Frederic Chantreuil
email: frederic.chantreuil@unc.nc

Kévin Fourrey
email: kevin.fourrey@lecnam.net

Isabelle Lebon
email: isabelle.lebon@unicaen.fr

Thérèse Rebière
email: therese.rebiere@lecnam.net

Frédéric Chantreuil

Magnitude and evolution of gender and race contributions to earnings inequality across US regions

Keywords: income inequality, decomposition, Shapley value, racial inequality, gender inequality. JEL Codes: C71, D63, J15, J71

Introduction

Race and gender inequality is a hot topic in the United States. A nationwide poll conducted by CNN and the Kaiser Family Foundation in 2015 found that 49% of US citizens think racism is "a big problem" in society today, whereas the figure was "only" 28% in 2011. Also in 2015, a Gallup's Minority Rights and Relations survey showed 67% of Americans supported affirmative action for women while slightly fewer, 58%, for minorities. These issues are not new. Title VII of the Civil Rights Act of 1964 prohibits employment discrimination based on race, color, religion, sex and national origin. In 1965, President Johnson signed an executive order promoting affirmative action that ensured that all applicants and employees would be considered with no distinction of race, creed, color or national origin. Since then, these measures have been strongly criticized by conservatives, and several states have opted to take action against positive discrimination policies in public institutions. The importance that these issues have maintained in public debates underlines the need for convenient indicators that are easy to use by public policy makers.

Existing income decomposition literature most often focuses on decomposing the evolution of income inequality over time or the difference in inequality between two groups (see [START_REF] Fortin | Decomposition methods in economics[END_REF] for a literature review or Montes-Rojas et al. 2017 for a recent application). Our contribution, on the other hand, proposes a practical Shapley decomposition tool that is able to decompose the inequality of income distribution by individual characteristics (hereinafter referred to as attributes). Our methodology has the advantage of not requiring choosing a modality of reference for categorical variables (e.g. level of education, gender or race) known to affect the results (see Oaxaca and Ransom 1999 for a discussion). Moreover, contrary to the distribution decomposition by income sources proposed by [START_REF] Lerman | Income Inequality Effects by Income Source: A New Approach and Applications to the United States[END_REF], our methodology is suitable for a decomposition by attributes and is applicable to a wide range of inequality indexes: Gini, Theil, Atkinson, etc. Finally, our decomposition does not stand on parametric models and their related hypotheses (as it is the case in Bjorklund et al., 2012). To the best of our knowledge, there is no other decomposition instrument in the literature for carrying out inequality index decomposition by attributes that presently provides all of these advantages. Our contribution is an application of this new Shapley income decomposition methodology in which we concentrate on the discriminatory part of inequalities observed through two subjective factors in income differences -race and gender -that contribute to the inequality in earnings (wages and self-employment revenues), as defined by the Gini index within the population of Blacks and Whites taken together for each of the geographical administrative divisions of the United States over the period 2005-2017. Literature on racial and gender inequality often concentrate on the income gap between groups, i.e. comparing different ethnic groups or women versus men. From this point of view, [START_REF] Sites | What Do We Really Know About Racial Inequality? Labor Markets, Politics, and the Historical Basis of Black Economic Fortunes[END_REF], and [START_REF] Couch | Black-White Wage Inequality in the 1990s: a Decade of Progress[END_REF] show that racial income inequalities in the United States were significantly reduced following the passage of the Civil Rights Act of 1964 and other measures aimed at reducing labor market discrimi-nation, but have not changed significantly since 1974, with the Black-White wage gap remaining around 30% to the late 1980s, while a new convergence was observed in the 1990s. More recently, the median annual income of a family in 2014 was $76,658 for whites and $43,151 for blacks, the second-largest ethnic minority after Hispanics (Economic Report of the President, 2016). As a matter of allocation, decades of sociological research has shown that Black-White inequality in local areas is greater where the Black population is relatively large (see, for instance, [START_REF] Huffman | Racial Wage Inequality: Job Segregation and Devaluation across U.S. Labor Markets[END_REF]). Using the decomposition techniques of [START_REF] Juhn | Accounting for the Slowdown in Black-White Wage Convergence[END_REF], [START_REF] Couch | Black-White Wage Inequality in the 1990s: a Decade of Progress[END_REF] found that greater occupational diversity among Black workers partly explained the reduction of the racial income gap during the 1990s. Convergence is partly due to equalization in the attainment of education and experience and also to the distribution of employment across industries and occupations, rather than being a purely ethnic matter. Sections of the literature explain part of the recently observed income inequality by the reduction in unionization rates. Blacks' private-sector unionization rates surpassed those of White workers for decades (see [START_REF] Jones | Union advantage for black workers[END_REF]. Using data from the CPS , [START_REF] Rosenfeld | Organized Labor and Racial Wage Inequality in the United States[END_REF] showed that African-Americans join unions for protection against discriminatory treatment in nonunion sectors. Private-sector union decline has exacerbated Black-White wage inequality (see also [START_REF] Rosenfeld | What unions no longer do[END_REF], especially among female workers. The gender wage gap has been intensively investigated for decades, but still remains an area of active research (see [START_REF] Blau | The gender wage gap: Extent, trends, and explanations[END_REF]. While the long-term trend shows a significant reduction in the wage gap, convergence has been slower and uneven since the 1990s. Income decomposition methods show that the gender distribution with respect to occupation and industry is the main factor explaining this gap nowadays. Still, gender differences in work interruptions and working times are significant sources of income gaps.

Our contribution differs from the previous literature by concentrating on racial and gender inequality as defined by the contribution of race or gender to overall inequality. In doing so, applying decomposition methods to measure earnings inequality seem to provide an attractive framework for the appraisal of inequality associated to these two characteristics. Among income decomposition methods, those inspired by the Shapley value seem particularly interesting since they provide an explanation of income inequality by determining the contribution of various income sources (i.e. labor and capital incomes), or the contribution of different sub-populations to overall inequality. 2 The Shapley decomposition methods developed so far cannot, however, be used to estimate the share of overall wage inequality that is due to a characteristic such as race or gender. In fact, if the two sub-populations are made up of Blacks and Whites respectively, the results of the decomposition will give the contribution of Blacks to overall income inequality on the one hand and the contribution of Whites to overall inequality on the other, i.e. inequality observed within each sub-population. But this contribution does not reflect inequality between sub-populations, i.e. between Blacks and Whites. Hence, this decomposition framework does not provide for the determination of the contribution of race to income inequality. In order to make up for this drawback of the "classical" Shapley decomposition rule, [START_REF] Chantreuil | Gender contribution to income inequality[END_REF] extended this framework to a third dimension, namely the decomposition of income inequality by attributes. Defining the earnings received by an individual as the sum of several elements, each element representing the part of earnings resulting from each individual's attribute, the Shapley decomposition rule then offers a simple way to determine the contribution of each of individuals' attributes to overall earnings inequality. Here, the order of attributes matters. Our contribution adds to this latter literature as we do not pre-assume the attributes' ordering.

The Shapley decomposition method enables us to distinguish inequality arising from several attributes, some of which are linked to intrinsic individual productivity such as the length of professional experience -for which age is a proxy -and education level, from inequality linked to discriminative factors such as race and gender. Focusing on the latter, we find that the share of earnings inequality attributable to racial affiliation is about 1% to 4%, depending on the 9 designated divisions of the United States Census Bureau, while the contribution of gender to the total observed inequality in the Black and White populations taken together is much larger, at 9% to 13%. Time comparison shows that the contribution of race to earnings inequality has tended to increase in all administrative divisions over the past decade, whereas that of gender tends to have been decreasing recently.

This paper is organized as follows: Section 2 presents the data and empirical evidence on earnings inequality in the United States over the period 2005-2017, focusing on gender and race. Section 3 outlines the Shapley decomposition methodology according to conditional decomposition. Section 4 analyzes the results. Section 5 empirically discusses the incidence of the ordering of attributes. Finally, section 6 contains some concluding comments.

Data and empirical evidence

We used annual earnings, defined as wages plus self-employment incomes in the American Community Surveys (ACS-PUMS, about 1% of the total United States population) from 2005 to 20173 . To be able to draw unambiguous conclusions, highlighting the impact of racial factors that can be observed, we limited our earnings inequality study to the comparison of those determined to be Black or White only; we exclude from our analysis other ethnic groups, as well as those Black and White persons that declared themselves as Hispanic.

Individuals are described according to several characteristics: the geographical administrative division where they live (see Appendix A for a list of states per division); their gender; their level of education (lower than high-school certification, high-school graduation plus some college but without a degree-level qualification, undergraduate degree, and graduate degree or more); and cohort (one group every 4 years from 25-29 years old up to 60-64 years old).4 Individuals over 65 years old, or individuals with declared annual earnings below one hour of the federal minimum wage for each year, are dropped from the data set. Finally, individual US states are grouped into 9 regional sub-sets according to the United States Census Bureau's designated divisions. The total sample over the period 2005-2017 is more than 12.3 million observations. Table 1 shows the distribution of Black and White populations per administrative division together with the 2017 average yearly earnings. In 2017, Blacks accounted for 16.76% of the Black-White population at the Federal level. They earned on average 66% of Whites' earnings. This federal average, however, hides large regional disparities. In line with [START_REF] Huffman | Racial Wage Inequality: Job Segregation and Devaluation across U.S. Labor Markets[END_REF], Table 1 shows that the Black population earns on average much less than the White population in most administrative divisions where the Black share of (Black and White) population is large. The highest gap is observed in the West South Central division where the Black population is the second highest with a 22.4% share. On the contrary, the smallest gap is observed in the Mountain division, where the Black share is only 6.18%. In this region, Blacks indeed earn 72% of Whites' earnings. New England seems to present a different pattern, with a low share of Black (8.35%) and a Black-White earnings ratio of 65%. This geographical distribution itself hides large disparities, since administrative divisions cover many different states (see Appendix A for the list of states per division). Nevertheless, racial differences consistently disadvantage the Black population.

Table 2 shows the 2017 average yearly earnings of Blacks and Whites for men and women by administrative division. Here, regional disparities are again noticeable. The highest differences for men are in the New England, Middle Atlantic and West South Central regions, where Blacks earn 58% of Whites' earnings. Concerning Black women, they suffer relatively less than men but the Black-White gap is calculated on much lower earnings. The highest differences are in New England, where Black women earn 76% of what White women earn, and in the East and West South Central regions (78%). In all divisions, Black women get the lowest average earnings, which represents half of what White men earn at the federal level. Reading: At the Federal level, in 2017, Black men earned on average 59% of White men's earnings. Black women earned on average 79% of White women's earnings, but earnings were much lower for White women than for White men (ratio of 65%) compared to women versus men in the Black population (ratio of 86.2%).

