2.1.1. Bulk Silicate Earth and Its Derivatives
 The central concept in any type of global mass balance argument is the bulk silicate Earth (BSE), which is also known as the primitive mantle (PM). BSE refers to the solid Earth excluding the core, and it may be regarded as the primordial mantle formed after core formation but before the extraction of continental crust. By definition, all of present-day silicate reservoirs have been derived from BSE, so they must add up to BSE. We can readily recognize two major silicate reservoirs: the continental crust (CC) and the depleted mantle (DM), which have long been considered to be complementary to each other [e.g., Hofmann, 1988, 2003]. That is, the continental crust is the product of mantle melting, whereas the depleted mantle is the solid residue. A number of trace elements are incompatible with solid phases, so they tend to be highly concentrated in a melt phase, which is the continental crust in the context here. The mass of continental crust is only ∼0.6% of that of the mantle, but the continental crust is highly enriched in incompatible elements, so its contribution to mass balance is substantial. The depleted mantle is depleted in those incompatible elements with respect to BSE but not depleted in compatible elements such as major elements.
 Note that although the oceanic crust (OC) is also generated by mantle melting, it is usually not considered as an independent chemical reservoir in terms of global mass balance because it is quickly recycled back to the mantle by subduction. At a normal condition, suboceanic mantle beneath mid-ocean ridges has a potential temperature (hypothetical temperature of mantle adiabatically brought to the surface without melting) of ∼1350°C and starts to melt at the depth of ∼80 km, resulting in an ∼6-km-thick oceanic crust and an ∼70-km-thick depleted mantle lithosphere (DML) [e.g., McKenzie and Bickle, 1988; Langmuir et al., 1992]. DM in global mass balance refers to the source mantle before this melting takes place, and DML is even more depleted than DM. By mass conservation, the sum of OC and DML should be equivalent to DM if the effects of degassing and seawater alteration are taken into account.
2.1.2. Missing Heat Source Paradox and Its Solutions
 Among a few global mass balance arguments, the missing heat source paradox is most relevant to the thermal budget, so it is explained in some detail here. The continental crust is estimated to have 1.3 ppm U, 5.6 ppm Th, and 1.5% K [Rudnick and Gao, 2003], the heat production of which is ∼7.5 TW (assuming the continental mass of 2.3 × 1022 kg). According to the model of Salters and Stracke , the abundance of those elements in the depleted mantle is 4.7 ppb U, 13.7 ppb Th, and 60 ppm K, and the corresponding heat production is ∼4.2 TW. Combined, these two reservoirs yield ∼12 TW. However, the bulk silicate Earth is commonly assumed to have 20 ppb U, 80 ppb Th, and 240 ppm K [e.g., McDonough and Sun, 1995], which give rise to the heat generation of ∼20 TW. Assuming that these values are reasonably accurate, the global mass balance does not seem to hold for these heat-producing elements. This is called the missing heat source paradox, and a key question is the following: where did we make a mistake?
 This failure of mass balance has been used to argue against whole mantle convection and to support the presence of a hidden reservoir in the deep mantle. This line of logic is understandable because the composition of the depleted mantle is based largely on the chemistry of mid-ocean ridge basalts (MORB). As mentioned in section 2.1.2, mid-ocean ridge magmatism is the result of mantle melting at shallow depths (<100 km), so a composition model for the depleted mantle is fundamentally biased to the upper portion of the mantle. The above mass balance thus implicitly assumes that the mantle is well mixed by whole mantle convection so the shallow mantle sampling is not an issue. As we just saw, however, this assumption does not lead to a self-consistent mass balance, so it seems that we should abandon the notion of well-mixed mantle and call for a deep reservoir that is more enriched in the heat-producing elements. As the classical layered-mantle model gradually lost its popularity, various alternatives have been proposed for this “hidden” reservoir. The reservoir may occupy a substantial portion of the lower mantle, and its interface with the upper layer is somehow seismologically invisible [e.g., Kellogg et al., 1999]. Or the reservoir may be distributed as a number of blobs [e.g., Manga, 1996; Becker et al., 1999; Helffrich and Wood, 2001], though it is uncertain why such blobs do not contribute much to mid-ocean ridge magmatism (to an extent that enriched MORB becomes much more common than observed). Or it may correspond to the thin, D″ layer at the base of the mantle [e.g., Coltice and Ricard, 2002; Tolstikhin and Hofmann, 2005], but the layer might be too enriched in heat-producing elements to have been gravitationally stable in the past.
