Lucien Wald

A CONCEPTUAL APPROACH TO THE FUSION OF EARTH OBSERVATION DATA

Data fusion is a subject becoming increasingly relevant as scientists try to extract more and more information from remotely sensed data using their synergy. A definition of data fusion is proposed, which allows to set up a conceptual approach to the fusion of Earth observation data by putting an emphasis on the framework and on the fundamentals in remote sensing underlying data fusion instead of on the tools and means themselves, as is done usually. Further definitions are given, which describe the objects intervening in any problem of data fusion. Fusion may be performed at different levels, simultaneously: measurement level (also improperly called pixel level), at attribute level, and at rule, or decision, level. It is shown that any process of fusion should deal with the selection of the representation space, the level of fusion and the processing to be applied onto the sources of information. The various architectures of fusion systems are presented. Their properties are discussed, including aspects in accuracy, time-consuming, operational constraints. From these basic architectures, more complex systems can be built, which are suitable to a given application.

Definitions

1.1

by the means of sensor fusion," in Proceedings of the 1 st conference "Fusion of Earth data: merging point measurements, raster maps and remotely sensed images", Thierry Ranchin and Lucien Wald Editors, published by SEE/URISCA, Nice,

Introduction

The quantity of information available to describe our environment increases rapidly. Archives are growing, as well as the number of space missions devoted to Earth observation. Many observation systems are presently available, including space-borne, imaging or not, sensors of optical or radar type, which provide various measurements, partly redundant, partly complementary. Data fusion is a subject becoming increasingly relevant as scientists try to extract more and more information from these measurements. Indeed, it is generally correct to assume that improvements in terms of classification error probability, rejection rate, and interpretation robustness, can only be achieved at the expenses of additional independent data delivered by more separate sensors. Data fusion allows to formalise the combination of these measurements, as well as to monitor the quality of information in the course of the fusion process.

Data fusion is a recent word. It means an approach to information extraction spontaneously adopted in several domains. However the operation by itself is not new in remote sensing: classification procedures are performed since long and are obviously relevant to data fusion. Data fusion means a very wide domain. It gathers a large number of methods and mathematical tools, ranging from spectral analysis to plausibility theory. Fusion is not specific to a theme or an application. On the contrary the tools used in a fusion process for a specific application may be tailored to that case.

A formal framework is mandatory for a better understanding of data fusion fundamentals and of its properties. It allows a better description and formalisation of the potentials of synergy between the remote sensing data, and accordingly, a better exploitation of these data.

Definition of data fusion

Data fusion is a formal framework in which are expressed means and tools for the alliance of data originating from different sources. It aims at obtaining information of greater quality; the exact definition of 'greater quality' will depend upon the application.

Data fusion is exploited by a large number of biological systems. An illustration is given by the human system which calls upon its different senses to perceive its environment. Acquired information is fused within the brain, which will use its memory, its experience, a priori knowledge and its reasoning capabilities to perform deductions and produces a representation of the environment. Data fusion is for example, used to improve results from classification, or control laws and their robustness. It is applied in various domains, ranging from image processing in medicine to management and control of industrial processes.

In data fusion, information may be of various nature: it ranges from measurements to verbal reports. This illustrates the difficulties encountered in data fusion. Some data cannot be quantified; their accuracy and reliability may be difficult to assess. In Earth observation domain, one may use some features held in a geographical information system (external knowledge) to help in classifying multispectral images provided by several sensors. In this particular case, some data are measurements of energy, and others may be symbols.

In the definition given above, quality has not a very specific meaning. It is a generic word denoting that the resulting information is more satisfactory for the « customer » when performing the fusion process than without it. For example, a better quality may be an increase in accuracy of a geophysical parameter or of a classification. It may also be related to the production of a more relevant information of increased utility, or to the robustness in operational procedures. Greater quality may also mean a better coverage of the area of interest, or a better use of financial or human resources alloted to a project.

This document mostly deals with the fusion of data from sensors. It is also called sensor fusion. In this case, information to be fused, are acquired by sensors that can be described precisely. Image fusion is a sub-class of sensor fusion.

Other definitions

According to this definition, spectral channels of a same sensor are to be considered as different sources, as well as images taken at different instants. Hence, any processing of data acquired by the same sensor is relevant to the data fusion domain. Examples in Earth observation are classification of multispectral imagery, computation of the NDVI (normalised difference vegetation index), or atmospheric correction of spectral bands using other bands of the same sensor. Any processing of time-series of data acquired by the same sensor or different sensors, is a fusion process.

The terms merging, combination will be used in a much broader sense than fusion, with combination being even broader than merging. These two terms define any process that implies a mathematical operation performed on at least two sets of information. These definitions are intentionally loose and offer space for various interpretations. Merging or combination are not defined with an opposition to fusion. They are simply more general, also because we often need such terms to describe processes and methods in a general way, without entering details. Integration may play a similar role though it implicitely refers more to concatenation (i.e. increasing the state vector) than to the extraction of relevant information.

Another domain pertains to data fusion: data assimilation or optimal control. Data assimilation deals with the inclusion of measured data into numerical models for the forecasting or analysis of the behaviour of a system. A well-known example of a mathematical technique used in data assimilation is the Kalman filtering. Data assimilation is daily used for weather forecasting.

Terms like measurements, attributes, rules or decisions, are often used in data fusion. These terms as well as others related to information are defined in the following. These definitions are those used in information theory and have been found in several publications.

Measurements are primarily the outputs of a sensor. It is also called signal, or image in the 2-D case. The elementary support of the measurement is a pixel in the case of an image, and is called a sample in the general case. By extension, measurement denotes the raw information. For example, a verbal report is a piece of raw information, and may be considered as a signal. In remote sensing, in the visible range, the measurements are digital numbers that can be converted into radiances once the calibration operations performed. If corrections for the sun angle are applied, one may get reflectances which are still considered as signal.

An object is defined by its properties, e.g., its colour, its materials, its shapes, its neighbourhood, etc. It can be a field, a building, the edge of a road, a cloud, an oceanic eddy, etc. For example, if a classification has been performed onto a multispectral image, the pixels belonging to the same class can be spatially aggregated. This results into a map of objects having a spatial extension of several pixels. By extension, the support of a signal (e.g., a pixel) may be considered as an object.

An attribute is a property of an object which describes geometrical, topological, thematic or other characteristics. Feature is equivalent to attribute. For example, the classification of a multispectral image allocates a class to each pixel; this class is an attribute of the pixel. The equivalent terms label, category or taxon are also used in classification. Another well-known example is the spatial context of a pixel, computed by local variance, or structure function or any spatial operator. This operation can be extended to time context in the case of timeseries of measurements. Equivalent terms are local variability, local fluctuations, spatial or time texture, or pattern. By extension, any information extracted from an image (or mono-dimensional signal) is an attribute for the pixel or the object. The aggregation of measurements made for each of the elements of the object (for example, the pixels or samples constituting the object), such as the mean value, is an attribute. Some authors call mathematical attribute such attribute deriving from statistical operations on measurements.

The properties of an object constitute the state vector of this object. This state vector describes the object, preferably in an unique way. The state vector is also called feature vector, or attribute vector. The common property of the elements of the state vector is that they all describe the same object. If the object is a pixel (or a sample), the state vector may contain the measurements as well as the attributes extracted from the processing of the measurements.

Works in pattern recognition have drawn an analogy with the syntax of a language. Terms of higher semantic content have been defined, such as rules and decisions. Rules, like the syntax rules in language, define relationships between objects and their state vectors, and also between attributes of a same state vector. Rules may be state equations, or mathematical operations, or methods (that is a suite of operations, i.e. of elementary rules). They may be expressed in elaborated language. Known examples of such rules are those used in artificial intelligence and expert-systems. Decisions result from the application of rules on a set of rules, objects and state vectors.

Usually, fusion of measurements results into attributes, and fusion of attributes into decisions. It is not always straightforward. Let take the case of the ARSIS concept which increases the spatial resolution of a multispectral image given another image of a better resolution not necessarily acquired in the same spectral bands 1 2

. It intends to simulate what would observed a multispectral sensor having a better spatial resolution. Accordingly, it simulates measurements through a fusion process and inference models. However, the results are not measurements, and are attributes. Since they are obtained at each pixel, and since the calibration is taken into account during the process, these attributes are similar to actual measurements. This illustrates that data fusion may be heterogeneous: the sources may not belong to the same semantic class (measurements, attributes, decisions).

Topological and processing issues

A fusion system can be a very complicated system. It is composed of sources of information, of means of acquisition of this information, of communications for the exchange of information, of intelligence to process the information and to issue information of higher content. The issues involved may be separated in topological and processing issues. Despite the interconnection between both issues in an integrated fusion system design, they can be decoupled from each other in order to facilitate the development of a systematic methodology of analysis and synthesis of a fusion system according to Thomopoulos 3 4

.

The topological issues address the problem of spatial distribution of sensors, the communication network between sensors and places of processing and decision-making, bandwidth and global architecture. Also at stake are issues for the exchange of information, the availability and reliability of information at the time of the fusion.

The cost of acquiring the information may also be relevant to the topological issues. In remote sensing, these issues are partly adressed by the space agencies and by the image vendors. It is also partly adressed by the customer, given its objectives and constraints, including the financial budget.

