or a wastewater cost tax/subsidy is compatible with a reduction of both land conversion and residential pollution over only a range of abatement efficiency ratios. Outside of these ranges, the policy maker faces conflicting environmental objectives.

Proof. Proposition 1 establishes that the impact of wastewater and transport costs on land conversion are monotonous. Proposition 2 provides the threshold for each cost parameter that defines the sign of their impact on aggegate water quality. Consequently, it is easy to derive the following :

• decreasing a or b reduces both land conversion and pollution load for γ s /γ e ∈ [0..γ a ];

for γ s /γ e > γ a , increasing a or b reduces pollution load at the expense of increased land conversion;

• increasing C e reduces both land conversion and pollution load for γ s /γ e ∈ [0..γ C e ];

for γ s /γ e > γ C e , decreasing C e reduces pollution load at the expense of increased land conversion;

• increasing t reduces both land conversion and pollution load for γ s /γ e ∈ [0..γ t ]; for γ s /γ e > γ t , decreasing t reduces pollution load at the expense of increased land conversion.

Policy design being outside the scope of this paper, this Corollary only aims at pointing out some situations in which pursuing one environmental objective may be carried out at the expense of another one through the implementation of a unique instrument. Considering the environmental issues at stake in this paper, water quality often pertains to higher decision levels than the city, such as regional water institutions : environmental quality standards are defined and monitored along the whole water bodies or at their outset.

Sprawl management is a more local issue. Consequently, an increase of the cost of septic to comply with tighter water quality regulation would have the added beneficial impact of decreasing sprawl for an abatement efficiency ratio below γ C e . Above this ratio, reducing water pollution load would necessitate a decrease in septic costs, and such a policy would entail more land conversion.

The next section extends the analysis of the environmental impacts of the urban form to the increasingly prominent feature of polycentric urban structures [START_REF] Anas | Urban Spatial Structure[END_REF]. This setting allows refining the analysis of potential interactions between policy instruments.

Land conversion and water quality in a polycentric city

In addition to the CBD, the city comprises two sub-centers (SBD) where some jobs are provided and around which households can settle either in a suburbian ring or an exurban area. Besides the empirical interest in polycentric structures worldwide, they also constitute an interesting research topic in particular because their environmental performance is still not fully understood [START_REF] Gaigné | Are compact cities environmentally friendly?[END_REF][START_REF] Legras | Urban form and sustainable development[END_REF]. Given our focus on water quality in this paper, polycentricity allows introducing a certain range of heterogeneities that may enrich the analysis of the relationship between sprawl and residential pollution load.

Consistent with the previous section, starting from a homogeneous case where all cost parameters are homogenous within the whole urban structure, we introduce heterogeneity on the following parameters : septic costs, sewer costs and commuting costs. This approach may constitute a first step in the analysis of strategic interactions between decisions makers within a urban structure, by providing the environmental consequences of a unilateral decision by the policy maker of the secondary urban area. As shown below, the consequences concern not only the area he is in charge of but also the whole urban structure.

Note that no explicit strategic interaction is accounted for here; our approach is positive by treating the question of how urban spread and water quality would be affected over the whole structure, and within each primary and secondary urban area, by a decision taken in the secondary city.

The Model

The firms established in the SBD have the same production function as those established in the CBD; however they incur fixed communication costs K > 0 to access some type of services that remain in the CBD preferably, such as banking, insurance, airports, etc. [START_REF] Cavailhès | Trade and the structure of cities[END_REF]. At the equilibrium, firms' profits are nul, so that the following relation links salary levels, where w s denotes the wage paid by SBD-established firms :

w -w s = K φ (8)
The wage difference between CBD and SBD based firms compensates for the communication costs borne by the latter. Lower wages in the SBD are consistent with lower urban costs incurred by households settled around the SBD.

As in the monocentric case, when choosing their residential locations agents decide whether to settle in suburbia or exurbia, leading to the equilibrium shares of agents in suburbia β I and β II , subscripts refering the the primary or secondary city. In this polycentric setting, they also have to choose between the primary and the secondary centers, resulting in a primary/secondary repartition share α : the higher α, the lower the polycentric level of the metropolis. The resulting urban structure has the characteristics illlustrated in Figure 1, where focus is given on the righ-hand side of the urban structure. The borders of the various areas are determined endogenously according to the equilibrium shares values.

Figure 1 about here.

Apart from the salary, all other parameters are assumed equal in the primary and the secondary urban areas; some of these assumptions are relaxed over the course of the section.