As discussed by [START_REF] Wilson | Black-white wage gaps expand with rising wage inequality[END_REF], the Black-White gap has increased since the 2000s. We focus on the recent time period (2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017) in order to highlight the change of slope that has occurred since 2010. Figure 1 shows the evolution of the Black-White and Women-Men earnings ratios over time. We can see that since 2010, Black-White earnings differences have increased (the Black-White ratio has deteriorated by about 4 percentage points). The Women-Men ratio increased sharply from 2005 to 2010, then stabilized, and finally increased again from 2015 to 2017, thus showing a clear reduction in Women-Men earnings differences (about 6% percentage points in 12 years). Concerning education, Table 3 shows that gender differences in the Black-White earnings gap exist for all education levels: they are much smaller for women, and smaller for the less educated. The difference in education level is partly explained by the type of jobs associated with low education levels, which are often paid the minimum wage or close to it. For each education level, women receive lower average earnings, notably Black women. This empirical evidence underlines the need to consider gender and education together with the racial component.

The income decomposition framework

The decomposition of income inequality into appropriate component contributions usually follows two main paths. The first studies situations in which different sources of total income are examined, 5 while the second considers the influence of population subgroups. 6 For both types of decomposition, the Shapley value has been proved to be useful in many applications; 7 nevertheless, the use of the Shapley decomposition rule by population subgroups such as race or gender does not lead to a clear-cut answer as to the question of the "real" contribution of such individual characteristics. [START_REF] Chantreuil | Gender contribution to income inequality[END_REF] resolved this problem by proposing a solution "assimilating the different dimensions of the status of individuals to a particular wage source in order to assess the contribution of each status". We call this proposed framework "income decomposition by attributes". 8 This method does not, however, deal with a question which remained unresolved in the conditional decomposition: that of the incidence of the ordering of attributes. To properly understand the manner in which distributions associated with each attribute are built, let us consider a brief example. Let us assume 8 individuals (i = A, B, ..., H) for whom we know the income w i as well as two characteristics (also referred to as attributes)

x and y, each having two possible modalities (x = x 1 , x 2 ; y = y 1 , y 2 ). The income of an individual i can be decomposed in two ways, as follows:

w i = w x i + (w x i ,y i -w x i ) + (w i -w x i ,y i ) (1)
or

w i = w y i + (w x i ,y i -w y i ) + (w i -w x i ,y i ) (2)
In equation (1) (respectively eq. ( 2)), the individual income w i is written as the sum of three terms:

• the income share associated with observed attribute x (resp. y): the average income of individuals where attribute x takes modality x i (resp. y i ),

• the income share associated with observed attribute y (resp. x): the difference between the average income of individuals where attribute x (resp. y) takes modality x i (resp. y i ) and attribute y (resp. x) takes modality y i (resp. x i ),

• the income share associated with unobserved attributes: the difference between individual i's income and the average income of individuals presenting the same modalities as individual i for attributes x and y.

5 See [START_REF] Fei | Growth and the Family Distribution of Income by Factor Components[END_REF], [START_REF] Shorrocks | Inequality Decomposition by Factor Components[END_REF], and [START_REF] Lerman | Income Inequality Effects by Income Source: A New Approach and Applications to the United States[END_REF].

6 See [START_REF] Bourguignon | Decomposable income inequality measures[END_REF], [START_REF] Cowell | On the structure of additive inequality measures[END_REF], [START_REF] Shorrocks | The Class of Additively Decomposable Inequality Measure[END_REF][START_REF] Shorrocks | Inequality decomposition by population subgroups[END_REF], [START_REF] Foster | Poverty orderings[END_REF], or [START_REF] Cowell | How much inequality can we explain? A methodology and an application to the United States[END_REF].

7 An incomplete list of applications of the Shapley value to inequality decomposition includes [START_REF] Sastre | Shapley inequality decomposition by factor components: Some methodological issues[END_REF], [START_REF] Israeli | A Shapley-based decomposition of the R-Square of a linear regression[END_REF], [START_REF] Bargain | Analysing the effects of tax-benefit reforms on income distribution: a decomposition approach[END_REF], [START_REF] Devicienti | Shapley-value decompositions of changes in wage distributions: a note[END_REF] and [START_REF] Charpentier | Income inequality games[END_REF]. This table shows that figures associated with attribute x and with attribute y differ according to the order in which the attribute appears in the income decomposition. The share of income inequality (as defined in what follows by the Gini index) associated with each attribute therefore differ slightly depending on the ordering of attributes, as shown in the following results in Table 5. In Table 5 the Gini index is 0.22 and the relative contribution of unobserved attributes to income inequality is 6.85%, i.e. 6.85% of the Gini value remains associated with unobserved characteristics. The share of the observed inequality associated with attribute x is 60.48% (resp. 32.67% for attribute y) in the first decomposition and 64.67% (resp. 28.48% for attribute y) in the second decomposition. Because there is no a priori reason to choose one decomposition over another, in the following we choose to evaluate the share of the observed inequality associated with each attribute as the mean of the share obtained in each decomposition ordering as shown in Table 5. We empirically discuss the incidence of the ordering of attributes in Section 5.

4 Results: contribution of gender and race to earnings inequality

We now apply the methodology outlined above to US observations, determining the average contribution to earnings inequality of several individual characteristics and their development over time. For each year between 2005 and 2017, and each geographical division, the earnings of each individual are treated as explicable in terms of four attributes capable of generating earnings differences either objectively (i.e. based on individual productivity): age (as a proxy for professional experience), education; or discriminatively: gender, race.9 These four attributes represent 24 (=4!) permutations of the ordering of attributes that we must consider in order to obtain the mean contribution of each attribute to earnings inequality. Overall results are presented in Appendix B. A robustness check using the bootstrap percentile method evaluates the validity of our results in Appendix C.

We first concentrated on the evolution of the Gini index. We then analyzed the relative contribution of each of the attributes at the federal level before focusing on race and gender by geographical divisions.

Change in the Gini index

Table 7 of Appendix B reports the mean contribution of each attribute at the federal level and the geographical division level between 2005 and 2017 as a percentage of the overall earnings inequality observed, as defined by the Gini index. As shown in Figure 2, this total earnings inequality measured by the Gini index varies over the period at the federal level and for each administrative division. The Gini index tended to increase uniformly from 2005 to 2013-2014, irrespective of the division. At the end of the period studied, inequality tended to stabilize at a higher level and was even trending towards a slow reduction. Figure 2 also shows that the relative value of regional earnings inequality is somewhat stable over time. Figure 3 shows the regional map for 2017. It can be noted that seaboard regions (which are among the most populous) are more inegalitarian, since they have a higher Gini index. West, North and East North Central as well as Mountain have the lowest Gini index. Not surprisingly, education accounts for most of the earnings inequality among the four selected attributes, with a relative contribution varying from about 15% to 20% of earnings inequality, depending on the year and the geographical division. Then comes gender (about 9% to 14%), age (about 6% to 9%) and finally race (about 1% to 4%). Such a result shows that there are racial and gender discrimination with respect to earnings, but that the inequality observed between Blacks and Whites is mostly linked to non-racial characteristics.

Relative contributions of each attribute at the Federal level

Figure 4 shows the time variation of each of the four attributes at the federal level. The share of earnings inequality associated with attributes observed is mainly related to education and age, two individual characteristics that legitimately affect earnings since they relate to productivity. Higher certification is indeed associated with higher skills, and greater age with greater experience. By contrast, gender and race are not objective criteria when dealing with productivity. These two characteristics nevertheless account for more than 12% of earnings inequality in a context where total earnings inequality (Gini index) of the Black and White populations taken together is tending to increase. Figure 4 shows that the contribution of education to earnings inequality was approximately the same in 2005 and in 2017 (about 18%), but is not stable over time. The subprime crisis may have induced a significant increase in earnings inequality associated with education, since educated workers were mostly less affected than those with lower educational qualifications. This effect diminished with the economic recovery. By contrast, the contribution of age to earnings inequality tends to increase, except during the three most recent years. Gender earnings inequality falls with the share of inequality associated with gender dropping by 2 to 3 percentage points (from about 12% to about 9%) between 2005 and 2017. From 2010 onward, however, this reduction seems to slow at the Federal level. Concerning race, its contribution was stable from 2005 to 2008, then fell for two years before consistently increasing up to 2017. It might be supposed that the rise in earnings inequality between Blacks and Whites also originates in the financial and economic crisis. This phenomenon did not reverse during the recovery, however, and even seems to be getting worse.

Contribution of gender and race by divisions

Let us now concentrate on race and gender. Figures 5 and6 let us to observe the earnings inequality associated with these two attributes between 2005 and 2017. The Federal trends are confirmed at the administrative division level -an increase in inequality associated with race in the most recent years can be observed in almost all geographical divisions together with a reduction in relative gender inequality over the entire period. For the gender contribution, this fell by at least 2% in every division during the period studied. Middle Atlantic, South Atlantic and Pacific, which correspond to most of the East and West coasts, are the least affected by inequality associated with gender (lower than 10% in 2017). Other divisions display a gender share of from 10.5% to 11.5% in 2017.

Comparing Figure 3 of the Gini index map, the contribution of race (Fig. 7) and that of gender (Fig. 8) highlight some additional points. Coastal divisions, which are the most heavily affected by earnings inequality, are also the least affected by gender inequality. West South Central is the only division which has to deal with both a high level of earnings inequality and a large share of earnings inequality associated with gender. It is also one of the divisions where the contribution of race is high.

10 Results for the Mountain division are to be treated with caution since this division has a very low share of Blacks in the Black and White population (3.95%, see Table 1). Studying sub-sample sizes shows that the Mountain division has several sub-samples with less than 5 observations per subgroup (accounting for 223 of the 573 observations belonging to a subgroup with fewer than 5 observations), especially in the Black population with least educational qualifications and in those with the highest. Now let's take a look at the contributions in values (referred to as absolute contributions) to earnings inequality, which can be used to account for total inequality as measured by the Gini index. The relative contribution is the percentage of the Gini index explained by each of the attributes. The absolute contribution is the "level" of inequality directly explained by each of the attributes. Considering these absolute contributions adds to the analysis since an identical share of the Gini index can be associated with a greater/smaller inequality value both in space and in time. Figures 9 and 10 display the absolute contribution of race and gender, respectively, with respect to the Gini index, for each US administrative division in 2017 . These two figures highlight specific features of several divisions. West North Central is distinguished by the lowest level of inequality and also by a racial contribution that is among the lowest, with a limited gender contribution. Symmetrically, the figures from the West South Central division show that a high level of inequality tends to go along with a high absolute contribution of both race and gender. Middle Atlantic and Pacific, which display the highest earnings inequality level, do not, however, conform to this trend: the absolute racial contribution to inequality is moderate, and the gender contribution is among the lowest. Finally, South Atlantic and Mountain both display a moderate level of inequality but with contrasting patterns with regard to the racial and gender contributions. In the South Atlantic division, the absolute contribution of gender is among the lowest and that of race among the highest, while in the Mountain division race has the lowest contribution and gender has a contribution among the highest.