 The very first step we should have taken instead may be to examine the accuracy of the mass balance calculation that caused the paradox. As discussed in the following, every component in this mass balance has substantial uncertainty. First of all, the most recent compilation for continental crust composition by Rudnick and Gao  assigns 30% uncertainty for trace elements, so the heat production in the continental crust is in the range of 5 to 10 TW. If one looks more closely at how the average composition of continental crust is estimated, even greater uncertainty may be more realistic. The continental crust is typically divided into three layers such as upper, middle, and lower crust, and all layers exhibit considerable regional variability. Global averaging is thus a difficult task. The 30% uncertainty adopted by Rudnick and Gao  is for the upper crustal composition, which may be estimated most reliably because a large number of data are available. The deeper layers are expected to have larger uncertainties, which are left unquantified because of the paucity of data. One may argue that such uncertainty may not concern us because the deeper layers are often considered to be depleted in heat-producing elements. The conventional wisdom of heat production exponentially decreasing with depths [Lachenbruch, 1970], however, is just a model and does not seem to be supported by observations [e.g., Jaupart and Mareschal, 2003]. Also, intracrustal differentiation, which is likely to have occurred in the evolution of continental crust [e.g., Rudnick, 1995], may deplete the lower crust of K but probably not of U and Th, because these two elements are highly concentrated in accessory minerals such as zircon, which tend to be unaffected by crustal melting [e.g., Watson and Harrison, 1984]. Existing estimates for the lower crustal composition vary by an order of magnitude in terms of trace elements, and the “best estimate” according to Rudnick and Gao  is based on crustal xenolith data, though they also acknowledge that it is unclear how well xenolith compositions could represent average lower crust and also that xenolith data are spatially limited. Given the potential of the continental lower crust therefore, the total heat production of continental crust may even be higher than 10 TW. Mantle geotherms based on xenolith data [e.g., Finnerty and Boyd, 1987; Russell and Kopylova, 1999] may be used to bound crustal heat production, but such constraints are not available globally. Though the heat production should not exceed the observed continental heat flow, this upper bound itself has been revised from 12 TW [Pollack et al., 1993] to 14 TW [Jaupart et al., 2007] on the basis of more recent heat flow data. The distribution of existing heat flow measurements is highly heterogeneous both on continents and on ocean basins [e.g., Pollack et al., 1993, Figure 1], so the recent upward correction is not surprising.
 Estimating the composition of the depleted mantle suffers from more compounded difficulties. At shallow depths, the depleted mantle usually differentiates into oceanic crust and depleted mantle lithosphere by melting, so the composition of the (unmelted) depleted mantle must be indirectly estimated based on these chemically differentiated phases. The oceanic crust itself is layered because of chemical fractionation, and only the very top portion (i.e., MORB) is usually accessible for sampling. So the estimate based on MORB has to take into account the effects of chemical fractionation as well as mantle melting. This indirectness may be compensated, however, by the fact that the oceanic crust is the product of mantle melting over the depths of several tens of kilometers. MORB samples thus have a potential to effectively probe the shallow upper mantle, with its small-scale heterogeneities automatically homogenized by melting. Perhaps a more serious issue is the considerable regional variations observed in the trace element chemistry of MORB, which tends to show the so-called “enriched” signature (higher concentrations of more incompatible elements) near hot spots such as Iceland [e.g., Schilling et al., 1983]. Samples from these anomalous regions have traditionally been disregarded when discussing the depleted mantle, because hot spots have often been considered to originate in a deep geochemical reservoir that is different from the depleted mantle. One may also be tempted to apply this type of screening to limit data variability and to estimate an average with small uncertainty. The origin of hot spot magmatism is, however, still debated (section 2.2.1), and this screening may not be warranted. The concentration of heat-producing elements in the depleted mantle has been estimated to be 8 ppb U, 16 ppb Th, and 100 ppm K by Jochum et al.  and 4.7 ± 1.4 ppb U, 13.7 ± 4.1 ppb Th, and 60 ± 17 ppm K by Salters and Stracke , but these estimates are biased to the depleted end-member of MORB samples because of this screening. An alternative approach by Workman and Hart  based on the trace element systematics of abyssal peridotites (exhumed pieces of DML) may be able to avoid this type of bias, but it suffers from different kinds of indirectness; their approach depends on the BSE composition as well as assumed isotopic evolution due to continental growth. The 30% error assigned by Salters and Stracke  translates to the heat generation of 2.9–5.5 TW, but it would be corrected upward if data screening is loosened [Langmuir et al., 2005]. At the same time, expanded data would certainly exhibit much more variability. A careful uncertainty analysis has not been published, but it would be vital to derive realistic bounds on heat production. An estimate without uncertainty is of little value when scrutinizing global mass balance.