Equilibrium urban structure and environmental impacts with homogeneous parameters

The resolution strategy is the same as in the monocentric case, with the addition of spatial equilibrium conditions between the primary and the secondary areas that allow deriving the equilibrium polycentrism degree α. To do so, besides the budget constraints established in section 4 for residents of the primary city, we also consider the following for secondary dwellers (refer to Appendix 3 for the expressions of the bid rents) :

BC sII : w s = t|x -x 0II | + R sII (x)h s + pq sII (x) + a + bq sII (x) BC eII : w s = t|x -x 0II | + R eII (x)h e + pq eII (x) + C e
Equating the utility levels over the whole urban structure, we derive the following equilibrium population shares :

α * = 1 3 1 + 2(w -w s ) th s N (9) β * I = 3ph s ∆U 1 (h e -h s )(2(w -w s ) + th s N (10) β * II = β * I -(1 -β * I ) 3(w -w s ) th s N -(w -w s ) ( 11
)
where

∆U I = U I s (β I = 1) -U Ie (β I = 0)
is the is the utility difference between a totally suburbian city and a totally exurban city. To ensure that the share of population in suburbia is positive, this difference has to be positive, reflecting lower urban costs.

Due to our assumption of homogeneous wastewater management costs, they have no impact on the primary/secondary equilibrium repartition of the population. However, they impact on the suburban/exurban population shares around each BD : increasing sewer costs induce more settlement in exurbia; and vice versa.

Transport cost have an effect on all types of population shares : higher commuting costs lead to a more monocentric metropolis with more suburbanites. The mechanisms at stake are the same as in section 4 for the suburbia/exurbia residential choice. Due to the wage difference between the primary and the secondary centers, the budget constraint is less tightened by an increase in commuting costs for agents that settle around the CBD.

Compared to section 4, the environmental impact of the various costs that shape the urban form result from a residential choice that not only concerns living in suburbia or exurbia, but also living and working around the CBD or the SBD. Within the restricted framework of this paper, this choice only reflects the interplay between commuting, consumption, housing and wastewater management costs, some of which are assumed homogeneous over the metropolis or certains areas for the sake of readability of the results.

The total width of the city is now expressed as the location of the eastern border x eII :

S p = x eII = Nα * [β * I h s + (1 -β * I )h e ] + N(1 -α * )[β * II h s + (1 -β * II )h e ] ( 12
)
Proposition 3. Decreasing sewer costs, increasing septic costs, or increasing commuting costs, leads to less aggregate land conversion. Less land is converted around each business district, with less exurban conversion but more suburban conversion.

Proof. Refer to Appendix 2.

Given the clear impact of costs on population shares, it is not surprising that their impact on land conversion is also clear-cut. From proposition 3, we know that to achieve the objective of less aggregate land converted, the policy maker may have recourse to any of the following policies : taxing transport or septic costs, or subsidizing septic costs. In all cases, this induces less land converted around each business district. Furthermore, this gives rise to more land converted to suburbia, but less converted to exurbia, around each BD.

Residential pollution load is expressed as the sum over all agents of their pollution emissions, proportional to their consumption levels :

E p = γ s h s x s x 0 q s (x)dx + 2 x 0II
x sIIL q sII (x)dx + γ e h e x e

x s q e (x)dx + +2

x sIIL

x e q eII (x)dx ( 13) Proposition 4. Increasing septic management costs increases pollution load from suburban areas and reduces that from exurban areas. The impact on total, primary and secondary pollution loads depends on the abatement efficiency ratio between sewer and septic in a comparable manner as in the monocentric setting.

Proof. Refer to Appendix 3.

Since septic costs only affect the suburban/exurban population shares, and that they are a fixed component of the budget constraint, their impact in terms of water quality is quite straightforward in a polycentric setting. Indeed, they always increase the share of agents in suburbia, hence their total emission load, and reduce that of exurbia. The resulting effect depends on the abatement efficiencies of the wastewater technologies.

Proposition 5. Increasing sewer management costs always increases pollution load from suburban areas. Their impact on exurban pollution load depends on the consumption level gap between exurban and suburban households : when it is high, increasing the cost of sewer increases the pollution load from exurbia due to a transfert of population from suburbia; when it is low, an increase in the cost of sewer has the impact of reducing pollution load from exurbia due to a high reduction in consumption for all households. The impact on total pollution load, and on primary and secondary pollution loads, depends on the abatement efficiency ratio between sewer and septic.

Proof. Refer to Appendix 3.