Incidence of the ordering of attributes

In order to evaluate the validity of our results, the impact of the ordering of attributes for both race and gender is discussed. We note that the order does not significantly reduce the contribution of one attribute or another.

As mentioned in Section 3, the value associated with an attribute appears in the earnings disaggregation. The first attribute takes the mean value of one of its different modalities, while the following attributes are only successively associated with mean differences. As a consequence, an attribute located in the first position logically captures a higher proportion of earnings than the following attributes. One might think that it also captures more of the total inequality than when it has a lower position, or at the extreme. This is the case that appears in the example presented in Section 3; however the results presented in Table 6 show that this effect is not systemic.

Table 6 presents, on the left-hand side, the racial contribution when race (R) is the first attribute to be considered (then comes education (E), age (A) and gender (G) in that order) and when race is the last attribute to be considered (following education, age and gender, in that order); and on the right-hand side is the gender contribution when gender (G) is the first attribute and when gender is the last attribute.

Table 6: Incidence of attributes' ordering on the relative contribution of race and gender to earnings inequality at the federal level, as a percentage of earnings inequality as defined by the Gini index Race Gender REAG EAGR GREA REAG

The results in Table 6 confirm some effect of the ordering of attributes on the relative contribution to earnings inequality obtained by the Shapley decomposition method. The impact of race is reduced when considered in the last position as compared with the first position. This reduction (of about 1 percentage point in 2017) does not however challenge our main finding: the earnings spread between blacks and whites has an impact on total inequality that is not insignificant.

Regarding gender, its position in the ordering of attributes also slightly modifies the explained part of total inequality, the magnitude of the impact remaining similar. Notice that the relative contribution of gender is higher when it is placed in the final position. This observation calls into question our previous supposition, that the first position does not automatically correspond to the maximum contribution of an attribute.

Such a result can easily be explained. Earnings associated with the gender attribute is spread over many more modalities when placed in the final position rather than in the first position. In fact, earnings are decomposed into 160 modalities (different age classes, education levels, two races, and two genders) rather than 2 modalities (male/female). If all other attributes are first taken into account, and if the gap between earnings values associated with men and with women is large, the gender contribution to earnings inequality may increase when placed in the final position.

At this stage we observe, for all the orders given in Table 6 and for those resulting from the set of all possible permutations, that race and gender as discriminating factors explain a significant part of overall earnings inequality. Bootstrap testing allows us to confirm the significance of these attributes (see Appendix C).

Conclusion

Income inequality decomposition à la Shapley enables us to derive the contribution of an individual characteristic to the total earnings inequality observed in the US over the period 2005-2017. An important share of this inequality can be explained by determinants of productivity, i.e. the level of skills and the duration of professional experience that we capture through education and age. Other significant factors are less objective. We concentrate on the discriminational part of the inequalities observed, i.e. inequalities associated with ethnic concerns or gender issues.

In a context where earnings inequalities increased over the period 2005-2017 at the federal level and for each administrative division, we show that the share of inequality explained by race has also increased. In 2017, we find that the racial contribution to earnings inequality is about 1% to 4% for all of the nine United States Census Bureau designated divisions. Our results confirm those of [START_REF] Huffman | Racial Wage Inequality: Job Segregation and Devaluation across U.S. Labor Markets[END_REF]: the contribution of race to earnings inequalities is much higher in areas densely populated with Blacks. But race is not the most important source of earnings inequality. Gender indeed explains a much higher share of earnings inequality (about 10%) whatever the geographical division. Unlike race, however, inequalities associated with gender were noticeably lower between 2005 and 2017.

There seems to have been a general stability to developments in earnings inequality during the period of study with regard to the contribution of race and gender to this inequality; there is a general stability from one geographical division to another. This means that the relative situation of the divisions persists over time. We identified geographical areas lastingly affected by strong earnings inequalities, with a marked impact of gender and race, such as in West South Central. By contrast, we identified a low level of inequalities, with the weak impact of the two attributes in, for instance, West North Central.

Our analysis does not account for disparities between states. Work at the state level is compromised because of the size of sub-samples. We might expect different results for the states belonging to the South Atlantic division, for instance, which include both former Confederate states (Virginia, North and South Carolina, Georgia and Florida) and Union states (DC, Delaware, Maryland, West Virginia). One way to bypass the sub-sample size issue would be to re-aggregate states according to their economic and historical characteristics. But the choice of factors driving re-aggregation would be open to criticism. Such an issue is left for further research.

Apart from our caution regarding geographical constraints, our results have several public policy consequences. While it does exist, the direct impact of race on earnings remains moderate. This means that the earnings gap between Blacks and Whites finds its origins in characteristics associated with ethnic minorities (low education, poverty, etc.), rather than being a matter of race in itself. In terms of public policy, such results call into question affirmative action in public institutions in general, but would make it more important in education. Our work also highlights the magnitude of gender in earnings inequality, justifying public action in favor of gender equality.

Figure 1 :

 1 Figure 1: Black-White and Women-Men's earnings ratios from 2005 to 2017

8

  [START_REF] Chantreuil | A note on the decomposability of inequality measures[END_REF] offers a complete methodological note on the Shapley method of decomposition. See the unpublished appendix available from the authors upon request for a short presentation.

Figure 2 :

 2 Figure 2: Change in the Gini index from 2005 to 2017

Figure 3 :

 3 Figure 3: Division map for Gini's index (year 2017)

Figure 4 :

 4 Figure 4: Change in the average relative contribution of attributes to earnings inequality as a percentage at the Federal level

Figure 5 :Figure 6 :

 56 Figure 5: Change in the average relative contribution of race to earnings inequality as a percentage

Figure 7 :Figure 8 :

 78 Figure 7: Division map of the contribution of race to earnings inequality (year 2017)

Figure 9 :

 9 Figure 9: Absolute contribution of race (Y axis) with respect to the Gini index (X axis) (year 2017)

Table 1 :

 1 Black share and average earnings of Blacks and Whites in 2017

		Black	Annual earnings ($) B/W earnings
		share	Black	White	ratio
	New England	8.35% 46,061	71,010	65%
	Middle Atlantic	16.89% 46,268	70,552	66%
	East North Central	12.74% 38,468	57,025	67%
	West North Central	7.66% 37,531	55,527	68%
	South Atlantic	27.84% 41,606	62,002	67%
	East South Central	21.82% 34,070	51,634	66%
	West South Central	22.40% 40,464	63,410	64%
	Mountain	6.18% 42,824	59,386	72%
	Pacific	10.61% 50,340	74,281	68%
	FEDERAL	16.76% 41,632	62,968	66%
	Population : Blacks and Whites only, 25-65 years old, employees and self-employed, 2017.
	Reading: At the Federal level, in 2017, 16.76% of the Black and White population was Black. Blacks
	earned on average 66% of Whites' earnings.

Table 2 :

 2 Men and women's earnings by race and geographical division in 2017

		Men		Women
	Division	Black White B/W	B/W	Black White B/W	B/W
		diff	ratio		diff	ratio
	New England	49,202 85,229 36,026	58%	43,112 56,393 13,281	76%
	Middle Atlantic	49,103 84,584 35,481	58%	43,799 55,409 11,611	79%
	East North Central 41,330 68,886 27,556	60%	36,108 44,128	8,020	82%
	West North Central 40,443 66,072 25,629	61%	34,884 44,155	9,271	79%
	South Atlantic	44,711 74,160 29,449	60%	39,030 48,612	9,581	80%
	East South Central 36,981 61,937 24,956	60%	31,668 40,363	8,695	78%
	West South Central 44,798 77,680 32,882	58%	36,729 47,135 10,406	78%
	Mountain	45,968 71,252 25,284	65%	38,841 45,674	6,832	85%
	Pacific	53,889 87,795 33,906	61%	46,676 58,541 11,865	80%
	FEDERAL	44,946 75,549 30,603	59%	38,760 49,079 10,319	79%

Table 3 :

 3 Men and women's earnings by race and education levels in 2017

		Men			Women
	Education level	Black White B/W	B/W	Black White B/W	B/W
			diff	ratio		diff	ratio
	< High-school de-	27,066	40,028 12,962 67.6% 22,349 24,374	2,025 91.7%
	gree
	≥ 12th grade, no col-	37,174	53,253 16,079 69.8% 30,238 33,877	3,639 89.3%
	lege degree
	Undergraduate level	56,192	87,284 31,091 64.4% 44,761 52,789	8,029 84.8%
	(2-4 years of college)
	Graduate level (≥ 5	87,322 133,476 46,153 65.4% 67,535 78,765 11,230 85.7%
	years of college)
	Reading: At the Federal level, in 2017, Black men with less than a high-school diploma earned on average
	67.6% of White men's earnings, while Black women earned on average 91.7% of White women's earnings.

Table 4 :

 4 Table 4 presents an example with virtual incomes. The first part of the table displays characteristics associated with each of the 8 individuals, the second part presents the income figures associated with equation (1), the third part presents the incomes figures associated with equation (2). Example of income disaggregation by attributeIncome associated with eq. (1) Income associated with eq. (2) i w ix i y i w x i w x i ,y i -w x i w i -w x i ,y i w y i w x i ,y i -w y i w i -w x i ,y i

	A 1000 x 1 y 1 1225	-200	-25	1575	-550	-25
	B 1050 x 1 y 1 1225	-200	25	1575	-550	25
	C 1350 x 1 y 2 1225	200	-75	2250	-825	-75
	D 1500 x 1 y 2 1225	200	75	2250	-825	75
	E 2200 x 2 y 1 2600	-475	75	1575	550	75
	F 2050 x 2 y 1 2600	-475	-75	1575	550	-75
	G 2900 x 2 y 2 2600	475	-175	2250	825	-175
	H 3250 x 2 y 2 2600	475	175	2250	825	175

Table 5 :

 5 Relative contributions and Gini index Relative contrib. Relative contrib. Relative contrib.

	Gini

Table 7 :