The processing issues address the question of how to fuse the data, i.e. select the proper measurements, determine the relevance of the data to the objectives, select the fusion methods and architectures, once the data are available, and according to the specifications issued by the project under concern. These issues are mathematically expressed in Pau 5 .

Some properties

Several problems are to be solved prior to any process of fusion 6 7

. The information entering a fusion process should present several properties which are now described. Some notations first:

• S is the set of NS sensors and sources of information S={S i , i=1...NS}

• (z S )
t is the set of measurements made by the NS sensors at instant t • (X S )

t is the representation of the object at instant t by the NS sources A representation of the object is the set of measurements, or attributes, or rules describing the object, completely or not. In principle, a representation consists in all the knowledge available about this object, given the set of sensors S. A representation includes the state vector of the object together with the relevant rules.

Alignment

Define a common representation (X S ) on the basis of the measurements (z S ) t , and the representations (X S ) t . Differently said, a common co-ordinate system (e.g., geographical space and time) should be found in which the sources data as well as the global knowledge can be represented. The data are said aligned, and the relevant operations are called alignment operations or process. This is called alignment, or conditioning, or positional data fusion.

For example, geocoding airborne or space-borne images is part of the alignment operation. Similar mathematical techniques can be applied to other types of images in different domains, such as medicine or industrial process.

Alignment should provide a general frame of referencing that can applied to homogeneous (commensurate) as well as heterogeneous (non-commensurate) data. This is a difficult problem, and there is no general theory. Even in the simple case of measurements of radiances, which are commensurate, it may still be not straightforward. Though having the same space reference, two sources may not refer to the same object (landscape). In the Meteosat case, the water vapour channel does not provide any information on the ground, while the visible and infrared channels do. Another example in oceanography is the fusion of observations of sea surface temperature, which are relevant to the very surface of the ocean, and of ocean colour, which are depth-integrated. This concept of alignment is extended to a wider reference space (representation space) which also includes standardisation of units, calibration of sensors and atmospheric corrections, etc., if necessary. The alignment problem calls upon physics, and is certainly the problem in data fusion which is the most relevant to the concerns of the remote sensing community.

Association

Let be two sets of sensors S(1) et S(2). Each provides a representation, (X S(1) ) t and (X S(2) ) t . Let be S, the union set of sensors. Assume information is aligned for this set S. Associating the two representations (X S(1) )

t and (X S(2) )

t requires that they refer to the same object. The union of the representations is called association or concatenation.

Association is made by an increase of the size of the state vector of the object.

Data concatenation is accomplished easily and straightforward by juxtaposing all the data into the state vector, hence augmenting it. An example is given by a time-series of images from the geostationary satellite Meteosat.

The raw data are processed by Eumetsat, and are spatially superimposable once delivered to the customer. In that case, at each pixel, one can define a state vector by the concatenation of all the observations made at this pixel in the period under concern. Because the data provider has performed the alignment of data, the customer deals in this case with concatenation and subsequent analysis.

Some examples have been given in the previous section, where sources are not exactly referring to the same object. In that case, though the sources are aligned, the representations cannot be associated.

In some case, the problem can be the selection of sub-sets of sensors, which are the most relevant for a given problem. A metric should then be defined for the comparison between sensors, and the choice of the most appropriate ones.

Fusion of attributes

Assume the information has the properties of alignment and association. Fusion of attributes consists in merging the attributes of a same object, derived from two representations (X S(1) ) t et (X S(2) ) t obtained by means of the sensors S(1) and S(2), in order to obtain new attributes in the space of sensors S = S(1) ∪ S(2).

Fusion of analysis

Assume the information has the properties of alignment and association. Fusion of analysis consists in aggregating representations (X S(1) ) t and (X S(2) ) t , into a new representation (X S ) t , then in generating an analysis / interpretation of the object for further use at instant (t+1), or at step i in an iterative process.

Fusion of representations

Fusion of representations is defining and performing meta-operations applicable to representations (X S(1) ) t and (X S(2) )

t to obtain a new representation (X S ) t

. For example, fusion of classification. Fusion of representations includes fusion of decisions. This fusion of representations may be performed at any moment, i.e. combined with other types of fusion.. This implies that fusion may operate at any of the three semantic levels, with possible crossings between levels:

• measurements (fusion of measurements)

• attributes (fusion of attributes) • rules (fusion of decision or rules)

Representing a fusion process and architectures

Representing a fusion operations

Several formalisms have been proposed. It is usally proposed to consider three levels in data fusion: pixel, attribute and decision 8 9 . They are also called low level, middle level and high level. It presents two drawbacks. The pixel is only a support of information and has no semantic significance. Measurements or observations would be more appropriate. But overall, it does not consider fusion processes dealing simultaneously with these different levels. The various nature of the information to be fused has already been emphasised. The definition of fusion of representations also stresses that fusion can operate at the three different levels with possible mixing. Accordingly, the formalism of Houzelle, Giraudon 10 is preferable and is adopted here. It allows all semantic levels (measurements, attributes, decisions) to be simultaneous inputs of a fusion operation.

A fusion operation can be decomposed into elementary operations. Each elementary can be represented by the means of the fusion cell in Figure 2.1.1. Actually, this cell may also represent very complex operations.

. Formalisation of an elementary fusion operation by a fusion cell

Sources of information, i.e. the measurements provided by the sensors and more generally the original information, are the main inputs of the fusion cell. The auxiliary information bring additional information, resulting from the specific processing of a source, or from another fusion operation. External knowledge is also additional information, whose objective is mainly to constrain or guide the fusion process by e.g., imposing the respect of a priori knowledge. In iterative processing, including time-dependent operations, results may become inputs to the fusion operation in a subsequent step or instant. They will act as auxiliary information, since they are not original sources. Let give a simple example. The sum is a fusion operation. Let be two measurements of similar type: a and b. The sum d: d = a + b can be represented like in Fig. 2.1.2a. 8 M. Mangolini, Apport de la fusion d'images satellitaires multicapteurs au niveau pixel en télédétection et photointerprétation. Thèse de Doctorat, Université Nice -Sophia Antipolis, France, 174 p., 1994. 9 C. Pohl, and J. L. van Genderen, "Multisensor image fusion in remote sensing: concepts, methods and applications", International Journal of Remote Sensing, vol. 19, n° 5, pp. 823-854, 1998. 10 S. Houzelle, and G. Giraudon, "Contribution to multisensor fusion formalization," Robotics and Autonomous Systems, vol. 13, pp. 69-85, 1994. The engine of a modern vehicle works along this scheme. The process laws are called 'engine cartography'. The sensors measure e.g., temperature, pressure and flow in different places. The result may be the quantity of gas to be injected into the combustion chamber.

Copyright Groupe

The process laws may take into account the history of each measurement or result. In that case, history becomes an input as an annex knowledge. Figure 2.1.3 may also represent the monitoring of an object, e.g., of the trajectory of a rocket. This requires the outputs of the fusion cell to become inputs for the following instant, as auxiliary information (see dotted lines in Figure 2.1.1). The fusion cell may consists in a Kalman filter.

Mapping from satellite images is the last example of this section (Fig. 2.1.4). Several images of different natures (optics, radar) are inputs to the fusion cell. The fusion method is a classifier, and outputs are maps of classes and confidence level. From the image of best spatial resolution (SPOT-P in this example), an image of texture is extracted. This image is an auxiliary information, and will help to classify the original measurements. If geographical information is available, too, it will increase the quality of the results if it is an input of the cell as an annex knowledge. In the following example (Fig. 2.1.5), colour images should be transmitted. The three channels are called R (red), G (green), and B (blue). The images are originally coded in 24 bits (3x8). Compression should be applied before transmission and compressed images are coded in 8 bits. The compression / re-coding algorithm calls upon rules, which are fixed but changes should be brought if necessary. The algorithm should also respect the main contours and some of the coloured transitions. Accordingly, one input of the algorithm is an index ID, computed from a mathematical combination of the wavelet coefficients (C1, C2) and of a quantity Q. The wavelet coefficients are obtained by two iterations of a wavelet transform WT applied to the intensity I of the images. The quantity Q is defined as follows

• Q = R-G if the saturation is greater than S 0 • Q = R-B otherwise
The threshold S 0 is fixed but changes should be brought if necessary. The architecture is shown in For the same example, the architecture may also be represented using a more condensed scheme, as in Figure 2.1.6.

Figure 2.1.6. Another scheme for the same operation for compression

Architectures

A fusion architecture describes the set of sensors, how they are assembled, and how they are used, together with mathematical techniques and processing, in order to perform a fusion operation. Usually three types of architectures are defined: centralised, decentralised and hybrid.

The centralised architecture exploits in a single location, simultaneously or not, the set of data acquired by the set of sensors. In Fig. 2.2.1, Si are the n sources. A source can be a set of measurements, attributes or decisions. All sources are inputs of the single fusion cell. The results R and quality parameters Q are obtained by the processing of all sources available at that moment. Of course, this architecture may include auxiliary information and external knowledge. The advantage of the centralised architecture is that it theoretically provides an optimal result, since decision is made taking into account the whole knowledge available. However, if a particular sensor has a large error rate or a low signal-to-noise ration, according to the fusion operation, it may happen that this sensor contaminates the whole data set, and leads to a decrease of the quality of the decision, compared to what would have been achieved without it.

In Earth observation, such cases may be encountered, as e.g., imaging radar whose image quality is a function of various parameters, such as the rainfall before the instant of acquisition, or the surface state of the water bodies.