Increasing a always decreases the suburban pollution load since it decreases both the consumption level and the suburban population. A double effect is at play to explain the variation of E e with a : a reduced consumption level of suburban households due to an increase of their wastewater bill which translates into less exurban consumption to comply with the spatial equilibrium; and a population transfert effect which increases the population emitting in exurbia. When the consumption levels gap between exurbia and suburbia is sufficiently high, increasing a leads to a higher level of exurban pollution.

Proposition 6. Increasing commuting costs always decreases pollution load from exurban areas. Their effect on suburban pollution load depends on the commuting costs level and how they affect both consumption and suburbia effects. The impact on total, primary and secondary pollution loads depends on the abatement efficiency ratio between sewer and septic.

Proof. Refer to Appendix 3.

Increasing t always decreases the exurban pollution load since it decreases both the consumption level and the exurban population. A double effect is at play to explain the variation of E s with t : a reduced consumption level and a population transfert effect which increases the population emitting in suburbia. The tradeoff between these effects explain the positive or negative effect on the suburban pollution load.

Relaxing some homogeneity assumptions

In this section we address the impact of introducing heterogeneity in some of the model's parameters, heterogeneity that could result from a policy maker's decision4 . The parameters used for the simulations are stipulated in Appendix 4.

Let's consider first a change in the septic management costs in the SBD. The most recent French legislation on individual wastewater systems introduces mandatory controls by the municipal authorities in the areas which have been designated as not covered by a collective wastewater system. These mandatory controls have increased the cost of relying on septic. Furthermore, costs have proven to vary between municipalities. Figure 2 illustrates the impact that an increase in secondary septic costs have on the urban structure, and on the width and pollution load of various areas : whole urban structure, primary and secondary urban areas, and total suburbia and exurbia.

Figure 2 about here.

First we can show that altering secondary septic costs has no impact on the primary city nor on the polycentrism degree. Consequently the environmental impacts of this cost alteration will only channel through a change of suburban/exurban repartition in the secondary city (see Section 4). Depending on the value of the abatement efficiency ratios, a policy maker willing to reduce the pollution load from its residents will either increase the septic cost, with the added benefit of less land converted, or decrease this cost at the expense of more land converted. Since the primary city is not affected, these results also apply to the whole urban structure. Figure 3 illustrates how an alteration of sewer costs in the secondary suburban area affects the urban structure and its environmental performance. Contrary to the previous case, the whole city structure is affected : altering secondary sewer costs alters both suburban/exurban and primary/secondary population shares. Lowering this cost leads to a more polycentric structure, with more suburbanites around the SBD and less around the CBD.

Increasing secondary sewer costs increases land conversion in around both cities towards more exurbia. According to the abatement efficiency ratio, this leads to either less, or more, pollution load in both cities. Consequently, a policy maker that would subsidize sewer costs to reduce the contribution of its city to water pollution (assuming that sewer is more efficient than septic), would also induce a decrease in pollution load from the primary city; furthermore, both city widths would be reduced.

Figure 4 about here.

Finally, Figure 4 shows the impacts of heterogeneous commuting costs. Increasing the per kilometer cost of commuting in the secondary city induce a more monocentric and suburban urban structure. While this effectively reduces the secondary city width, this also leads to a more spread primary city, which is not the focus of policy design5 . Furthermore, both pollution loads are reduced, whatever the efficiency ratio.

Concluding remarks

In this paper, the link between water quality and urban spread is addressed through the analysis of household residential location and wastewater technology choice. We do so within the simplified framework of an urban economics model of a monocentric (and then polycentric) city, with two fixed, but different, residential lot sizes according to the location in suburbia or exurbia. This assumption of fixed but heterogenous lot sizes is a departure from standard urban economics models, where the lot size is either endogenously determined or unique over the whole urban structure. It allows the derivation of analytical results and still describes the stylized fact of decreasing lot size with distance from the city center [START_REF] Legras | Urban form and sustainable development[END_REF].

We derive the impact of wastewater management and commuting costs on the shape of the city, namely the repartition of population between suburbia and exurbia, to grasp to which extent residential location choice is affected by the commuting and wastewater bills. Then we show how these costs also have the potential to alter the environmental performance of the city, with respect to urban spread and water quality : reducing city width doesn't necessarily imply a reduction in residential pollution load. Section 4 puts in perspective the crucial role of the abatement efficiency gap between sewer and septic technologies in the understanding of the link between urban spread and water quality.