 7 Average relative contribution of attributes to income inequality as percentage and Gini index

	Geog., year	Race Gender	Age Education Unobserved GINI index
	Federal, 2005	2.50	12.47 6.03	17.84	61.16	0.4427
	Federal, 2006	2.59	12.07 6.28	18.15	60.91	0.4458
	Federal, 2007	2.55	11.85 6.40	18.24	60.97	0.4482
	Federal, 2008	2.60	11.57 6.61	17.67	61.55	0.4496
	Federal, 2009	2.44	10.82 6.66	19.18	60.92	0.4515
	Federal, 2010	2.33	10.30 6.82	19.16	61.38	0.4511
	Federal, 2011	2.43	10.30 7.00	18.99	61.28	0.4539
	Federal, 2012	2.57	10.21 7.37	18.66	61.18	0.4548
	Federal, 2013	2.74	10.12 7.40	18.11	61.64	0.4618
	Federal, 2014	2.82	10.00 7.77	18.01	61.40	0.4613
	Federal, 2015	2.97	10.01 7.79	17.83	61.39	0.4617
	Federal, 2016	2.97	9.84 7.76	17.93	61.50	0.4608
	Federal, 2017	3.10	9.64 7.85	17.77	61.65	0.4598
	New England, 2005	1.13	13.21 6.09	18.45	61.12	0.4370
	New England, 2006	1.18	13.20 6.14	18.30	61.19	0.4401
	New England, 2007	1.11	13.01 6.73	18.47	60.68	0.4429
	New England, 2008	1.20	12.39 6.92	18.09	61.40	0.4471
	New England, 2009	1.36	11.85 7.13	19.56	60.10	0.4479
	New England, 2010	1.24	11.42 7.20	19.40	60.74	0.4483
	New England, 2011	1.26	11.41 7.37	19.36	60.60	0.4449
	New England, 2012	1.40	10.88 7.80	19.23	60.70	0.4546
	New England, 2013	1.26	10.55 8.04	18.90	61.24	0.4564
	New England, 2014	1.42	10.55 8.50	18.62	60.92	0.4617
	New England, 2015	1.73	10.57 8.27	18.44	60.99	0.4547
	New England, 2016	1.73	10.37 8.54	18.11	61.25	0.4508
	New England, 2017	1.76	10.01 8.43	18.70	61.10	0.4506
	Middle Atlantic, 2005	2.37	12.18 5.48	18.92	61.05	0.4553
	Middle Atlantic, 2006	2.49	11.62 5.55	19.30	61.04	0.4544
	Middle Atlantic, 2007	2.49	11.01 5.62	19.34	61.55	0.4600
	Middle Atlantic, 2008	2.49	10.89 6.07	18.52	62.03	0.4605
	Middle Atlantic, 2009	2.36	10.31 6.33	19.96	61.04	0.4644
	Middle Atlantic, 2010	2.27	10.16 6.15	19.72	61.70	0.4588
	Middle Atlantic, 2011	2.35	10.14 6.50	19.42	61.61	0.4625
	Middle Atlantic, 2012	2.44	9.85 6.71	19.24	61.75	0.4588
	Middle Atlantic, 2013	2.57	9.63 7.13	18.85	61.83	0.4713
	Middle Atlantic, 2014	2.81	9.38 7.35	18.59	61.87	0.4697
	Middle Atlantic, 2015	2.89	9.53 7.54	18.73	61.31	0.4704
	Middle Atlantic, 2016	2.77	9.50 7.48	18.94	61.32	0.4684
	Middle Atlantic, 2017	3.05	9.17 7.47	18.39	61.92	0.4698
	East North Central, 2005	1.79	14.00 6.18	17.37	60.65	0.4282
	East North Central, 2006	1.98	13.26 6.74	17.89	60.13	0.4320
	East North Central, 2007	1.84	13.30 6.64	18.42	59.81	0.4365
	East North Central, 2008	1.98	12.78 6.60	17.58	61.07	0.4350
	East North Central, 2009	1.68	11.53 6.63	19.52	60.64	0.4424
	East North Central, 2010	1.69	10.68 6.95	19.60	61.08	0.4408
	East North Central, 2011	1.89	10.75 7.39	19.48	60.49	0.4432
	East North Central, 2012	1.93	10.78 7.78	18.73	60.79	0.4456
	East North Central, 2013	2.08	10.94 7.48	18.53	60.98	0.4499
	East North Central, 2014	2.10	10.75 7.84	18.34	60.97	0.4454
	East North Central, 2015	2.27	11.10 7.72	18.10	60.80	0.4487
	East North Central, 2016	2.36	10.76 7.54	18.26	61.07	0.4489
	East North Central, 2017	2.48	10.90 7.72	18.17	60.72	0.4435
	West North Central, 2005	1.21	13.75 6.44	16.33	62.27	0.4175
	West North Central, 2006	1.34	13.19 6.32	16.88	62.27	0.4215
	West North Central, 2007	1.32	13.06 6.66	17.01	61.95	0.4223
	West North Central, 2008	1.40	13.02 6.47	16.70	62.42	0.4286
	West North Central, 2009	1.33	11.91 7.15	18.01	61.61	0.4265
	West North Central, 2010	1.37	11.22 7.12	18.32	61.97	0.4255
	West North Central, 2011	1.20	10.90 7.45	17.70	62.75	0.4260
	West North Central, 2012	1.38	11.71 7.22	17.44	62.25	0.4308
	West North Central, 2013	1.44	11.15 7.51	16.84	63.05	0.4313

See Chantreuil and Trannoy (2011, 2013) and[START_REF] Shorrocks | Decomposition procedures for distributional analysis: a unified framework based on the Shapley value[END_REF] for the definition of the Shapley decomposition rules determining the contribution of different income sources or different sub-populations to overall income inequality.

Using the IPUMS-CPS from University of Minnesota with weekly earnings from 1989 to 2015, gives similar results over the period 2005-2015, but it is open to criticism arising from a much weaker sample size.

We exclude individuals between 20 and 24 from our analysis since the sample contains few observations with a graduate degree or more.

We concede that earnings also depends on occupation. This attribute is excluded from the analysis, however, implying that the impact of occupation on earnings inequality is captured by the residual due to the limited size of each subset of the sample.

Note that bootstrapping in the ACS database requires a large amount of memory and calculation time. Such a calculation would not have been possible without the use of the ATOS BULL Myria computer available at CRIANN, Normandy, France.

C Robustness: confidence interval

We performed a robustness check of the results obtained in Section 4 using the bootstrap percentile method. The contributions obtained via the Shapley decomposition method do not a priori follow a normal distribution. We therefore performed a bootstrap sampling to calculate a simple percentile confidence interval. For 2017, at the Federal level, we performed 1,000 resamples with replacement of the initial database and, for each sample, calculate the (absolute) contribution as defined in Section 4. Bootstrap Confidence Intervals are reported in Table 8. 11 The 2017 absolute contribution is found to be within 95% of the CI, thus demonstrating the robustness of our results regarding the existence of the two discriminational characteristics (race and gender) when decomposing income inequalities.


domain: spi phys

text:
Jean-François Ripoll

Mick H Denton

David P Hartley

Geoffrey D Reeves

David Malaspina

Gregory S Cunningham

Ondrej Santolík

Scott A Thaller

Vivien Loridan

J F Fennell

J.-F Ripoll

D L Turner

W S Kurth

C A Kletzing

M G Henderson

A Y Ukhorskiy

J.-F Ripoll

Scattering by whistler-mode waves during a quiet period perturbed by substorm activity

Keywords: Electron, wave-particle interactions, radiation belts, moderate substorm activity, substorm, whistler waves

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Earth's electron radiation belts consist of two toroidal belts of energetic charged particles (~0.1 to ~10 MeV) surrounding Earth, with the inner belt sitting between 1 and 3 Earth radii (RE) and the outer belt typically lying between 3 and 7 RE in the equatorial plane. Between the two radiation belts lies a zone often devoid of electrons, called the slot region. This ideal representation is highly dynamic in both space and time; first, because dynamic/disturbed conditions bring complex dynamic mono-(e.g. [START_REF] Reeves | Electron acceleration in the heart of the Van Allen radiation belts[END_REF] or multi-belt structures (e.g., [START_REF] Baker | A Long-Lived Relativistic Electron Storage Ring Embedded in Earth's Outer Van Allen Belt[END_REF] and, second, because Earth's radiation belt location is energy dependent (e.g. [START_REF] Lyons | Equilibrium structure of radiation belt electrons[END_REF]. In this work, we specifically address the energy dependence of the radiation belts, between 2 and 6 RE in the equatorial plane, during a period of quiet state perturbed by substorm activity (e.g. [START_REF] Friedel | Substorm onsets observed by CRRES: Determination of energetic particle source regions[END_REF][START_REF] Turner | Energetic electron injections deep into the inner magnetosphere associated with substorm activity[END_REF] and preceding a high-speed solar wind-stream (HSS) (Denton et al., 2017). We focus on the characterization of the fast slot formation and the outer belt decay, which leads to a drastic reduction in the flux level of the whole radiation belt system (e.g. [START_REF] Reeves | Energy-dependent dynamics of keV to MeV electrons in the inner zone, outer zone, and slot regions[END_REF]Ripoll et al., 2016a). Understanding the variability of the electron radiation belts, to the point of predictability, is one of the great outstanding challenges in space weather research (cf. review in Ripoll et al.. (2020) and references therein). All progress made will converge toward new advances in the hardening of electronic spacecraft systems and their protection from severe space weather conditions. This study is based on the analysis of high-quality electron flux observations from NASA's Van Allen Probes [START_REF] Mauk | Science objectives and rationale for the Radiation Belt Storm Probes mission[END_REF]. Data analysis is supported by computations made with the CEVA code (e.g. Reveillé et al., 2002) that is based on the quasi-linear Fokker-Planck formalism. We feed the CEVA code with event-specific pitch-angle diffusion coefficients (Ripoll et al., 2016a[START_REF] Ripoll | Effects of whistler mode hiss waves in March 2013[END_REF](Ripoll et al., , 2019a(Ripoll et al., , 2019b) ) that are computed from in-situ measurements of plasmaspheric hiss waves and the plasma density observations made by the Van Allen Probes. This computational step requires massivelly parallel computations in order to perform the computation of diffusion coefficients at each satellite outbound pass at time, t, and location, L, i.e.

, , , , , , knowing the various locally measured wave properties, generically denoted here , for i=1…4, and the plasma density, , , for any electron of energy, E, and equatorial pitch angle, . An alternative methodology is the computation of a single diffusion coefficient from statistical mean properties, , , , ,

denoting a temporal average) which has low numerical cost but neglects correlations between the properties of the ambient domain and, therefore, introduces some error.

This method is applied, for instance, to study hiss waves effects in [START_REF] Ma | Characteristic energy range of electron scattering due to plasmaspheric hiss[END_REF]. Our effort of keeping the plasma/wave coupling is motivated by the will to reach the maximum accuracy possible, though approximations always remain (cf. discussion in section 4). This event-driven method has proven successful for an event of 10 days during a quiet period of March 2013 (Ripoll et al., 2016a[START_REF] Ripoll | Effects of whistler mode hiss waves in March 2013[END_REF](Ripoll et al., , 2019a(Ripoll et al., , 2019b)). The new questions we address herein are: can this event-driven method be successful when applied to another, more complex and dynamic, event? How does substorm activity affect the results, their accuracy, and the method?