In most cases, using radar images as inputs to a fusion operation will be highly profitable. In some cases, it may decrease the quality of the result: if a wind is strong, rice fields cannot be perceived at certain growth states,

Copyright Groupe Télédétection & Modélisation -Ecole des Mines de Paris / Armines The centralised architecture has some drawbacks with respect to processing. It requires all the data to be present on the processing site, which implies large communication bandwidth. It also imposes a heavy processing load on the computer, which renews at any change of an input.

The decentralized architecture offers a large flexibility and modularity, and is often adopted for these reasons. It is also called autonomous because it involves independent processing of each source of information (or group of sources) until the fusion of some representation of higher semantic level takes place at a later stage (Fig. 2.2.2).

Figure 2.2.2. Decentralised architecture It should be selected when communication problems are at stake: small bandwidth, unsecure communications, which may be broken, etc. If the acquisition rate of information (sources) is very different between all sources, it may also be adopted to avoid re-processing all the sources while a few have changed, which is the case in the centralized scheme. The decentralized architecture will be adopted in risky domains, such as a battlefield or industrial hazards.

Each source Si enters a fusion cell, which may also include auxiliary information and external knowledge. As said before, a source Si is a set of inputs, which are composed of measurements, attributes, and / or decisions. The local fusion cells (F1, F2... Fn) result into results Ri and quality parameters Qi. These results and quality parameters are transmitted to the final fusion cell F. The results Ri are the inputs of this process. The quality parameters Qi are auxiliary information and will help in deciding the weight of a source in the final process.

One may note that each fusion process Fi is performed locally, using local intelligence. The fusion processing usually reduces the amount of information to be transmitted to the final fusion process. This accommodates for low communications bandwidth. One may also note that this scheme is more robust to the loss of a source of information than the centralized scheme. From a practical point of view, it is easy with such architecture to remove, or not to take into account, a sensor whose confidence is questionable. It is much more difficult with a centralized architecture. In the case of strongly asynchronous information acquisition, i.e. very different time sampling of information from each source, this architecture gathers the locally fused information at the final central point, and thus does not need to renew the whole process at each acquisition time of the most rapid source.

The sources are processed independently from the others. Accordingly the results locally available Ri a lower quality and a lower information content than that would have been achieved with a centralized architecture.

Other architectures may be designed, which is combination of centralized and decentralized architectures. They are called hybrid architectures and have various forms (Fig. 2.2.3). Such architectures involve fusion of the sources at different semantic levels and at different processing stages. Depending upon the combination, such architecture is more or less close to a centralized or decentralized architecture, and so are its properties (advantages and drawbacks).

As a conclusion regarding the architectures, each architecture has advantages and drawbacks. They should be selected on a case by case basis. Trade-off involve many factors 11 , including the availability of smart sensors that perform data preprocessing, the availability of communications links and their bandwidth, and the computational abilities of the central processor / decentralized processors.

Figure

  Figure 2.1.1. Formalisation of an elementary fusion operation by a fusion cell

  Télédétection & Modélisation -Ecole des Mines de Paris / Armines sources F

Figure

  Figure 2.1.2a. Example of a sum

Figure 2

 2 Figure 2.1.2b. Example of a sum depending from an auxiliary information

Figure 2

 2 Figure 2.1.2c. Example of a sum depending from an annex information

Figure 2

 2 Figure 2.1.4. Mapping from satellite images

  Figure 2.1.5. Another example of architecture, for data compression

Figure 2

 2 Figure 2.2.1. Centralised architecture. Si are the sources, R and Q the results and quality parameters

  due to the wavelets make them confusing with other objects in the landscapes. It is more profitable to adopt another architecture.

  have a fairly low information content, depending upon the sources. It further results in the fact that the final result R has Copyright Groupe Télédétection & Modélisation -Ecole des Mines de Paris / Armines

Figure 2

 2 Figure 2.2.3. Hybrid architecture

  11 D. Hall. Mathematical techniques in multisensor data fusion. Artech House, Boston, London, 1992. Copyright Groupe Télédétection & Modélisation -Ecole des Mines de Paris / Armines

Copyright Groupe Télédétection & Modélisation -Ecole des Mines de Paris / Armines Data fusion-L. Wald -Lecture notes

T. Ranchin, L. Wald, and M. Mangolini, "The ARSIS method: a general solution for improving spatial resolution of images

L. Castagnas, "De la synergie des images satellitaires SPOT / ERS au travers de deux exemples de fusion". Thèse de Doctorat, Univ.Nice -Sophia Antipolis, France, 139 p., 1995.

L. F. Pau, "Sensor data fusion," Journal ofIntelligent and Robotics Systems, vol. 1, pp. 103-116, 1988.Copyright Groupe Télédétection & Modélisation -Ecole des Mines de Paris / Armines


Salomé Lachat

Indicators of success and assumptions in technical assistance projects

Keywords: Activity, Assumption, Assumption out Result, Service provider, Sources of verification, ToR

Logical frameworks are planning aids for projects pertaining to technical assistance. They ask for identifying and defining indicators of success and assumptions, which are included in the project ToR and in the project follow-up reports. Their consequences, in particular legal, remain underestimated by service providers, donors and recipients. Indicators of success cannot be considered separately. They should be attached to the sources of verification. To be useful, indicators of success and sources of verification must avoid any confusion with assumptions or expectations and meet various criteria like objectivity, robustness, and perenniality.

Les cadres logiques sont des aides à la planification de projets d'assistance technique. Ils imposent l'identification et la définition d'indicateurs de succès et d'assertions qui sont repris dans les cahiers des charges et les rapports de suivi des projets. Leurs conséquences, notamment juridiques, restent sous estimées par les prestataires de service, donateurs et donataires. Les indicateurs de succès ne peuvent être considérés isolément. Il faut les rattacher aux sources de vérification. Pour être utiles, les indicateurs de succès et les sources de vérification doivent éviter toute confusion avec des assertions ou des expectations et répondre à différents critères comme l'objectivité, la robustesse, la pérennité.

Indicators of success and assumptions are a management information system twined with logical frameworks organizing technical assistance projects.

Preparing and implementing a technical assistance project, like for example, a project pertaining to upgrading a legislation, imply establishing a temporal succession of five events, ie assumptions, activities, results, specific objectives and an overall objective. These events are gathered within a logical framework, or logframe, or project planning matrix. The first one (assumptions) is in the last column of the matrix and the four others are in the first column entitled logic of intervention.

For example, the following series of events may occur when performing a technical assistance project aiming at upgrading a law:

The logical framework and the initial methodology were first described by L. Rosenberg and L. Posner in 1979. The latter methodology was taken again by GTZ in 1987 before spreading to USaid, then to the World Bank and other donors, to be finally adopted by the projects relative to technical assistance of the European Union (EuropeAid) 1 .

Usually a set of matrices connecting inputs (assumptions, activities) with outputs (results, particular objectives, overall objective) is adopted while some donors are promoting the use of one matrix only 2 . It is accepted that where these outputs are achieved, they would automatically cause achieving the objectives of technical assistance project.

Accordingly, in technical assistance contracts, a first matrix (upper matrix) distinguishes a long term overall objective to which contribute the short term specific objectives stemming from the results. It clarifies the passage from results to objectives. It means that there is a mechanistic design of the changes occurring automatically when all of the necessary ingredients are present.

1 L. Rosenberg & L. Posner are represented as inventors by FAO that indicates the role of GTZ in the creation of the project planning matrix; see: FAO, FAO corporate document repository, Annex 13: The logical framework, http://www.fao.orglWairdocs/x5404e/5005e0p.htm . See also the ZOPP guide published by GTZ, http://hq.unhabllat.org/governance/htmlbookslzopp_e.pdf. According to other sources, http://www.leamusa.com, the inventor would be R. Moses Thompson in 1979. 2 For example, DIFID, Guidelines on humanitarian assistance, May 1987, proposes a sole 4x4 matrix with a final assumption in the first row. In DIFID, Humanitarian guidelines for NGOs, 2007, p. 21, both the indicators of success and the final assumption in the first row are "not essential".

Assumption: The Government wishes that the country satisfies the conditions for becoming a member of an international Convention (For example TRIPS). It welcomes a technical assistance project.

Activities:

A service provider is selected for writing a draft law A service provider writes a draft law The draft law is endorsed by the recipient (The domestic patent office) Results:

The draft law is delivered to the Government The draft law is upgraded by the Government The upgraded draft law is endorsed by the Government The upgraded draft law is submitted to the Parliament Specific objective:

The Parliament passes a law in compliance with the provisions of the International Convention Overall objective:

The volume of external trade of the country increases Each matrix (lower matrix) of a set of matrices proposes to reach at least one result based on activities and assumptions. It is usual to prepare several lower matrices, each of them corresponding to a result as it may be seen at figure 1 below.

Each result calls for a lower matrix. The same results appear in upper and lower matrices. This vision is effective when objectives are physical like, for example, assistance for fighting against a disease, training, upgrading a legal system… It is it much less effective when it is wished to increase governance, transparency, fighting against corruption… Another difference between upper and lower matrices lies in their columns, even if all of them have four columns. The upper matrix proposes the logic of intervention of the donor that is interested in reaching objectives. Accordingly it stipulates the indicators of success, their sources of verification, and assumptions relative to the passage from results to specific objectives contributing to the foreseen overall objectives. The lower matrices reveal the logic of intervention accepted by the service provider. It presents the conditions of passage from a performed activity to the following activity which is to be performed. This may occur because an activity was successfully performed or because the commitments of the recipient and/or the donor to meet certain assumptions are met.