The extension of the analysis to the case of a polycentric urban structure allows introducing heterogeneity with respect to some parameters. This approach constitutes a first step in the analysis of strategic interactions between decisions makers within a urban structure, by providing the environmental consequences of a unilateral decision by the policy maker of the secondary urban area. Once again, no explicit strategic interaction is accounted for here; we simply investigate how urban spread and water quality are affected over the whole structure, and within each primary and secondary urban area, by a decision taken in the secondary city.

This analysis rests on a series of simplifying assumptions that would need to be adressed in future research. First, we consider contiguous settlement in both suburbia, as is usually the case, and exurbia, which is less standard. This allows us to derive the environmental performance of the urban strutcure analytically. Second, sewer infrastructure is held externally : we don't analyse the provision of infrastructure per se and we don't introduce any heterogeneity within types of technology. Indeed, in the case of both sewer and septic systems, different technologies exist with different levels of performance regarding water pollution abatement. An extension of this work could endogeneize infrastructure provi-sion, in the vein of [START_REF] Quaas | Pollution-reducing infrastructure and urban environmental policy[END_REF], and allow for abatement efficiency to vary with the level of investment, especially in the case of sewer. This would introduce strategic interaction between policy makers in charge of, say, exurbia, suburbia, the primary or the secondary urban area. Finally, the depiction of the hydrological system at stake is absent from the analysis, since we only consider aggregate pollution to an undefined water body. The consideration of pstream/downstream interactions or surface/underground water links would enrich the analysis of the link between urban spread and water quality.

and wastewater management costs.

To ensure β * < 1, the following condition must be met :

v [ln(h e ) -ln(h s )] > 1 (p + b) C e -(a + bq * e ( xs )) (C2)
where xs = x s (β = 1) = Nh s is the eastern limit of a totally suburban city. The RHS of the expression represents the difference of utility due to a higher lot size in exurbia; the LHS represents the difference in wastewater management cost between exurbia and the eastern limit of a totally suburbian city. In other words, β * is less than unity, i.e. there is an exurbian zone in the city, when the gain in utility from a higher lot size more than compensates for the difference in wastewater management costs.

Appendix 2 : impacts of the various cost parameters in the monocentric case

The impacts of the cost parameters are derived as follows :

∂β * ∂a = -p (b + p)tN(h e -h s ) < 0 , ∂β * ∂C e = 1 tN(h e -h s ) > 0 , ∂β * ∂b = -q * s ( xs ) (b + p)tN(h e -h s ) < 0 and ∂β * ∂t = w -C e -pq * e -p p+b tNh s t 2 N(h e -h s ) > 0
The latter expression is positive by the exurban budget constraint and the fact that p p+b tNh s < tNh e .

The derivation of the impact of the cost parameters on S is as follows :

∂S ∂a = p t(p + b) > 0 , ∂S ∂C e = - 1 t < 0 , ∂S ∂b = pq * s ( xs ) t(p + b) > 0 and ∂S ∂t = -N(h e -h s ) ∂β * ∂t < 0
The impacts of the cost parameters on the aggregate pollution load are as follows : The term attached to γ s is negative by compliance with (C1). The various area limits are given by the following expressions :

∂E ∂b = q s (
∂E ∂Ce = 1 (h e -h s )t γ e v ln h e h s -(γ e -γ s )q * s ( xs ) ∂E ∂t t 2 (h e -h s )(p + b) 2 = γ s M 1 + γ e M 2 M 1 = (w -a) (p + b)(
x

Figure 3

 3 Figure 3 about here.

ForFigure 1 :

 1 Fig 1 C eI = 0.3 0.1 1 γ s = 0.8, γ e = 0.1 γ s = 0.1, γ e = 0.8 Fig 2 0.3 a I = 0.1 t I = 1 γ s = 0.8, γ e = 0.1 γ s = 0.01, γ e = 0.8 Fig 3 0.3 0.1 1 γ s = 0.9, γ e = 0.3 γ s = 0.1, γ e = 0.8

Figure 2 :

 2 Figure 2: Impact of heterogeneity of septic costs. Clockwise from top left corner : urban structure, urban spread, pollution load with high γ s /γ e , pollution load with low γ s /γ e .

Figure 3 :

 3 Figure 3: Impact of heterogeneity of sewer costs. Clockwise from top left corner : urban structure, urban spread, pollution load with high γ s /γ e , pollution load with low γ s /γ e .

Figure 4 :

 4 Figure 4: Impact of heterogeneity of commuting costs. Clockwise from top left corner : urban structure, urban spread, pollution load with high γ s /γ e , pollution load with low γ s /γ e .