Section 2 of the article describes the event chosen for this study, including the presentation of both the whistler-mode waves and the plasmasphere activity. In Section 3 we discuss the numerical method used to compute the event-specific pitch-angle diffusion coefficients. Section 4 shows and discusses NASA's Van Allen Probes flux observations and the quasi-linear simulations of the scattering associated to the whistler-mode wave activity that sustains our analysis. We explain that the moderate substorm activity contributes to decreasing plasmaspheric density, which, in turn, enhances pitch angle diffusion from whistler-mode waves. However, this effect is found insufficient to compensate for the lowering of whistler-mode wave amplitudes that occurs simultaneously. There is nevertheless an enhancement of the mean scattering above L~4.7 for electrons up to 1 MeV. This is attributed to the strengthening hiss activity which occurs in between the substorm injections.

Case analysis: 30 of June 2014 -10 of July 2014

In this study, we examine the evolution of the radiation belts during a 10-day period in 2014, from 30 June to 10 July. This period corresponds to a period of calm-tostorm geomagnetic conditions, for which the upcoming storm at the end of the event is driven by high-speed solar wind-streams (HSSs). This event was selected in Denton et al.

(2017) who studied the evolution of the ions composition during a 10 day interval preceding the HSS and the HSS itself. High-speed solar wind-streams are well described in Borovsky & Denton (2009[START_REF] Borovsky | On the heating of the outer radiation belt to produce high fluxes of relativistic electrons: Measured heating rates at geosynchronous orbit for high-speed stream-driven storms[END_REF], 2016), [START_REF] Denton | Magnetosphere response to high-speed solarwind streams: A comparison of weak and strong driving and the importance of extended periods of fast solar wind[END_REF][START_REF] Albert | Quasi-linear diffusion coefficients for highly oblique whistler mode waves[END_REF] and [START_REF] Denton | An improved empirical model of electron and ion fluxes at geosynchronous orbit based on upstream solar wind conditions[END_REF] and Hartley et al., 2013 and references therein. This type of event leads to the injection of fresh plasma into the inner magnetosphere, via geosynchronous orbit (GEO), [START_REF] Friedel | Plasma sheet access to the inner magnetosphere[END_REF][START_REF] Korth | Plasma sheet access to geosynchronous orbit[END_REF] within a few hours of HSS arrival at the magnetopause (Denton & Borovsky, 2009). In our study, we will focus on the whole period that is reasonably quiet but perturbed by moderate substorm activity. On July 9 activity picked up with the HSS and the analysis stops on July 10.

Substorm activity

Figure 1 contains plots of the geomagnetic conditions, including AE (top) and Kp (bottom) geomagnetic indices. The AE index is well-correlated with the occurrence of substorms-events that are known to inject plasma into the inner magnetosphere, either in isolation or during multiple substorm onsets [START_REF] Mauk | Correlationof Kp With the Substorm-Injected Plasma Boundary[END_REF][START_REF] Reeves | Numerical Tracing of Energetic Particle Drifts in a Model Magnetosphere[END_REF][START_REF] Sergeev | Event study of deep energetic particle injections during substorm[END_REF]Henderson, Reeves, et al., 2006;Henderson, Skoug, et al., 2006;[START_REF] Turner | Energetic electron injections deep into the inner magnetosphere associated with substorm activity[END_REF]. During this time period, the AE index remains mostly at very low levels, except for two excursions indicative of substorms (June 30

and July 3-4). The Kp index (plotted here as a mean value every 8h) and SYM/H index (Figure 1, 4 th image) were also at very low levels. The maximum Kp was reached twice on June 30 and July 10 (Kp = 3), and the minimum SYM/H occurred on July 3 (SYM/H=-20 nT). Apart from these excursions, both indices highlight the very low levels of magnetospheric activity (Kp ≤ 2 and SYM/H~0) during this interval.

The substorm activity during this time is more complex than it appears from the AE index and Van Allen Probes. There are a number of substorms (and injections) that can be seen from the electron flux reported from 6 satellites of the LANL GEO constellation in Figure 2. Comparison with Van Allen Probes flux is also provided in the 2 bottom panels. Van Allen Probes apogee MLT was ~9 on July 1. Days of quietness with steady flux are July 1, 5, 6. Small injections occur on July 2 near 12:00 while Probe A was at perigee and Probe B was at very low L and on July 4 while both Probes are in the inner zone. These two injections appear to be single injection followed by drift echoes.

July 3 has a longer interval of substorm activity observed both by LANL satellites at GEO and by Van Allen Probes. At the beginning of July 7, we observe another injection that is fairly typical of an isolated substorm, followed by drift echoes. This injection is also visible in the Van Allen Probes data that are located at higher L-shell. Some of these substorms affect the plasmapause location more than others. Substorm injections on July 3 and 7 modify the density and the wave activity as shown in the next subsection.

There is also activity on July 8 which perturbation on the density is conversely quite mild and not well apparent (cf. next section and Figure 3). July 9 has two more isolated substorms when Van Allen Probes were at low L. These substorms did seem to move the plasmapause inward as well. The HSS event starts on July 10 with the beginning of the flux enhancements visible in Figure 2.

Plasma density

NASA's Van Allen Probes consist of twin spacecraft on near-equatorial elliptic orbits. The orbital period is ~9 hr, the MLAT coverage |MLAT| < 20°, and the orbital apogee is ≈ 6 RE. Full precession of the apogee through all MLTs occurs approximately every 2 years. In this work, we use the data from two waves instruments, the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) [START_REF] Kletzing | The electric and magnetic field instrument suite and integrated science (EMFISIS) on RBSP[END_REF] and the Electric Fields and Waves (EFW) instrument [START_REF] Wygant | The electric field and waves instruments on the radiation belt storm probes mission[END_REF], as well as data from the Magnetic Electron and Ion Spectrometer (MagEIS) particle instrument [START_REF] Blake | The Magnetic Electron Ion Spectrometer (MagEIS) instruments aboard the Radiation Belt Storm Probes (RBSP) spacecraft[END_REF], all onboard the Van Allen Probes [START_REF] Mauk | Science objectives and rationale for the Radiation Belt Storm Probes mission[END_REF].

The plasma density enters directly in the dispersion relation that controls the strength of wave-particle interactions. An accurate knowledge of its value is therefore important for the computation of the electron diffusion and scattering performed in the next section. For instance, [START_REF] Albert | VLF transmitters and lightning-generated whistlers: 2. Diffusion of radiation belt electrons[END_REF] found that the lifetimes for 1-2 MeV electrons for L=[1.5,2] was a factor of 4-5 times greater using a low-density model than a high-density model. Because plasma density influences so greatly diffusion coefficients, it is a source of a great amount of uncertainty in electron flux prediction [START_REF] Camporeale | On the propagation of uncertainties in radiation belt sim-ulations[END_REF]. The plasma density for our event period is derived by procedure that combines Van Allen Probes data and models when data are lacking. The density is first either derived directly from the upper hybrid resonance frequency [START_REF] Kurth | Electron densities inferred from plasma wave spectra obtained by the waves instrument on Van Allen Probes[END_REF] measured by EMFISIS, or when EMFISIS density is unavailable, derived from spacecraft floating potential data calibrated against plasma densities determined from the upper hybrid resonance frequency [START_REF] Ripoll | Effects of whistler mode hiss waves in March 2013[END_REF]. Spacecraft floating potential is measured by the EFW instrument. Combining both techniques allows to obtain the density the majority of the time (as shown and discussed below and in Figure 3). Nevertheless, if the density from either technique is unavailable, the procedure automatically replaces the unknown density by the statistical model of [START_REF] Albert | Analysis of quasi-linear diffusion coefficients[END_REF] within the plasmasphere and the [START_REF] Sheeley | An empirical plasmasphere and trough densitymodel: CRRES observations[END_REF] expression outside. In these cases, the plasmapause location is needed to differentiate between the two models. This is defined from the 100 #/cc level extracted from both EMFISIS and EFW database or, if the 100 #/cc level (+/-30%) is also undefined, we use the empirical plasmapause model of [START_REF] Carpenter | An ISEE/whistler model of equatorial electron density in the magnetosphere[END_REF] with the knowledge of the Kp* index in order to separate the dense plasma from the low plasma density. When the density is obtained from the measurements, as here in most cases, there is no need for defining a plasmapause in the modeling.

Figure 3(top left) shows the plasma density from 1 to 10 of July 2014. We accumulate all densities which fall into a given time bin i, t in [ti, ti + Δt), and into a given L bin j, L in [Lj, Lj + ΔL]. This gives a 2-D matrix [t, L], which has a set of values in each cell (i, j). For each cell, we then calculate the mean value of the density (and of all quantity mentioned in the following). We choose a time scale of Δt = 8 h, as a compromise between the time resolution (the lowest minimum time step between perigee and apogee is Δt = 4.5 h.), a day integer fraction, and a not too prohibitive computational time. This discretization gives 30 time samples for 10 days. We also choose to treat 43 points in L shell, from L = 1.3 to L = 5.5 (by steps of ∆L = 0.1), at which values have been averaged over the available latitudes and MLT of the probes, resulting in a fine grid of 43 × 30 = 1290 nodes, within which all models will be evaluated. This method was first established in Ripoll et al., (2016a[START_REF] Ripoll | Effects of whistler mode hiss waves in March 2013[END_REF]. Figure 3(left) shows that a lack of density measurements occurs below L=1.7, where the density model of [START_REF] Albert | Analysis of quasi-linear diffusion coefficients[END_REF] replaces the missing observations, without any impact on this study. We will also show below that the [START_REF] Sheeley | An empirical plasmasphere and trough densitymodel: CRRES observations[END_REF] model is never used for this event as the Van Allen Probes density is always defined below L=5.5.

The inner plasmasphere, defined here as a density larger than 100 #/cc, extends up to L~5.5 for the quietest times. Figure 3 (top right) shows the few cells below L=5.5

in which the density level goes below 100 #/cc. During the main substorm activity associated with the AE peaks [START_REF] Mauk | Correlationof Kp With the Substorm-Injected Plasma Boundary[END_REF][START_REF] Reeves | Numerical Tracing of Energetic Particle Drifts in a Model Magnetosphere[END_REF][START_REF] Sergeev | Event study of deep energetic particle injections during substorm[END_REF]Henderson, Reeves, et al., 2006;Henderson, Skoug, et al., 2006;[START_REF] Turner | Energetic electron injections deep into the inner magnetosphere associated with substorm activity[END_REF] occurring during short durations (<8 hours) on July 3, 4, and 5, we observe short local compression or erosion of the plasmasphere down to L~4.5. The incoming HSSs on July 10 is also visible. Comparing Figure 2 and 3 confirms the substorms on July 3 to 4, 7 and 9 affect the density. These perturbations of the plasmasphere by substorm injections have direct consequences on the whistler-mode wave activity and will be discussed further in the next section.