According to a logical framework, activities are performed by a service provider usually selected by the donor, while the results, the specific or overall objectives are expected from the mere project performance without assistance of the service provider. The service provider agrees with the goals of the project. On the basis of this consent, donors want that service providers share the risk of absence or weakness of the results. Service providers may refuse this request despite the potential threat of being blacklisted by the donors.

A mechanistic design of the implementation of project ToR forces to monitor whether the expected results are automatically present according to the logic of intervention. This means checking whether particular objectives and the overall objective, were reached. Results belong to the lower matrices while objectives appear in the upper matrix as it may be seen at figure 2.

An objective tool for assessing the actual results is preferred for avoiding disputes between recipient, service provider, and donor. However, a result is not always measurable per se.

For example, a new law may have been passed but it does show effect only after several years. The quality of a law is assessed through its use3 . The mere fact that a law is passed and becomes an integer part of the legal system of a country is an indicator proving that the service provider has carried out the necessary activities but is neither an assessment of the quality of this law, nor a guarantee that changes will occur.

Another example: it is necessary that a certain percentage of the population be vaccinated before one disease moves support, which is only possible if there is no virus mutation. The percentage of the vaccinated population is an indicator proving that the service provider has carried out the necessary activities.

The service provider is not stripped. It is required from the expert writing the ToR that the indicators are objectively verifiable. They identify the source of verification.

For example, will statistics from an independent source be used for checking whether an indicator of success is fulfilled, recommended statistics sources will be put down in the logical framework.

Objectively verifiable indicators of success are indicators of the performance of the activities for which the service provider is remunerated within the framework of the project. As of the response to the invitation to tender for performing the project, potential service providers can dispute the indica- tors of success contained in the ToR if they submit substitutes. Moreover, at the beginning of a project there is usually an inception phase which asks for an inception report offering the opportunity to propose objectively verifiable indicators of success substituting for the indicators of the ToR. The inception report is supported by a plan of execution that precisely describes the activities to be undertaken by the service provider. This plan of execution is binding on the service provider.

Under these conditions, all of the parties to a project, ie the donor that finances the project, the recipient for the benefit of which the project is carried out, the expert charged of drafting the project ToR, and the service provider that carries out the project ToR seek objectively verifiable indicators of success. The description of the indicators of success (second column), i.e. the statement of the conditions to be met, is supported by the indication of the sources of verification (third column) within the logical framework. These sources will have to be accessible when the indicators of success are used.

It is advisable to lay down the logical and technical rules to which indicators of success and assumptions must satisfy to be valid. This will lead to submit new support matrices in addition of the two sets of matrices recommended for working with logical frameworks. As a result, new tools for managing a technical assistance will be designed.

Logical rules

A logical framework is read from bottom to top -like a house which is built from its foundations to its roof -, and from left to right. With logical frameworks, foundations are the initial assumptions which authorize beginning the performance of the activities leading to results. Once an activity (a series of activities) has/have been performed, indicators of success authorize undertaking the next activity (series of activities). The upper matrix of a logical framework supposes that at least one specific objective is mechanically obtainable as soon as the results identified as necessary are reached because the activities programmed in the lower matrix have been performed. The upper matrix may imply indicators of success and assumptions. However no activity is to be performed by the service provider.

The following lines are structured as follows: The first section describes the two categories of indicators of success. The second section presents the characteristics of assumptions.

Indicators of success

Indicators of success are tools that tell whether performed activities have achieved the results that are needed for reaching the objectives stipulated by the upper matrix. Objectives describe the new bal- ance that the project seeks to reach. They are the raison d'être of the project and are defined by the donor, possibly according to a proposal made by the expert writing the ToR, and accepted by both the recipient (more precisely endorsed by the recipient) of the technical assistance and the service provider. The donor must be able to show the recipient that it undertook all necessary steps so that the objectives are achieved. Accordingly, a project implies indicators of success and sources of verification accepted by all of the parties.

An indicator of success reports at best on the present, but may report more generally on a passed transitory state.

Indicators of success are:

-Either authorizing the passage from an activity to another after the successful performance of the first one by the service provider using the resources provided by the donor, -Or documenting that a result or an objective has been reached. No activity is to be performed.

Accordingly, two categories of indicators of success appear according to whom (either the service provider that undertook the activities or the donor that selected the right logic of intervention) is successful.

Morphology of indicators of success

The typical morphology of an indicator of success is of the type "If indicator of success X relative to activity Y is satisfactorily fulfilled using sources of verification recommended by the project ToR, then the service provider may begin carrying out activity Z". It follows the well known "if X is true, then do Z" pattern.

Sometimes conditions of the type "If the condition set by the indicator of success X is satisfactorily fulfilled, and if the condition set by the indicator of success Y is met, then activity Z may be undertaken" are used.

For example it may be read "if ten judges attended five hours basic training in fighting against counterfeiting, and if twenty customs officers attended fifteen hours of basic training in seizures by customs, training may be deemed carried out, and may actually understood being an indicator of success allowing a study tour".

Usually, logical frameworks do not use "If…, then… -statements" because the reader knows that the second columns of the matrices stipulate conditions (no need of "if-statement"). Neither the "thenstatement" (allowing to performing the next activity) is needed because it is the only reason why indicators of success are called for.

Success in performing activities

A first kind of indicators of success may be identified. They are pertaining to the commitment of the service provider to reach the results foreseen in the inception report if the prescribed activities are carried out. The proof that results are reached is brought by the use of indicators of success.

For example, a draft amendment of the company law is submitted to the Parliament by the ministry of Justice. The indicator of success will be the actual proposal for an amendment put forward to the ministry of Justice within the time window foreseen by the ToR. The service provider cannot be responsible for the local administrative incompetence which buries the proposal.

The service provider is responsible for the performance of the activities and the result, because the absence of identification of an activity necessary to reach a result is ascribable to the service provider as soon the inception report is produced. This latter inception report applies even if the ToR propose an incomplete succession of activities to be undertaken in order to reach a certain result.

The service provider answers to the invitation to tender by a technical proposal often independently elaborated from the experts that should perform the fieldwork. Experts may use the inception report for identifying the possible absence of certain activities. This inception report is submitted for acceptance to the donor and the recipient.

It follows that the passage from activities to results, which corresponds to the lower matrices of the logical framework, -Has been worked out by a first expert that has been drafting the ToR on behalf of the donor, -Was possibly criticized by the writers of the tender document, and -Was subject to the comments of the experts in charge of the actual performance of the project.

The definition of new or at least upgraded indicators of success was possible until the delivery of the inception report. In other terms, all precautions were taken for identifying relevant Indicators of success and their sources of verification.

Success in meeting objectives

Beside indicators of success relative to the performance of the activities listed in the inception report, there are indicators of success showing that objectives were achieved. These indicators of success are essential to the relation between donor and recipient. The service provider cannot be liable for the transition from results to objectives that are stipulated in the project ToR. Moreover, modifying this articulation often implies an additional investment that can be carried out by the donor only. Accepting the ToR involves the adhesion of the service provider to the mechanistic effect providing the foreseen objectives once the results are reached. It is an expectation, generally checked with experts' statements and experience, but subject to assumptions, therefore strongly hazardous. The service provider must inform the donor if it identifies a risk not to achieve the foreseen objectives; by doing this, it lessens its liability.

Indicators of success relative to the objectives to be reached are stipulated by the donor who selected an expert to write the project ToR drawing up the list of results necessary to mechanically obtaining the objectives. Usually, no subsequent control by other experts is organized. The donor is generally not able to dispute the cogency of the mechanical passage between results and objectives. The service provider does not have any influence on the indicators of success of the upper matrix of the logical framework.

The service provider is only liable for achieving the passage from an activity to the next one and for reaching the foreseen results but not for reaching any objective. It can only commit itself on the activities which it undertakes and to the related results. It cannot be liable of the absence of mechanical effect of the results if they are achieved. Contesting the mechanical effect may lead to differences with the donor.

Assumptions

Assumptions are external conditions that are not -in principle -within the reach of the service provider implementing the project ToR. Assumptions should hold true, be explicit, and demonstrate a low degree of uncertainty. If this were not the case, the project may become difficult to perform.

The expert in charge of drafting the project ToR must be sure of their feasibility. Accordingly, a list of the incurred risks and the means to reduce them should be put down. It means that not only the

Using logical frameworks for preparing project ToR asks for matrices describing indicators of success supported by sources of verification, and assumptions

resources foreseen for performing the activities are relevant but also that both indicators of success and assumptions are realistic.

Another aim of the assumptions is indicating to the recipient the coercions it incurs if it does not meet its commitments described by the said assumptions.

Morphology of assumptions

The typical morphology of an assumption is of the type "If assumption X is satisfactorily fulfilled by the recipient, then it is possible to consider that the service provider begins carrying out activity Y". It follows the well known "if X is true, then do Y" pattern. An assumption lays out for the future. It implicitly defines activities of the donor. It does not refer to activities performed by the service provider. In other words, they should be considered as a new input.