  Given the budget constraint equation defined in section 5, we obtain the bid rent functions by equating the urban costs within each area of the city :

	R sIIL (x) =	t h s	(x -x sII ) +	t h e	(x sII -x e )
			R eIIL (x) =	t h e	(x -x e )
	vp ln	h e h s	-C 2 ln	h e h s	(C e -a-vp ln	h e h s	)-b(th s N) 2
	Appendix 3 : polycentric setting
	R sI (x) =	t h s	(x s -x) +	t h e	(x e -x s )
			R eI (x) =	t h e	(x e -x)

e ) + pa + wb + t 2 N 2 (ph s (h sh e )h e (p + b)) M 2 = (w-a) (pb)(p + b)vp ln h e h s -C e (p + b) + pa + wb +v(p+b)

  s = Nαβ I h s ; x e = x s + Nα(1 -β I )h e ; x sIIL = xe + + 2w s -3a -tNh s -3pv(p + b) ln h e ebq * s + vp ln h e h s -C e + a) t(h eh s )(p + b)Proposition 6 is derived from the following results : (ww s )(w + 2w s -3a) h s t 2 (p + b)

	∂E p ∂a	=	∂E p s ∂a	+	∂E p e ∂a	,	∂E p 1 ∂a	=	1 3	∂E p ∂a	-	2 3	γ s (w -w s ) h s t(p + b)	and	∂E p 2 ∂a		=	2 3	∂E p ∂a	+	2 3	γ s (w -w s ) h s t(p + b)
	1 + (x s + 2x 0II -2x sIIL ) N(1 -α)(1 -β II )h e ∂q * s ∂a < 0 2 N(1 -α)β II h s ∂(x s + 2x 0II -2x sIIL ) ∂a x oII = x sIIL + 1 2 Wastewater management and commuting costs affect land consumption in the following ∂E p s ∂a = γ s q * s h s manner : ∂S p ∂a = 3p t(p + b) > 0 , ∂S p ∂C e = -3 t < 0 , ∂S p ∂b = 3pq * s ( xs ) t(p + b) > 0 and ∂S p ∂t = -b(w h s t(p + b) 2 < 0 ∂E p e ∂a = γ e 3(pq ∂E p ∂t = ∂E p s ∂t + ∂E p e ∂t , ∂E p 1 ∂t = 1 3 ∂E p ∂t -2 9 γ s and ∂E p 2 ∂t = 1 3 ∂E p ∂t + 2 9 γ ∂E p s ∂t = γ s h s q * s ∂(x s + 2x 0II -2x sIIL ) ∂t ∂q * s + (x s + 2x 0II -2x sIIL ) ∂t
	Proposition 4 is derived from the following results : ∂E p ∂C e = ∂E p s ∂C e + ∂E p e ∂C e , ∂E p 1 ∂C e = 1 3 ∂E p ∂C e ∂E p e ∂t = γ e h e q *	and	∂E p 2 ∂C e	=	3 2	∂C e ∂E p
							∂E p s ∂C e	= γ s	w c + 2w s -3a -tNh s t(h e -h s )(p + b)	> 0
					∂E p e ∂C e		= -γ e	w c h s t(h e -h s )(p + b)	< 0
	Proposition 5 stems from the following results :

c + 2w s ) + 3pa -3C e (p + b) + 3vp(p + b) ln h e * s (ww s )(w + 2w s -3a) h s t 2 (p + b) e ∂(-x ex s + 2x sIIL ) ∂t + (-x ex s + 2x sIIL ) ∂q * e ∂t <

0

Appendix 4 : parameters used in the simulations

rather that sprawl, since we consider contiguous settlement both in suburbia and in exurbia.

Residential lot sizes also constitute an interesting policy basis, however it is difficult to sign analytically their environmental impacts.

The results of this section are presented graphically; however, apart from the impact of heterogeneous commuting costs on pollution loads, the analytical derivations are available from the author upon request.

Note that a general increase of commuting costs would have the same impact in terms of sprawl for both cities.

Appendix 1 : conditions for 0 < β * < 1

The denominator of β * is positive. The positivity of its numerator is ensured by the following condition :

The RHS of the above inequality is composed of two terms : the difference in wastewater management costs and the difference in commuting costs between the eastern limits of a totally exurban and a totally suburban city. In other words, β * is positive, hence there is a suburban area in the city, if the difference in utility derived from the lot size between exurbia and suburbia does not compensate for the difference in commuting costs