Figure 3 (bottom left) provides an alternative computation of the plasma density based on observations of plasmaspheric hiss [START_REF] Hartley | Determining plasmaspheric densities from observations of plasmaspheric hiss[END_REF]. This new approach has two direct interests. First, it provides a verification of the density level and a replacement value when direct measurements are unavailable. Though this method does contain some approximations, such as the assumption of parallel propagation, it has been shown to provide results consistent with other methodologies [START_REF] Hartley | Determining plasmaspheric densities from observations of plasmaspheric hiss[END_REF]. Second, it provides an indicator of the hiss activity; if the density can be inferred from the hiss waves, it follows that hiss must be present at that time and location. We can see that hiss waves were present below L=5 for the whole period, except during the substorm activity on the July 3 and 4 and before the HSS on July 10. We will confirm these interpretations in the next section from the wave observations. Limitation of the computations to L=5 is not mandatory per se but guarantees greater accuracy of the density values as the plasmasphere erosion and the absence of hiss occurs above L=5, particularly as magnetospheric activity increases. be called the denser part of the plasmapause region. This region is often located above L~4.7 for this event, and above L=5 outside of the substorm injection, and is widely spread over L-shell, except at the time of substorm injections. We will show below that this region contains whistler-mode waves, likely whistler-mode hiss waves due to their low mean frequency and to the high density shown in Figure 3(bottom left) (though checking the microscopic structure is beyond the scope of this study), which will contribute to electron scattering in the outer belt. [START_REF] Malaspina | Variation in plasmaspheric hiss wave power with plasma density[END_REF] also found a correlation between whistler-mode hiss wave power and the plasma density (for L>2), independent of geomagnetic activity; hiss power increases with density. For instance, in Ripoll et al., (2019b) this correlation was illustrated for various AE index, i.e. various substorm conditions indicative of the magnetic activity (see also [START_REF] Malaspina | The distribution of plasmaspheric hiss wave power with respect to plasmapause location[END_REF]. Density and hiss power are thus coupled, with good knowledge of their values at all locations and all times being critical to obtaining the best accuracy.

Wave properties

We use wave measurements from the EMFISIS instrument, restricting the measurements to right-hand polarized waves with ellipticity [START_REF] Santolik | Magnetic component of narrow-band ion cyclotron waves in the auroral zone[END_REF] above 0.2, and to a frequency range typical for plasmaspheric hiss from 50 Hz to 2 kHz [START_REF] Santolik | Propagation analysis of plasmaspheric hiss using Polar PWI measurements[END_REF][START_REF] Hartley | Statistical properties of plasmaspheric hiss from Van Allen Probes observations[END_REF]. These criteria are still not sufficient to select whistler-mode hiss waves only and can lead to the selection of whistler-mode chorus waves at high L-shell during disturbed time. The knowledge of the density and the plasmapause location can, in turn, help discriminate whether the whistler-mode waves are indeed hiss or chorus. But this task still remains complex when the plasmasphere density is at intermediate levels in the plasmapause region, between ~100 #/cc and ~10#/cc. We overcome that complexity by 1) identifying as best as possible the plasmasphere and, therefore, the hiss location, 2) perform our computation of whistler effects without trying to distinguish whistler-mode hiss waves from whistler-mode chorus waves. The latter task is uncommon and made possible in this work because the knowledge of all wave and plasma properties is locally available such that no statistical model is involved for either statistical hiss or chorus wave properties. It should be noticed that high frequency whistler mode waves occurring in low-density regions (below 30 /cc) are assimilated to chorus waves in this study, although the microscopic structure of these waves has not been checked. The main limitation of our method comes from the upper frequency we impose on the whistler-mode waves (2 kHz), which results in the exclusion of higher frequency waves (e.g. upper band chorus waves) from this analysis. Missing the upper band of the chorus frequency spectrum leads to an underestimation of pitch angle diffusion at low energy (<100 keV) for L>4.5. This is not a concern for this study since we focus on >100 keV electrons.

Figure 4 shows the whistler-mode wave properties for the period of interest, and that enter in the computation of hiss pitch angle diffusion effects (cf. [START_REF] Ripoll | Effects of whistler mode hiss waves in March 2013[END_REF] for their definition). Figure 4 (left) shows the mean value of the whistler-mode wave frequency for each node. In the rare case of large wave normal angles, above ~60°, pitch angle diffusion drastically reduces and electron lifetime increases [START_REF] Ripoll | Electron lifetimes fromnarrowband wave-particle interactions within the plasmasphere[END_REF][START_REF] Albert | Quasi-linear diffusion coefficients for highly oblique whistler mode waves[END_REF]. We find some similarities with the wave observations made in [START_REF] Ripoll | Effects of whistler mode hiss waves in March 2013[END_REF] for a quiet event in March 2013.

Computation of diffusion coefficients

Validity of the approximations used in the model

In this study, bounce averaged diffusion coefficients Dα0α0(L, E, α0, t) are computed following the equations of Lyons et al. (1972), which uses both the low frequency (ωm/ωce0 < 1) and the dense plasma (ωmωce0/ωpe 2 ≪ 1) approximations. To demonstrate the validity of these assumptions, these quantities are plotted in Figure 5 (center) and (right). Both of these approximations have been shown to be reliable in the plasmasphere [START_REF] Albert | Analysis of quasi-linear diffusion coefficients[END_REF][START_REF] Glauert | Calculation of pitch angle and energy diffusion coefficients with the PADIE code[END_REF] due to the dense plasma, large magnetic field, and the dominance of low-frequency whistler-mode waves. Furthermore, they have been shown to be more generally applicable to the plasmasphere during quiet times, when the plasmasphere expands out to L~5.5 [START_REF] Ripoll | Effects of whistler mode hiss waves in March 2013[END_REF](Ripoll et al., , 2019a)). Figure 5 demonstrates that these approximations still hold during the moderate perturbations brought by substorm activity. Only on July 3 rd (date of the largest substorm) and on July 10 th (date of the HSS) for L>5.2 (4 cells), are these approximations close to being violated, with ωmωce0/ωpe 2 >0.01. We note that this may introduce some inaccuracy of our results at these dates and locations. This approximation is applicable primarily due to the low frequency of the whistler-mode waves, which compensate for the low density associated to the erosion of the plasmasphere during the substorm activity. During the various substorms on July 3, 4, and 7, as well as during the HSS on July 10, we note that ωpe/ωce0 (Figure 5, left) drops by a factor ~4, from 20-25 down to ~6, a clear indicator of the plasma density decrease.

Diffusion coefficients

The computation of the bounce-and drift-averaged equatorial pitch-angle diffusion coefficient is performed for each 8 h interval at each L-shell (0.1 L resolution) based on the observed wave properties described in section 2. Mathematical expressions of the diffusion coefficients can be found in (Lyons et al. 1972;[START_REF] Albert | Analysis of quasi-linear diffusion coefficients[END_REF] are computed for 60 energies (E varying from 0.07 MeV to 6MeV) at every node (t, L), i.e., 75 600 times (43L × 30t × 60E). A grid of 256 pitch angles in [0, 90°] and n cyclotron harmonics is used, with the harmonic number determined by an estimate (Mourenas and Ripoll, 2012). A massively parallel supercomputer is required to generate this database of 19.3M diffusion coefficients, with a total cost of computation of ~12 hours/processor for ~1300 processors, i.e. ~15600 hours for our ~10-day event.

The result is a data-derived, time-varying diffusion coefficient, , , , = , , Δ , , , , Δ , , , , , , which depends on (t, L)

through the mean wave frequency, , the wave frequency width, Δ , the mean wave normal angle, , the wave normal angle width, Δ , and the plasma density, .

Dependence on energy, E, and equatorial pitch-angle, α0, is inherited from the resonance and dispersion relations used for deriving . Time-varying diffusion coefficients are plotted in Figure 6 at L=3.5 and L=4.5, with each panel showing the first 8h for each day of the event (307 200 values, 1.6% of the database). It is evident that the diffusion coefficients are highly variable from one day to the next. This is due to the variability of the wave properties and the plasma density. Typically, the wave normal angle influences the diffusion profile; a low wave-normal angle produces cyclotron diffusion which increases with pitch angle, whereas an intermediate wave-normal angle produces cyclotron diffusion that is approximately constant with respect to pitch angle. We observe higher diffusion coefficients, for instance on the 07/01 and on 07/05, which coincides with non-perturbed times. It is important to note that wave properties are correlated together as well as correlated with the plasma density as previously discussed (and in [START_REF] Malaspina | The distribution of plasmaspheric hiss wave power with respect to plasmapause location[END_REF][START_REF] Malaspina | Variation in plasmaspheric hiss wave power with plasma density[END_REF]Ripoll et al., 2016a[START_REF] Ripoll | Effects of whistler mode hiss waves in March 2013[END_REF]Ripoll et al., , 2019aRipoll et al., , 2019b)).

Computing time-varying diffusion coefficients, as done here, allows us to keep all these correlations. The alternative method (not chosen here) of averaging each wave property and the plasma density separately before plugging them into a single diffusion coefficient, i.e. , Δ , , Δ , , , or , Δ , , Δ , , , (e.g. [START_REF] Zhu | New hiss and chorus waves diffusion coefficient parameterizations from the Van Allen Probes and their effect on long-term relativistic electron radiation-belt VERB simulations[END_REF] in which . denotes a temporal or both a temporal and a spatial averaging, cannot conserve the various correlations between the various properties and introducing de facto an error. Solving for the diffusion coefficients at small time and spatial scales allows us to avoid any artificial numerical smoothing that inevitably occurs if one uses averaged wave and plasma properties.

Figure 7 shows the averaged diffusion coefficients used for the simulation, at 4 selected L-shells. This diffusion coefficient is computed from the temporal average of the 30 time-varying diffusion coefficients at each , , :

Figure 7 shows that pitch angle diffusion occurs at lower energy as L increases. At L=2.5, one can easily distinguish cyclotron diffusion (the upper left green region) from the Landau diffusion above ~75°, both separated by a wide region of extremely weak (Landau) diffusion (in blue). This region of extremely weak diffusion shrinks with increasing L as cyclotron diffusion reaches both higher pitch angles and lower energies.

Lifetime

Electron lifetimes, i.e. the e-folding time scales, τ, in days, are computed from the averaged diffusion coefficients following the original method of Lyons et al. (1972). That method is discussed and explained in greater detail in Ripoll et al., (2016b) and shall not be recalled here. It has been proven extremely robust when used to generate lifetimes from several millions of diffusion coefficients in [START_REF] Ripoll | Electron lifetimes fromnarrowband wave-particle interactions within the plasmasphere[END_REF].