Service provider and recipient are not on an equal footing since the donor cannot demand that the recipient fulfills its commitments. It can only prompt the recipient to meet the assumption.

A distinction should be made between two types of assumptions, i.e. between:

-Assumptions relative to tasks that may be performed by the service provider (assumption within the reach of the service provider -AWR) if the recipient does not fulfill its commitments, and -Assumptions implying tasks that are beyond its reach (assumptions out of the reach of the service provider -AOR) because they should be performed by the recipient or a third entity.

For example, the recipient is the patent office of a certain country that receives a technical assistance for drafting a law while the draft law should be passed by the Parliament (AOR). It is out of the reach of the service provider to prompt the recipient to prompt the government to prompt the Parliament to pass the law. Some experts 4 understand that activities and results are under full control of the service provider while assumptions are beyond its control. The reality is different: indicators of success should show that specific and overall objectives are reached while assumptions may be present in any row of the logical framework except the upper one. It is necessary to give a minimum flexibility 5 to the logical framework through accepting that assumptions out of the reach of the service provider may become within its reach in some circumstances. This should avoid stopping the technical assistance project.

Unlike with indicators of success, "if…, then… -statements" are not developed for assumptions.

Where to find assumptions?

Assumptions are to be found in two places of the logical framework, ie in the upper matrix and in the lower matrix. 4 See Keerti Bhusan Pradham, The logical framework approach, undated. http://www.pitt.edu/~super7/16011-17001/16211.ppt. 5 For a critic of the logical framework, see J. MacArthur, The logical framework -A tool for the management of project planning and evaluation, in The realities of managing development project, Farhad Analoui (Ed.), Aldershot, Hants, UK, pp 1994

Indicators of success are internal to the project while assumptions are external to the project

In a lower matrix, the first assumption (bottom of the fourth column) is the initial assumption which is to be met before the beginning of the technical assistance.

A series of assumptions (fourth column) is to authorize beginning a new activity following an activity that has been carried out independently from any other condition set by an indicator of success. However, indicators of success of the previous activity should be met.

The last assumption of lower matrices (above, fourth column) is relative to the results reached by the performance of the activities.

Indicators of success and assumptions do not belong to the same category, because if the indicator of success reveals that an activity was performed by the service provider (or sub-contracted by the service provider that remains fully liable), the assumption does not depend on its efforts but on those of the recipient. Accordingly, because assumptions and indicators of success cannot cumulate and act simultaneously, it is not suitable to join together indicators of success and assumption under the same logical condition imposed to the service provider.

For example, the two conditions and the consequence are: Condition (1): if ten judges selected by the service provider attended a five hours basic training organized by the service provider, and Condition (2): if twenty customs officers were selected by their administration to attend five hours of basic training organized by the service provider, Consequence (3): then it is possible that both judges and customs officers undertake a study tour in the European Union.

Condition (1) is within the reach of the service provider; it is an AWR according to the definition above. Condition (2) is out of the reach of the service provider that cannot be liable for its absence of fulfillment by a third party. Condition ( 2) is an assumption about the positive and speedy answer of the third party to a request of the service provider; it is an AOR according to the definition above.

Usually, if such confusion of assumptions and indicators of success is identified in the project ToR (it should be at the latest when delivering the inception report), the service provider should propose to cut out the flow of activities in order to reveal two flows of activities that should be independently undertaken. Each flow is leading to a result / an intermediate result. Another solution is introducing the assumption as an initial assumption to be met before beginning the project activities; hence it is not necessary to repeat it in the flow of activities.

Then, in the upper matrix, assumptions relate to the passages from the results to the specific objectives, then from specific objectives to overall objectives. It is worth noticing that no activity is considered by the project ToR to ensure or only to facilitate this passage. It is about a mechanistic effect which can prove to be contingent. Assumptions in the upper matrix are AOR according to the definition above.

AOR -assumptions describe how one deus ex machina transforms results having implied activities into actual specific objectives. Intensity and quality of the efforts are not identified. If activities may be carried out during a certain time whose duration is set in advance, jumping from results to objectives is not the same issue because the transition is contingent. The intervention of the service provider being limited in time, it seems difficult to impose to him an indicator of success in the upper matrix. In other words, indicators of success and assumptions proposed in the upper matrix are directed only to the recipient. Their statement means that the recipient may receive a technical assis-tance but that this technical assistance might be ineffective without the will of the recipient to fully benefit from the latter technical assistance. Usually, a reference is made to the political will of the recipient in the list of risks.

Usefulness of assumptions

Assumptions are objective elements because either they are or are not fulfilled. However their definition is not useful if the condition is not measurable, at least with a Boolean option (yes/no).

For example, significance of an AOR -assumption of the type: "the government supports the implementation of the project" is strongly contestable, because it does not state which actions of support are reasonably foreseen to carry out the transition from results to achievement of the objectives. It means that the service provider is to identify the sources of verification that the writer of the project ToR was unable to identify.

A second example: Training the staff of a Patent office in sorting out patents according to the Classification of Strasbourg will not be possible as long as this staff is not recruited and that time necessary for this training is allocated. The expression "as long as" is not suitable with an administrative body of a Sovereign state. The formulation will rather be of the kind "The necessary time is allocated to the personnel recruited for attending to training". This formulation must be deconstructed to find the conditional form.

Technical rules

Independently of logical rules, technical rules are to be identified. It is necessary to establish if an activity was actually carried out to obtain a certain result. It is generally an all or nothing game. Questions are of the kind: Have relevant activities been undertaken? Has the foreseen result been reached? This measurement cannot be qualitative, because one does not know how to qualify the performance of an activity. It was (possibly with difficulty) or was not carried out but, what matters, is precisely that it was carried out.

Indicators of success when writing the project ToR, and assumptions when performing the project ToR, ask for identifying sources of verification. It is suggested to identify indicators of success and sources of verification together.

Accordingly, a basic indicator stipulates the kind and the nature of change. Its phrasing is close to the description of the activity used in the first column of the logical framework. Then the basic indicator may be upgraded through progressively adding conditions. The first set of conditions is usually relative to the extent of the change. The second set of conditions is giving hints about the quality of the change, while the last set of conditions is relative to the timing of the change. Place and cost may be added if necessary.

Sources of verification give a description of where to find the data to verify the indicators of success.

The following questions are to be asked:

-What data? -Where is the data? -Is the data readily available? -Is special data gathering required?

-Is the data acquisition affordable to the project?

Indicators of success for which no suitable sources of verification could be identified should be replaced by relevant indicators.

The complexity of the sources of verification is an important variable because it means a cost increase. Some sources of verification may be the mere reference to an already performed task. The cost of gathering such evidence is very law. Conversely, monitoring and final reports involve specialized external staff that has to be paid by the donor.

For example, a follow-up report relates facts that may be a proof of the performance of a certain activity while a monitoring report is a secondary source of description of the facts.

Figure 3 shows how indicators of success and sources of verification may be identified. The added sets of conditions are underlined. The matrix should be read from bottom to top and from left to right like the logical framework. Sometimes, this matrix is called "QQT-Matrix" where Q stands for quantity, for quality, and T for time.

The same kind of matrix may be used for assumptions and their sources of verification.

Figure 3: Identifying indicators of success and sources of verification

Indicators of success are sometimes presented as an overabundant statement of the inputs enabling the service provider to perform the activities. This view may be rebuttable because spending the investment foreseen for performing an activity does not mean that the activity will be fully performed or that the said activity could have been performed at lesser cost. Hence, there is a possibility to compare the allocated resources with the actually tapped resources. This builds a MIS -management information system. Delivering technical assistance asks for using the whole allocated resources in order to avoid complains against the service provider and to avoid its liability. If any changes in the inputs they should be submitted and approved in the inception report.

Accordingly, it is suggested to use a 6 x 4 matrix as shown figure 4 that eases the follow up of a technical assistance project.

Assessing indicators of success

An indicator of success should show an identical behavior in a similar project. Its capacity to establish in various environments whether the same results have been obtained where starting from the activities undertaken is essential. If one expert knows that an indicator of success is not satisfactory, it will not recommend it in order to avoid burdening on the project credibility. Any dispute on the quality of the project ToR by the service provider must be avoided after the inception report so that, in turn, the recipient cannot dispute the donor choices.

In the lower matrix of a logical framework, an indicator of success precedes the sources of its verification when reading from left to right. The extreme right-hand column should recall assumptions. Very often, no assumption is provided, because it is implicitly accepted that -Success in undertaking an activity is an obvious condition to the passage to the following activity, and that -Assumptions are initial assumptions to be met before beginning the project implementation.

The extreme right-hand column in the upper matrix relates to assumptions.

Neither the logical rules, nor the technical rules take into account the cost of using an indicator of success. Estimating the cost, high or weak, acceptable or exorbitant, is eminently subjective. Donors expect that experts choose the indicators of success according to their lower price. This generally implicit directive is foreseen by the ToR to be met by the expert in charge of drafting the project ToR.

Indicators of success may be assessed from various points of view. They should be objective. For example, their quality may be assessed taking into account their relation to the sources of verification they refer to, or through their capability to prove that certain activities were actually successfully performed.

Indicators of success should be objective

An indicator of success should not be questionable unless it is irrelevant per se or the sources of verification are irrelevant. It cannot be limited to what is important6 but should stipulate the necessary condition(s) for jumping from an activity to the next one.

Any expert must be able to use indicators of success to objectively document a monitoring. No personal or emotional element should modify actual facts that are documented in the source of verification, ie indicators of success must be usable by any expert and should lead to the same findings. The condition contained in the indicator of success does not have only to be verifiable but is to be checked. Neither the implementing expert nor the monitoring expert may modify an indicator of success to assert that the result is reached. The procedure must be simple.

For example, as regards laws and implementing regulations upgrade, an indicator of success is often submitting a draft law to the parliament. Three comments shall be made: -Submitting a bill to the Parliament is not an indicator of success but an assumption of the type AOR because the service provider is not in a position to submit the draft to the parliament but the Government.

-The draft law prepared by the service provider and the recipient might have been modified by the government.

-The draft law quality cannot be taken into account. A quality criterion can only be used if ascertained beforehand. In the absence of a statement describing quality standards and their modus operandi, the quality assessment is likely to be subjective and the indicator of success may be disputed. In practice, indicators of success are all the more effective that they are simple and take numerical data into account.

Sources of verification should not be deceptive

Checking the adequacy between results and indicators of success is exceptional, because it is not explicitly foreseen by the donors' reference guidelines. The issue is assessing whether fulfilling a certain indicator of success relative to an activity is enough to undertake the next activity. However, the service provider has more than just a commitment to use its best endeavours, because it must inform the donor project manager about any obstacle preventing from obtaining the expected result in order to undertake palliative measures.

An example of deceptive indicator of success may be presented. Teaching prior art research techniques may be necessary for increasing the capabilities of the staff of a patent office. If the indicator of success states that a certain number of persons are to be trained and that the source of verification is the participants' list, it may happen that the participants register the first day and disappear immediately after registration, when their supervisor is gone away. Obviously such an indicator of success bound to a source of verification is not relevant.

Characteristics of the sources of verification affect the quality of the indicators of success.

In principle, a source of verification must be objective and perennial. Without these two characteristics, it is not possible to propose a reasonable indicator of success since it would become subjective and fugacious in contradiction with the objectivity of its statement. Written and published sources are generally accepted as sources of verification. These sources may be criticized by any interested party. Acceptable sources may not be effective because of the absence of publication or introduction of an intern defect like a change of series in statistics.

The robustness of an indicator of success is essential because it will be used after the project completion when assessing its evaluation. As time goes by certain characteristic of a project fade away while others are reinforced.

Counterexamples should not be known

The search for counterexamples is possible but rare because requiring important efforts and time. Pressures exerted by donors on service providers hinder that the latter refuses a reference to an indicator of success in the absence of duly indexed counterexample. The time between the drafting of the ToR with its logical framework and the use of indicators of success is determining: short, the counterexample can remain undisclosed; longer, the counterexample may appear.

Assessing the performance of the project (Monitoring)

Assessing the performance of a project is a specific task that is usually sub-contracted by the donor to a service provider independent from the service provider in charge of the project TOR implementation. In most cases, the selected service provider in not a technical assistance expert, but an expert specializing in project monitoring. Specific methodologies have been developed. They ask for interviews and paper evidences.

Authorizing the passage to the next activity

Successfully performing a certain series of activity transforms expected intermediate results into achieved intermediate results. The proof of this transformation is with the use of indicators of success supported by sources of verification. When the service provider understands that the expected intermediate results are achieved, it may turn to performing the next activity described in the logic of intervention. A result is achieved when all of the activities foreseen by the logic of intervention have been performed.

As shown in figure 5 above, the passage from activity n to activity n+1 asks for two main steps:

-Indicator(s) of success (internal to the project)

o Identifying the sources of verification of the indicator(s) of success provided by the project ToR, o Checking the characteristics of the achieved results with the indicator(s) of success using the sources identified by the project ToR without any risk of conflict of interest because the service provider did not put down the project ToR, o Recording (paper evidence) that the indicator(s) of success is/are met -Assumption(s) (external to the project) o Identifying the results that should be achieved by the performance of the assumption(s) o Identifying sources of verification of the performance of the assumptions with no risk of conflict of interest because the service provider did not performed the activities necessary to fulfill the assumption(s), o Qualifying the assumption(s) (AWR or AOR) o Checking whether the assumption(s) has/have been fulfilled o Recording (paper evidence) that the assumption(s) has/have been fulfilled

Both indicators of success and assumptions are needed to ensure that activity n+1 (next level activity) is not begun before activity n is fully performed.

The risk that an assumption may not be fulfilled during the performance of a technical assistance project is important because the pace of an administration can-

Monitoring and evaluation will be given high priority at all level. The indicators for monitoring and evaluation are most relevant when measuring the effective and timely implementation of all activities and impact of the project intervention (EuropeAid/129242/C/SER/CN)

not compare with that of a project. In order to avoid suspending a technical assistance project, the number of assumptions should be kept minimal as long as the activities of the service provider have not been fully provided.

Evidencing the actual performance of activities

A first and obvious rule is that a report is to be prepared for each indicator of success and/or assumption. Both are provisions of the project ToR. Putting down indicators of success into the project ToR means that the results to which they correspond are fully identified since the ultimate issue (including during the monitoring) is establishing whether these results have been actually achieved. But defining a result means also that the resources necessary for achieving that result are relevant and fully described. This identification is done by experts delegated by a first service provider that is expected to prepare the project ToR that will be performed by a second service provider. This second service provider has a commitment to achieve a result that is verifiable through using at least one indicator of success. It means that at the time of the posterior (final) evaluation of the provided services, only a fulfillment of the indicator is controlled by a third service provider. It is worth noticing that all of the service providers involved in a project should be independent from each other in order to avoid conflicts of interest.

Indicators become managing tools of the project since the service provider must fulfill them. The recipient of the technical assistance endorses these indicators of success: It will not be able to dispute them and claim that the technical assistance was not relevant.

The donor obtains a proof of correct use of the invested funds in a project since the indicators were fulfilled. If the objectives stated in the ToR were not reached, the cause is not to be sought in the implementation and/or the performance of the project but is in a defect of design of the project by the expert who wrote the ToR, that is to say in obstacles unforeseen or depending of the donor.

To be accepted within a logical framework, an indicator of success must meet several criteria. In addition to its objectivity, it must be able to avoid falsification.

FACIT

Figure 6 shows how ToR organize self-assessment by the service provider and monitoring by the donor. Monitoring depends on indicators of success. Two kinds of monitoring appear:

-Real time monitoring to assess the use of the resources by the service provider, -Final assessment of the project by the donor and the recipient. It assesses whether the overall objective has been met.

It is worth reminding that usually four ToR are involved in performing a technical assistance project (column 1). Each of these ToR implies selecting a specialized service provider. These service providers are independent of all other service provider in order to avoid any conflict of interest. A self-

Real time monitoring is to assess the use of resources by the service provider. Final assessment shows to the donor and the recipient that the project overall objective has been met

Figure 2 :

 2 Figure 2 : Transition from results to objectives

  . Cedric D. Saldanha & John F. Whittle, Using the Logical Framework for Sector Analysis and Project Design: A User's Guide, Asian development bank(ISBN 971-561-174-5), 1998.

  the professional representatives attended a training and half of the trainees have passed a qualifying exam equivalent to the EPO qualification exam during the first year of the technical assisthe professional representatives attended a training and half of the trainees have passed a qualifying exam equivalent to the EPO qualification exam Exam protocol Adding quantity (extend of the change, ie how much? how many?) A third of the professional representatives attended a training and half of the trainees have passed a

  Figure 4: A MIS -matrix

On assessment of law quality, see: S. Lachat, L'évaluation des lois de propriété industrielle et intellectuelle, Humanisme & Entreprise, June 2007, pp 17-40.

Conversely, http://www.fidafrique.net/IMG/pdf/Handout_Logframe01.pdf asserts « Indicators measure what is important".

assessment of the project ToR implementation is foreseen by the Project ToR through asking for an inception report that is intended to allow upgrading and updating or the project ToR at the beginning of the project implementation. Then, indicators of success and assumptions are structuring the passage from an activity to its follower according the logic of intervention (1 st column of the lower matrices). They also confirm that results were achieved.

Project ToR implementing service providers are not left alone in assessing the performed work. The donor selects a service provider that should monitor the project implementation and another service provider for a final assessment of the project when finished. Specific ToR are referred to.

Indicators of success used in a project relative to technical assistance are a fundamental element of the ToR. The attention given to them should be increased. Often, their consequences for the service provider, the donor, and the recipient are not enough taken into account. If their raison-d'être is to facilitate the monitoring of the activities performed by service providers, their definition, and later their use and results can become a sensitive issue. An imperfect drafting leads to unbalance the relation between the donor and the service providers. Differences may occur when the donor tries to impose indicators of success to the service providers, which are assumptions on the behavior of the recipient.

Four ToR

Four

Figure 6: ToR, assessments, and monitoring

A temporal difference can be highlighted: an assumption is directed to the future and remains to be fulfilled during the technical assistance project, while indicators of success document past events aiming at changing particular or general objectives.

Indicators of success are based on sources of verification which must be objective and perennial as of a certain moment of a certain time span. Indicators of success should not present a known counterexample and are to be univocal.


Manfred Lehn
email: lehn@mathematik.uni-mainz.de

Christoph Sorger
email: christoph.sorger@univ-nantes.fr

A SYMPLECTIC RESOLUTION FOR THE BINARY TETRAHEDRAL GROUP

Keywords:

We describe an explicit symplectic resolution for the quotient singularity arising from the four-dimensional symplectic represenation of the binary tetrahedral group.