The electron lifetime profile is particularly interesting and characteristic when plotted daily in the (L, E) plane (Meredith et al., 2006b;Ripoll et al., 2014aRipoll et al., , 2016b;;[START_REF] Albert | Quasi-linear diffusion coefficients for highly oblique whistler mode waves[END_REF]. There are two comments of importance that are evident from these lifetimes during our period of interest shown in Figure 7. First, electron lifetimes are similar in both shape and amplitude to those obtained during the quiet period of March 2013. The shape follows a complex and dynamic filamentary structure, directly inherited from the wave properties. Average electron lifetimes are of the order of 1-10 days when waves are active and at the energies where the resonance causes maximal diffusion. The absence of pitch angle diffusion from whistler-mode waves for inner belt electrons below ~800 keV [START_REF] Fennell | Van Allen Probes show that the inner radiation zone contains no MeV electrons: ECT/MagEIS data[END_REF] causes lifetimes in excess 100 days (see [START_REF] Ripoll | Particle Dynamics in the Earth's Radiation Belts: Review of Current Research and Open Questions[END_REF] for recent apparent lifetimes extracted from Van Allen Probes data). Theoretically, lifetimes can be much larger and the time to reach equilibrium, so that decay is steady enough to be described by lifetimes, can itself be quite large, in particular in the inner belt (Ripoll et al. 2016b). Large lifetimes explain why inner belt electrons below 1 MeV are so stable and can remain trapped for many months. It is worth noting that despite lightning-generated whistlers being commonly observed in the inner belt, with their amplitude recently reported form the Van Allen Probes measurements in (Ripoll et al. 2020b), they are disregarded in this study. This is due to the fact that their amplitudes are typically too weak to contribute significantly to fast losses occurring within 10 days [START_REF] Meredith | Slot region electron loss timescales due to plasmaspheric hiss and lightning-generated whistlers[END_REF][START_REF] Meredith | Relativistic electron loss timescales in the slot region[END_REF][START_REF] Ripoll | Local and statistical maps of lightning-generated wave power density estimated at the Van Allen Probes footprints from the World-Wide Lightning Location Network database[END_REF]. For quiet times with a greatly extended plasmasphere, effects on low-energy electrons (<300 keV) extend to the outer belt, to L = 5.5. Such a plasmasphere configuration is essential to understanding the decay of outer belt electrons from whistler-mode waves.

Whistler-mode waves predominantly act along a main diagonal line in the (L-shell, Energy) plane, which decreases with increasing L-shell [START_REF] Lyons | Equilibrium structure of radiation belt electrons[END_REF].

scattering, predominantly due to whistler-mode hiss waves. Electron lifetimes remain low as energy increases from 200 keV up to 2 MeV; at 1 MeV the lifetime is 3.5 days at L=3.0, 9 days at L=4.0, 7 days at L=5.0, and 21 days at L=5.5.

The second comment of importance concerns the two zones that are observable in the lifetime profile, one below L~4.7 and the other above, with low lifetimes seemingly shifted up in energy. In actuality, this is not a shift in energy since the lifetime decreases homogeneously in energy; lifetimes are simply lower. This effect is the net result of various effects acting oppositely. This reduction in electron lifetimes could be directly associated with a decrease in the plasma density. From Figure 3 (top), it is apparent that the lowest density does indeed occur above L~4.7 (cf. the black region).

This density decrease is attributed to the increase in the AE index and to the substorm activity, as explained in section 2. Therefore, we may postulate that the moderate substorm activity leads to a lower plasma density in the outer belt, above L~4.7, that in turn lowers the electron lifetime. But evaluating individual diffusion coefficients for Lshell at L=4.8, 4.9, 5., 5.1, by step of 8 hour (Figures not shown), we do not observe a noticeable increase in the diffusion coefficients associated with the substorm injection and attributed to the lowering of the density, though theoretically speaking lowering the density does lower the electron lifetimes (all other parameters unchanged). This scenario is possible, and likely happening to some extent, and could occur for other events. However, we find that pitch angle diffusion is actually lower (higher lifetimes)

during the substorm injections, despite the lower plasma density. This is due to the whistler-mode wave amplitude decreases that occur during July 3, 4, 7 and 10, each associated with substorm injections (cf. Figure 23). These reductions in wave amplitude contribute to lower pitch angle diffusion, which directly increases electron lifetimes. The decay of the whistler-mode wave amplitude becomes more important on average than the decay of the density. That is, the mean lifetime increases during substorm activity.

The decrease in the mean electron lifetime above L~4.7 can be explained by the enhancement of whistler-mode waves during the reasonably quiet times that occur between the substorm injections, such as on July 5 in Figure 6. The substorm activity is weak enough that the moderate compression (or moderate erosion) of the plasmasphere does not cause the whistler-mode hiss waves to vanish with it. We can conclude that pitch angle diffusion by whistler-mode waves has various characteristic features of opposite effect; first, during the injection, there is a lowering of the whistlermode wave amplitude which increases lifetimes and also a decay of the density which lowers lifetimes, and, second, in between the substorm perturbations, there is a strengthening of the whistler-mode waves which lowers the mean lifetime. When combined, all three effects lead to a mean enhancement of loss above L~4.7 in our event.

Finally, the density drop associated with the substorm injections is also likely to cause changes in the wave properties such as the mean frequency, at least when the chorus incursion occurs leading to a higher mean frequency. Would these features still be apparent if we had used statistical diffusion coefficients (or lifetimes) that cannot keep all plasma and wave properties correlated? et al., 2019a). Note that the background correction algorithm of [START_REF] Claudepierre | A background correction algorithm for Van Allen Probes MagEIS electron flux measurements[END_REF] has been applied to the MagEIS electron flux measurements shown here.

Observations and Fokker-Planck simulations

Van Allen Probes flux observations

The snapshots of Figure 8 are similar to the original format presented in [START_REF] Reeves | Energy-dependent dynamics of keV to MeV electrons in the inner zone, outer zone, and slot regions[END_REF], and also those used in Ripoll et al. (2016aRipoll et al. ( , 2016b[START_REF] Ripoll | Effects of whistler mode hiss waves in March 2013[END_REF]Ripoll et al. ( , 2019aRipoll et al. ( , 2019b)).

The main substorm injection of electrons is apparent on July 3, with higher flux in the outer belt observed below 200 keV (e.g. [START_REF] Turner | Energetic electron injections deep into the inner magnetosphere associated with substorm activity[END_REF]. The substorm of July 4 is too weak to be visible; the flux of new incoming electrons at L~5 is likely balanced by the strong loss of the older outer belt electrons below 200 keV. The substorm on July 7 leads to a slight increase of the flux at the lowest energies. The high-speed stream injection at the end of the event, on July 10, is also visible. There is a good concordance between the observed flux decay and the low electron lifetimes in the (L, E) plane. The slot region progressively enlarges, and the outer belt flux drastically decays during this 10-day period as substorm injections modify only the low energy structure. Since the beginning of the event, there is a characteristic quiet time S-shape to the inner boundary of the outer belt when drawn in the (L, E) plane reported by [START_REF] Reeves | Energy-dependent dynamics of keV to MeV electrons in the inner zone, outer zone, and slot regions[END_REF] and computed in Ripoll et al. (2016aRipoll et al. ( , 2016b[START_REF] Ripoll | Effects of whistler mode hiss waves in March 2013[END_REF]Ripoll et al. ( , 2019aRipoll et al. ( , 2019b)). When most of the high energy part of the outer belt has decayed after 10 days, with little flux left (omitting low energy injections below 200 keV), there exists a remnant pocket of trapped electrons, for L in [4.7, 5.2] and E in [1, 2] MeV, that constitutes the remains of the upper loop of the 'S', first discussed in Ripoll et al. (2019a).

Simulations

The temporal evolution of the gyro, bounce, and drift phase-averaged phase space density, , , , , in the presence of pitch-angle diffusion can be described by [START_REF] Schulz | Particle Diffusion in the Radiation Belts[END_REF][START_REF] Shprits | Review of modeling of losses and sources of relativistic electrons in the outer radiation belt I: Radial transport[END_REF]:

= 1 ,! " # # ,! $ 1
with # # computed in section 3 (cf. Figure 7) and = % sin 2 . T(α0) is proportional to the bounce period evaluated at the equatorial pitch-angle, α0. We use ,*+ = 0 at the loss cone, which itself is computed from a dipolar magnetic field. A Newman boundary condition is used at 90°, -. /,0 # -0 # = 0. Equation (1) neglects radial diffusion and off-diagonal terms in the diffusion tensor. The numerical method and code are described in [START_REF] Réveillé | Dynamic evolution of relativistic electrons in the radiation belts[END_REF]. The absence of radial diffusion makes it impossible to solve for any injections associated with the substorm activity. In addition, energy diffusion is neglected in this study since energy diffusion coefficients are negligible for whistler mode hiss waves (e.g. [START_REF] Albert | Efficient approximations of quasi-linear diffusion coefficients in the radiation belts[END_REF] on which that study focuses.

Only scattering from pitch angle diffusion, and how the effects of whistler-mode waves are affected by the substorm activity, are investigated in this section.

Figure 9 shows the omnidirectional flux, i.e. the phase space density integrated with respect of pitch angle, solution of Equation 1 at each L-shell and energy. In this first numerical experiment, we choose the initial condition as = 0, , = 123 4 ,*+ -123 4 . This function guarantees a smooth transition at the loss cone where f=0 and is homogeneous with respect to energy. This allows all particles to decay from their initial state and for us to better understand and quantify their loss. Results immediately show the two distinct regions, below and above L~4.7, with the region above L~4.7 showing more important scattering, as discussed in section 3.3. Interactions with whistler-mode waves are particularly strong below 150 keV for L>3.5, with the omnidirectional flux decaying by more than 5 orders of magnitude in 10 days (Figure 9 is saturated at level -5 in order to see details at higher energies). This low energy region (<150 keV) is highly dynamic; with injections required to be both numerous and fast in order to bring new electrons to lower L-shells and overcome the losses due to whistler-mode waves, as seen in the observations in Figure 8. More generally, Figure 9 describes the decay associated with the formation of a wide slot region. Above L~4.7, we see a significant decay of the omnidirectional flux in the outer belt up to ~1 MeV, similar to observations.