Let G be a finite group with a complex symplectic representation V . The symplectic form σ on V descends to a symplectic form σ on the open regular part of V /G. A proper morphism f : Y → V /G is a symplectic resolution if Y is smooth and if f * σ extends to a symplectic form on Y . It turns out that symplectic resolutions of quotient singularities are a rare phenomenon.

By a theorem of Verbitsky [START_REF] Verbitsky | Holomorphic symplectic geometry and orbifold singularities[END_REF], a necessary condition for the existence of a symplectic resolution is that G be generated by symplectic reflections, i.e. by elements whose fix locus on V is a linear subspace of codimension 2.

Given an arbitrary complex representation V 0 of a finite group G, we obtain a symplectic representation on V 0 ⊕ V * 0 , where V * 0 denotes the contragradient representation of V 0 . In this case, Verbitsky's theorem specialises to an earlier theorem of Kaledin [START_REF] Kaledin | On crepant resolutions of symplectic quotient singularities[END_REF]: For V 0 ⊕ V * 0 /G to admit a symplectic resolution, the action of G on V 0 should be generated by complex reflections, in other words, V 0 /G should be smooth. The complex reflection groups have been classified by Shephard and Todd [START_REF] Shephard | Finite unitary reflection groups[END_REF], the symplectic reflection groups by Cohen [START_REF] Cohen | Finite quaternionic reflection groups[END_REF]. The list of Shephard and Todd contains as a sublist the finite Coxeter groups.

The question which of these groups G ⊂ Sp(V ) admits a symplectic resolution for V /G has been solved for the Coxeter groups by Ginzburg and Kaledin [START_REF] Ginzburg | Poisson deformations of symplectic quotient singularities[END_REF] and for arbitrary complex reflection groups most recently by Bellamy [START_REF] Bellamy | On singular Calogero-Moser spaces[END_REF]. His result is as follows:

Theorem 1. (Bellamy) -If G ⊂ GL(V 0 ) is a finite complex reflection group, then V 0 ⊕ V *
0 /G admits a symplectic resolution if and only if (G, V 0 ) belongs to the following cases:

1. (S n , h), where the symmetric group S n acts by permutations on the

hyperplane h = {x ∈ C n | i x i = 0}. 2. ((Z/m) n ⋊ S n , C n ),

Applied to a minimal resolution of the

A m-1 -singularity C 2 /G, G ∼ = Z/m, this construction yields a small resolution Hilb n ( C 2 /G) → Sym n ( C 2 /G) → Sym n (C 2 /G). Similarly, (h ⊕ h * )/S n is the fibre over the origin of the barycentric map Sym n (C 2 ) → C 2 . Thus (h ⊕ h * )/S n is resolved symplecti- cally by the null-fibre of the morphism Hilb n (C 2 ) → Sym n (C 2 ) → C 2 .
It is the purpose of this note to describe an explicit symplectic resolution for the binary tetrahedral group.

The binary tetrahedral group

Let T 0 ⊂ SO(3) denote the symmetry group of a regular tetrahedron.

The preimage of T 0 under the standard homomorphism SU(2) → SO(3) is the binary tetrahedral group T . As an abstract group, T is the semidirect product of the quaternion group Q 8 = {±1, ±I, ±J, ±K} and the cyclic group Z/3. As a subgroup of SU(2) it is generated by the elements

I = i 0 0 -i and τ = - 1 2 1 + i -1 + i 1 + i 1 -i
The binary tetrahedral group has 7 irreducible complex representations: A three-dimensional one arising from the quotient T → T 0 ⊂ SO 3 , three onedimensional representations C j arising from the quotient T → Z/3 with τ acting by e 2πij/3 , and three two-dimensional representations S 0 , S 1 and

W 1 ∩ W 2 .
For j = 1, 2, let α j : Z ′ j → Z denote the blow-up along W j . Next, let W ′ j be the reduced singular of locus Z ′ j , and let

β j : Z ′′ j → Z ′ j denote the blow-up along W ′ j .
Theorem 2. -The morphisms σ j = α j β j : Z ′′ j → Z, j = 1, 2, are symplectic resolutions.

Proof. As all data are explicit, the assertion can be checked by brute calculation. To cope with the computational complexity we use the free computer algebra system SINGULAR 1 [START_REF] Greuel | Singular 3-0-4. A Computer Algebra System for Polynomial Computations[END_REF]. It suffices to treat one of the two cases of the theorem. We indicate the basic steps for j = 2. In order to improve the readability of the formulae we write q = √ -3.

Let C[x 1 , x 2 , x 3 , x 4 ] denote the ring of polynomial functions on S 1 ⊕ S 2 .

The invariant subring C[x 1 , x 2 , x 3 , x 4 ] T is generated by eight elements, listed in table 1. The kernel I of the corresponding ring homomorphism

C[z 1 , . . . , z 8 ] → C[x 1 , x 2 , x 3 , x 4 ] T
is generated by nine elements, listed in table 2. The curve C 2 is given by the

semiinvariant x 4 3 + 2qx 2 3 x 2 4 + x 4 4 .
In order to keep the calculation as simple as possible, the following observation is crucial: Modulo I, the Weil divisor W 2 can be described by 6 equations, listed in table 3. This leads to a comparatively 'small' embedding Z ′ 2 → P 5 Z of Z-varieties. Off the origin, the effect of blowing-up of W 2 is easy to understand even without any calculation:

1 A documented SINGULAR file containing all the calculations is available from the authors upon request.

Table 1: generators for the invariant subring C[x 1 , x 2 , x 3 , x 4 ] T :

z 1 = x 1 x 3 + x 2 x 4 , z 4 = x 2 x 3 3 -qx 1 x 2 3 x 4 + qx 2 x 3 x 2 4 -x 1 x 3 4 , z 2 = x 4 3 -2qx 2 3 x 2 4 + x 4 4 , z 5 = x 3 2 x 3 -qx 2 1 x 2 x 3 + qx 1 x 2 2 x 4 -x 3 1 x 4 , z 3 = x 4 1 + 2qx 2 1 x 2 2 + x 4 2 , z 6 = x 5 1 x 2 -x 1 x 5 2 , z 7 = x 5 3 x 4 -x 3 x 5 4 z 8 = x 1 x 2 2 x 3 3 -x 3 2 x 2 3 x 4 -x 3 1 x 3 x 2 4 + x 2 1 x 2 x 3 4 .
Table 2: generators for

I = ker(C[z 1 , . . . , z 8 ] → C[x 1 , . . . , x 4 ] T ). qz 3 1 z 5 -z 1 z 3 z 4 -2z 2 z 6 -z 5 z 8 , z 1 z 2 5 + 2z 4 z 6 + z 3 z 8 , qz 3 1 z 4 + z 1 z 2 z 5 -2z 3 z 7 -z 4 z 8 , z 1 z 2 4 -2z 5 z 7 -z 2 z 8 , -z 4 1 + z 2 z 3 -z 4 z 5 -3qz 1 z 8 , qz 2 1 z 3 z 5 -2z 3 1 z 6 -z 2 3 z 4 + z 3 5 -6qz 6 z 8 , z 2 1 z 4 z 5 + qz 3 1 z 8 + 4z 6 z 7 -z 2 8 , qz 2 1 z 2 z 4 -2z 3 1 z 7 -z 3 4 + z 2 2 z 5 -6qz 7 z 8 , 4z 2 1 z 4 z 5 + qz 3 z 2 4 -qz 2 z 2 5 + 4z 6 z 7 + 8z 2 8
Table 3: generators for the ideal of the Weil divisor

W 2 ⊂ Z. b 1 = z 3 z 7 + 2z 4 z 8 , b 2 = z 2 z 4 + 2qz 1 z 7 , b 3 = z 2 z 3 -4qz 1 z 8 , b 4 = z 3 2 + 12qz 2 7 , b 5 = z 1 z 2 2 -6z 4 z 7 , b 6 = z 2 1 z 2 -qz 2 4 .
the action of the quaternion normal subgroup

Q 8 ⊂ T on S 1 ⊕ S 2 \ {0} is free. The action of Z/3 = T /Q 8 on S 1 ⊕ S 2 /Q 8 produces transversal A 2 -
singularities along a smooth two-dimensional subvariety. Blowing-up along W 1 or W 2 is a partial resolution: it introduces a P 1 fibre over each singular point, and the total space contains a transversal A 1 -singularity.

The homogeneous ideal I ′ 2 ⊂ C[z 1 , . . . , z 8 , b 1 , . . . , b 6 ] that describes the subvariety Z ′ 2 ⊂ P 5 Z is generated by I and 39 additional polynomials. In order to understand the nature of the singularities of Z ′ 2 we consider the six affine charts U ℓ = {b ℓ = 1}. The result can be summarised like this:

The singular locus of Z ′ 2 is completely contained in U 2 ∪ U 3 , so only these charts are relevant for the discussion of the second blow-up. In fact, the corresponding affine coordinate rings have the following description:

R 2 = C[z 1 , b 3 , b 4 , b 5 , b 6 ]/(b 5 b 6 -2qz 1 ) 2 + b 4 (3qb 3 -b 3 6 ) is a transversal A 1 -singularity. R 3 = C[z 1 , z 3 , z 5 , z 6 , b 1 , b 2 , b 6 ]/J,
where J is generated by five elements, listed in table 4. Inspection of these generators shows that Spec(R 2 ) is isomorphic to the singularity (h 3 ⊕h * 3 )/S 3 , the symplectic singularity of Coxeter type A 2 that appears as case 1 in Table 4: generators for the ideal sheaf J of Z ′ 2 ⊂ C 7 in the third chart:

4z 1 b 1 + qz 3 b 2 + z 5 b 6 , z 1 z 5 + z 3 b 1 + qz 6 b 6 , z 2 1 b 6 -z 3 b 2 6 -4qb 2 1 -3z 5 b 2 , z 2 1 z 3 -z 2 3 b 6 -qz 2 5 -12z 6 b 1 , z 3 1 -z 1 z 3 b 6 + qz 5 b 1 + 3qz 6 b 2
Bellamy's theorem. It is well-known that blowing up the singular locus yields a small resolution. For arbitrary n, this is a theorem of Haiman [6, Prop. 2.6], in our case it is easier to do it directly. Thus blowing-up the reduced singular locus of Z ′ 2 produces a smooth resolution Z ′′ 2 → Z. It remains to check that the morphism α 2 : Z 2 → Z is semi-small. For this it suffices to verify that the fibre E = (α -1 2 (0)) red over the origin is two-dimensional and not contained in the singular locus of Z ′ 2 . Indeed, the computer calculation shows that E ⊂ P 5 is given by the equations 2 is irreducible and two-dimensional and dominates the singular locus of Z. Thus the second requirement is fulfilled, too.

The equivariant Hilbert scheme

Though the description of the resolution is simple and straight the method of proof is less satisfying. It is based on explicit calculation that given the complexity of the singularity we were able to carry out only by means of appropriate software. Remark that even for the classical ADE-singularities arising from finite subgroups G ⊂ SU(2) the actual resolutions of C 2 /G could only be described by explicit calculations. The difference to our case essentially is one of complexity: The dimension is four instead of two, there are 8 basic invariants satisfying 9 relations instead of Klein's three invariants with a single relation, and the singular locus is itself a complicated singular variety instead of an isolated point. The first construction that resolved the ADE-singularities in a uniform way was given by Kronheimer in [START_REF] Kronheimer | The construction of ALE spaces as hyper-Kähler quotients[END_REF] in terms of certain hyper-Kähler quotients. Later Ito and Nakamura [START_REF] Ito | McKay correspondence and Hilbert schemes[END_REF] used G-Hilbert schemes to the same effect.

Recall that given a scheme X with an action of a finite group G the equivariant Hilbert scheme G-Hilb(X) is the moduli scheme of zero-dimensional equivariant subschemes ξ ⊂ X such that H 0 (ξ, O ξ ) is isomorphic to the regular representation of G. There is a canonical morphism G-Hilb(X) → X/G which is an isomorphism over the open subset that corresponds to regular orbits. For a linear action G ⊂ SL(V ) it is known that G-Hilb(V ) is smooth if dim(V ) = 2 or 3 and provides a crepant resolution of V /G (see [START_REF] Bridgeland | The McKay correspondence as an equivalence of derived categories[END_REF]).

Thus ρ : H := T -Hilb(S 1 ⊕ S 2 ) → Z = (S 1 ⊕ S 2 )/T is a natural candidate for a resolution. As the generic singularity of Z is a transversal A 2 singularity that arises from a Z/3-action it is clear that ρ is a crepant resolution off the origin. However, it turns out that H has two irreducible components that are smooth and intersect transversely. One of them is the closure H orb of the locus of regular orbits, it dominates the quotient Z. This orbit component also appears as 'dynamical component' or 'coherent component' in the literature. Any other component of H must be contained in the fibre ρ -1 (0), though this is not true in general.

The two factors of the group C * × C * act on S 1 ⊕ S 2 via dilations on the first and second summand, respectively, and the polynomial ring C[S 1 ⊕ S 2 ] may accordingly be decomposed into irreducible T ×C * ×C * -representations, the first terms being An explicit calculation shows that the dimension of the tangent space is 4 in seven points (necessarily smooth points of H) and is 5 in six other points.

C[S 1 ⊕ S 2 ] = L 0 ⊕ (S 1 x ⊕ S 2 y) ⊕ R 0 x 2 ⊕ (R 0 ⊕ L 0 )xy ⊕ R 0 y 2 ⊕(S 1 ⊕ S 2 )(x 3 ⊕ y 3 ) ⊕ (S 0 ⊕ S 1 ⊕ S 2 )(x 2 y ⊕ xy 2 ) ⊕ . . .

The calculation of the quadratic component of the analytic obstruction or

Kuranishi map Hom

T (I i , C[S 1 ⊕ S 2 ]/I i ) → Ext 1 T (I i , C[S 1 ⊕ S 2
]/I i ) yields in all cases a reducible quadric with two distinct factors. This suffices to conclude that there are no further components of H, that H orb is smooth and that the two components H orb and P 2 × P 2 intersect transversely. By the universal property of the blow-up there is a commuting diagram

H orb ւ ց Z ′′ 1 Z ′′ 2 ց ւ Z ,
which conjecturely relates the two-resolutions by a Mukai-flop. In fact, we found the two resolutions first by contracting local models of H that are given as subschemes of a relative Grassmannian over Z. The calculations so far described are insufficient to formally prove that H orb and P 2 ×P 2 intersect along the incidence variety and that this intersection is the exceptional locus for the two contractions. However, recall that equivariant Hilbert schemes can be seen as special cases of quiver varieties ( [START_REF] Nakajima | Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras[END_REF], [START_REF] King | Moduli of representations of finite dimensional algebras[END_REF]) for the McKay quiver [START_REF] Mckay | Graphs, singularities and finite groups[END_REF]

  the action being given by multiplication with m-th roots of unity and permutations of the coordinates.

3 .

 3 (T, S 1 ), where S 1 denotes a two-dimensional representation of the binary tetrahedral group T (see below). However, the technique of Ginzburg, Kaledin and Bellamy does not provide resolutions beyond the statement of existence. Case 1 corresponds to Coxeter groups of type A and Case 2 with m = 2 to Coxeter groups of type B. It is well-known that symplectic resolutions of h ⊕ h * /S n and C n ⊕ C n /(Z/m) n ⋊ S n ∼ = Sym n (C 2 /(Z/m)) are given as follows: For a smooth surface Y the Hilbert scheme Hilb n (Y ) of generalised ntuples of points on Y provides a crepant resolution Hilb n (Y ) → Sym n (Y ).

S 2 .

 2 Here S 0 denotes the standard representation of T arising from the embedding T ⊂ SU 2 . This representation is symplectic, its quotient S 0 /T being the well-known Klein-DuVal singularity of type E 6 . The two other representations can be written as S j = S 0 ⊗ C j , j = 1, 2. They are dual to each other. It is as the subgroup of ⊂ GL(S 1 ) that T appears in the list of Shephard and Todd under the label "No. 4". The diagonal action of T on S 1 ⊕ S 2 provides the embedding of T to Sp 4 that is of interest in our context.Whereas the action of T on S 0 is symplectic, the action of T on S 1 and S 2 is generated by complex reflections of order 3. Overall, there are 8 elements of order 3 in T or rather 4 pairs of inverse elements, forming 2 conjugacy classes. To these correspond 4 lines in S 1 of points with nontrivial isotropy groups. Let C 1 ⊂ S 1 and C 2 ⊂ S 2 denote the union of these lines in each case. Then C 1 × S 2 and S 1 × C 2 are invariant divisors in S 1 ⊕ S 2 . However, the defining equations are invariant only up to a scalar. Consequently, their images W 1 and W 2 in the quotient Z = S 1 ⊕ S 2 /T are Weil divisors but not Cartier. The reduced singular locus sing(Z) is irreducible and off the origin a transversal A 2 singularity. It forms one component of the intersection

  b 1 , b 3 b 5 , b 3 b 4 , b 2 5 -b 4 b 6 and hence is the union of two irreducible surfaces. The singular locus of Z ′

  where x and y are formal symbols indicating the weight with respect to the C * × C * action. Using this decomposition one can see that H contains a further component isomorphic to P 2 ×P 2 : if I ∈ H is to be an ideal contained in the square of the maximal ideal generated by S 1 x⊕S 2 y, of the respectively three copies of S 1 and S 2 of total weight 3 two have to be contained in I.The possible choices amount to picking a line in a three-dimensional space for each of S 1 and S 2 . Of course, one still needs to check that every choice really leads to an admissable ideal.As the map ρ is proper, each equivariant closed subset of H must contain fixed points for the C * × C * -action. These correspond to T -equivariant bihomogeneous ideals I ⊂ C[S 1 ⊕ S 2 ]. Using the given decomposition of the coordinate ring it is not difficult to see that there are 13 such fixed points I i ∈ H. The tangent space to H at I i ist given by Hom T (I i , C[S 1 ⊕ S 2 ]/I i ).

  associated to the given action. As the referee suggests one might try to obtain the diagram above and resolutions of Z in a single stroke by a variation of the stability condition in the construction of the quiver variety. As the McKay quiver for the action of T on S 0 is the Dynkin graph of type E 6 , it is easy to deduce the graph underlying the McMay quiver for the action of T on S 1 ⊕ S 2 :