Whistler-mode waves have little effect on the remnant pocket of electrons observed in The second numerical experiment uses the same pitch angle profile, this time combined with the energy spectrum from the MagEIS spin-averaged L2 flux on the 06/30/2014 (6.54am-11.31am) (cf. first panel of Figure 8). The results of this simulation are plotted in Figure 10, and are the best representation of the losses that we believe are occurring in the slot region and in the outer belt during this event. Various important features are apparent: i) the inner belt, defined as ~1 MeV electrons below L=2 (at the edge of the simulation domain) or ~100 keV electrons below L=3.5, is unaffected by whistler-mode waves; ii) the slot region is formed in ~10 days along the diagonal and is primarily driven by pitch angle diffusion caused by whistler-mode waves (cf. early observations of 35-560 keV electrons from [START_REF] Lyons | The storm and post-storm evolution of energetic (35-560 keV) radiation belt electrons[END_REF]); iii) the outer belt flux decays strongly below ~1 MeV; iv) three complex substorm conditions (cf. section 3) explain why the significant loss reaches up to 1 MeV electrons in the outer belt; v) the remnant pocket of trapped electrons (L in [4.7, 5.2] for E in [1, 2] MeV) decays very slowly, extending to L=5.5 in the simulation due to lack of outward radial diffusion, and becomes increasingly apparent as the outer belt decays; and vi) the injections of low energy electrons (below ~150 keV and absent of the simulations) associated with substorm activity (up to 300 keV for the strongest substorm) are required to be numerous and fast (e.g. [START_REF] Friedel | Substorm onsets observed by CRRES: Determination of energetic particle source regions[END_REF][START_REF] Turner | Energetic electron injections deep into the inner magnetosphere associated with substorm activity[END_REF] to compensate the intense loss. Finally, comparing the results of Figure 10 with the observations of Figure 8, we see qualitative agreement for both the daily enlargement of the slot and for the significant reduction of the outer belt. Reproduction of these key features is highly encouraging, and demonstrates that this method has the ability to model reasonably electron scattering during quiet time dynamics, perturbed (or not) by moderate substorm activity.

The main limitations of this study relate to; i) the lack of inward transport (usually modeled by radial diffusion) that does not allow reproduction of the substorm injections, ii) the approximations used in the diffusion coefficients (cf. section 3.1), and iii) the magnetic local time (MLT) description that is subsequently discussed. The measurements of frequencies and wave normal angles used here are made locally in MLT, though they are applied to all MLTs. Only the wave amplitude is rescaled by the MLT dependence given by [START_REF] Spasojevic | Global empirical models of plasmaspheric hiss using Van Allen Probes[END_REF]. We believe variation of the wave properties (frequency and mean wave normal angle) to have only minor effects (unless changes are drastic, which is possible [START_REF] Hartley | Statistical properties of plasmaspheric hiss from Van Allen Probes observations[END_REF]). However, variations in the plasma density with MLT, and the possible development of higher frequency whistlermode waves, i.e. chorus, in regions of lower density (cf. Figure 3), could affect the accuracy of these simulations since decaying plasma density, caused by substorm activity, is found to have important repercussions. Another caveat of the method is that if the inward motion of the plasmapause is localized where we observe it (or if it is only an apparent motion because of field line stretching) then most of the electron drift orbit is still in the plasmasphere and experiences whistler-mode hiss in a way it may not have a significant effect on the drift-averaged pitch angle diffusion. Both of these problems have no current solution since they are related to the uniqueness of the observation at a

given MLT and the absence of availability of any other, leading to speculations about possible situation scenarios along the drift. Finally, there could be acceleration occurring in the rare regions of wave intensification identified in Figure 4 and in the low-density regions highlighted in Figure 3, but the phenomena are too limited in space or/and in time to believe they would significantly matter for this event.

The alternative of using statistical models sorted in MLT has little chance of increasing accuracy. Validation would also be an issue as the observed fluxes would be measured at the same MLT. What is encouraging is the synchronization of global indices (like AE here) with local observations, which goes in favor of the generalization of the observations. On the other hand, the location of the whistler-mode waves, which are not uniformly distributed over MLT, is an issue that unavoidably requires the introduction of an MLT-dependence. The only viable alternative, which does not yet exist, is multiple synchronized measurements at various MLT and at fixed L-shell. It is hoped that new radiation belt missions could aim to resolve this issue (e.g. [START_REF] Millan | Small satellites for space science A COSPAR scientific roadmap[END_REF]Ripoll et al., 2020). For now, the only viable approach is to include these necessary approximations in our models and to quantify the accuracy of the results against observations of the decaying radiation belt flux, as it has been done in this study.

Conclusions

In this work we study the dynamics of radiation belt electrons during a geomagnetically reasonably quiet period perturbed by moderate substorm activity and preceding a high-speed stream (HSS), with the aim of obtaining a global description of the pitch angle scattering occurring in the radiation belts in L-shell [2,6], and energy [0.1, 10] MeV. The goal is to assess whether the event-driven methodology, which has been successfully applied for quiet conditions (Ripoll et al., 2016a[START_REF] Ripoll | Effects of whistler mode hiss waves in March 2013[END_REF](Ripoll et al., , 2019a(Ripoll et al., , 2019b)), can also obtain a high degree of accuracy when the radiation belts are perturbed by moderate substorm activity.

For the event studied we show that the plasmasphere extends up to and beyond L = 5, which provides favorable conditions for whistler-mode waves reaching out, and scattering, outer belt electrons. Substorm injections result in low density plasma extending down to L~4.5, although these periods last less than 8h. Whistler-mode waves, mostly plasmaspheric hiss, predominantly act along a main diagonal line in the (L, E) plane, decreasing in energy with increasing L-shell. This diagonal is well marked for quiet events, but shifts up in energy at L~4.7 in this case due to the substorm activity, thus allowing the scattering of up to 1 MeV electrons in the outer belt. Whistlermode waves drive major loss phenomena which contributes to both the creation of a wide slot region and significant decay of the outer radiation belts below 1 MeV. A good qualitative agreement is found between the observed daily flux from the Van Allen Probes and the computed flux from quasi-linear resonant interactions between electrons and whistler-mode waves.

A main result of this study, brought thanks to the high temporal and spatial resolution of the event-specific pitch angle scattering model in use, is the mean enhancement of the electron pitch angle scattering above L~4.7, which we attribute to substorm activity. We have demonstrated that the observed substorm activity coincides with a short diminution of the plasmaspheric density, in a way that is moderate enough to conserve a plasmaspheric density of ~100 #/cc, or higher, at L=5.5 at times different than the injection time (+/-4h). This has two direct consequences. First, the conservation of the plasmasphere at high L-shell (above 5) allows whistler-mode hiss waves to remain, exist, and continue contributing to the lowering of the flux level in the outer belt. Second, while the lowering of the density associated with the substorm could increase pitch angle diffusion via wave-particle interactions, we find that this effect does not compensate the whistler-mode wave amplitude decay, which occurs simultaneously during the substorm injections. That is, pitch angle diffusion actually becomes weaker during the substorm injections (+/-4h). However, whistler-mode waves are enhanced in the periods between the substorm injections, due to the widely extended plasmasphere out to, and above, L=5.5. The combination of these opposing effects leads to pitch angle scattering by whistler-mode waves that is, on average over the 10-day period, enhanced above L ~4.7. This allows the scattering of up to 1 MeV electrons in the outer belt but is not sufficient to significantly decay the remnant pocket of relativistic electrons between 1 and 2 MeV located at L in [4.7, 5.2], for which both observations and simulations show an omnidirectional flux that decays only slightly in 10 days. July 3 and 10. The times at which the plasma density (cf. Figure 3) is perturbed by substorm injections (cf. Figure 2) are identified by red arrows.

Figure 3 (

 3 Figure 3(bottom right) shows the density is always obtained in our study from

  Figure 4(right) shows mean values of the wave power spectra, accounting for Magnetic Local Time (MLT) dependence of the wave amplitude using the[START_REF] Spasojevic | Global empirical models of plasmaspheric hiss using Van Allen Probes[END_REF] model. The mean wave vector angle, θm, calculated from data obtained by the Singular Value Decomposition method of[START_REF] Santolik | Singular value decomposition methods for wave propagation analysis[END_REF], and the mean wave vector width ∆θm are also plotted in Figure4. A filamentary temporal structure of both frequency and amplitude hiss wave data is very apparent. The mean frequency shows three populations, two for hiss and one for chorus. The first is between 200-400 Hz and the second is at higher frequencies between 600-800 Hz. Both correspond to whistler-mode hiss waves that occupy most of the domain of the study. Some rare incursions of chorus waves above L=5.2 are visible, with a mean frequency that exceeds ~1000 Hz. These chorus incursions are synchronized with the lowest densities observed in Figure3. Structures change in time and space in a dynamic and complex way as the satellite orbits around the Earth. Wave amplitudes vary by orders of magnitude. The stronger hiss waves are located in the plasmasphere interior, between L = 2 and L = 4. The mean wave normal angle and mean wave normal width are rather homogeneous, with a moderate global mean value of the wave normal angle close to 40°.

Figure 8

 8 Figure 8 shows 27 electron flux snapshots observed by the Van Allen Probes in

Figure 8

 8 Figure 8 at L in [4.7, 5.2] for E in [1, 2] MeV, with the omnidirectional flux decaying by

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: From top to bottom: AE index, solar wind flow speed, interplanetary magnetic

Figure 4 :

 4 Figure 4: Evolution of the whistler-mode wave properties during the first 10 days of July

Figure 5 :

 5 Figure 5: (left) The ratio of the plasma frequency to the equatorial gyrofrequency

Figure 6 :

 6 Figure 6: Diffusion coefficients in the (Energy, pitch angle) plane, shown for the first 8

Figure 9 :

 9 Figure 9: Evolution of the simulated omnidirectional flux during 12 days, plotted at the

Figure 10 :

 10 Figure 10: Evolution of the simulated omnidirectional flux from 06/30/2014 to

Acknowledgments.

This work was performed under the auspices of an agreement between CEA/DAM and NNSA/DP on cooperation on fundamental science. The authors acknowledge the International Space Sciences Institute (ISSI) and the participants in a 2020 ISSI workshop. Part of the research presented in this article was supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory under project number 20180731ER. Work at Iowa is performed under the support of JHU/APL contract 921647 under NASA Prime contract NAS5-01072. Work at CU/LASP is supported by NASA grant 80NSSC18K1034. O.S. acknowledges support from the Czech Academy of Sciences Praemium Academiae award and from the MEYS grant LTAUSA17070. Processing and analysis of the [HOPE, MagEIS, REPT, or ECT] data was supported by Energetic Particle, Composition, and Thermal Plasma (RBSP-ECT) investigation funded under NASA's Prime contract no. NAS5-01072. All RBSP-ECT data are publicly available at the Web site http://www.RBSP-ect.lanl.gov/

2014 (omitting July 6). We observe the slot region forming between the belts during 10 days and gradual removal of the outer belts. Substorm activity is also visible, e.g., on the July 3, with influx of electrons below ~200 keV.