 Finally, the composition of the bulk silicate Earth represents what we expect for the bulk silicate Earth. Compared to our estimates for the continental crust and the depleted mantle, therefore, it is most theoretical, depending critically on the validity of theoretical assumptions employed. A number of geochemists have estimated the BSE composition with different sets of assumptions (see a review by Lyubetskaya and Korenaga [2007a]), and the so-called pyrolite approach [e.g., McDonough and Sun, 1995] appears to be most robust as it requires only the following two assumptions: (1) the BSE should be found somewhere along the compositional trend displayed by mantle rocks, and (2) the ratios of refractory lithophile elements (RLE) in the BSE are chondritic. If either of these assumptions is invalid, it is impossible to estimate the BSE composition (not only by the pyrolite approach but also by other existing methods), so its uncertainty would be essentially unbounded. The first assumption implicitly requires whole mantle convection. Our mantle samples such as mantle xenoliths and massif peridotites are all from the upper mantle, so theoretically the composition of the lower mantle could be different from that of the upper mantle, and if so, the RLE ratios could not be imposed on the composition trend of the upper mantle samples because these cosmochemical constraints are valid only when considering BSE as a whole. For the estimated BSE composition to be justified, therefore, global mass balance must be self-consistent. This issue is explained in more detail in section 2.1.3. The second assumption is reasonable because refractory elements have high condensation temperatures, so they are unlikely to have been fractionated from each other during planetary formation processes, and lithophile elements do not enter the core. The RLE ratios are the most stable ratios among various classes of chondrites, and, indeed, the isotope data of terrestrial samples require that at least Sm/Nd, Lu/Hf, and Th/U (these are all RLEs) should be within a few percent of the chondritic value [e.g., Lyubetskaya and Korenaga, 2007a]. With these assumptions, we can quantify the uncertainty of the BSE composition by taking into account the uncertainty of observed compositional trend as well as that of chondritic RLE ratios, and Lyubetskaya and Korenaga [2007a] obtained ∼20% error for the BSE abundance of refractory lithophile elements. This uncertainty represents the influence of scatter in the upper mantle compositional trend under the imposed chondritic constraints. Lyubetskaya and Korenaga also found that the previous estimate of 20 ppb U and 80 ppb Th should be corrected slightly downward, and the new estimate with one standard deviation is 17 ± 3 ppb U and 63 ± 11 ppb Th. As K is not a RLE, it must be derived with additional constraints such as bulk Earth K/U and K/La, which further amplify uncertainty. The revised estimate is 190 ± 40 ppm K. The likely range of BSE heat production is then 13–19 TW.
 Note that these uncertainties in heat production do not correlate to each other because the compositions of these silicate reservoirs are estimated independently of each other. Using one standard deviation therefore, the range of combined CC and DM heat production is 7.9–15.5 TW, which overlaps with the range of BSE heat production (note also that the upper bound of 15.5 TW is likely to increase if the DM composition is estimated with more even data sampling). The missing heat source paradox is thus unproven and probably does not exist; it could well be an artifact caused by neglecting unavoidable uncertainties in geochemical inference. Given the uncertainties, it is still possible to postulate a hidden reservoir (e.g., by assuming 5 TW for CC, 3 TW for DM, and 19 TW for BSE), but it is no longer required.
2.1.3. Importance of Self-Consistent Mass Balance
 The self-consistent mass balance is vital to justify the estimated BSE composition because, as noted in section 2.1.2, whole mantle convection has to be assumed when deriving the composition. In other words, if mass balance is not self-consistent, the BSE composition is undefined. Interestingly, however, the notion of layered-mantle convection has coexisted with the BSE composition estimated by assuming whole mantle convection. A common layered-mantle model advocated by geochemists is illustrated in Figure 3b; the upper layer is the depleted mantle as observed through mid-ocean ridge magmatism, while the lower layer is identified as the primitive mantle [e.g., Jacobsen and Wasserburg, 1979; DePaolo, 1980; Allègre, 2002]. Note that the PM composition is identical to the BSE composition, both in major and trace elements, and is very similar to the DM composition in terms of major elements. The volume of the upper layer is determined so that the average composition of these three reservoirs (CC, DM, and PM) is equal to the BSE composition. This is, of course, one possibility, but the average composition can be (vastly) different from the estimated BSE composition both in major and trace elements (Figure 3c). In fact, if a layered-mantle model has to be true, the major element composition of the lower layer may be required to be different from the PM composition. In the classical layered-mantle model, the internal boundary coincides with the 660-km discontinuity, and it was speculated that the endothermic phase change expected at this discontinuity may be able to sustain the layering [e.g., Christensen and Yuen, 1984]. Later studies on mantle convection, however, suggested that it was difficult to maintain the layered state with the phase change alone [e.g., Tackley et al., 1993; Zhong and Gurnis, 1994], and this idea became obsolete particularly after the internal boundary was pushed down the 660-km discontinuity. To achieve a layered mantle in a physically plausible manner, therefore, a different major element composition (e.g., enriched in iron) seems necessary to make the lower layer intrinsically denser than the upper layer [e.g., Kellogg et al., 1999; Davaille, 1999].
Figure 3. Different chemical models of the bulk silicate Earth. (a) Whole mantle model, in which no vertical stratification is assumed for both major and trace elements. This is adopted as a reference model in this article and is equivalent to Figure 2b. (b) Layered-mantle model popular in geochemistry. The lower layer is enriched in incompatible trace elements, but the two layers are almost identical in terms of major element composition. (c) Layered-mantle model popular in mineral physics. The lower layer is different from the upper layer in major elements, so its trace element composition is likely to be different as well. CC is continental crust, DM is depleted mantle, and PM is primitive mantle.
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 A failure to close global mass balance therefore presents a serious challenge to our understanding of the bulk Earth chemistry. It does not only indicate the presence of a hidden reservoir, but it also makes the BSE composition (and thus heat production) unconstrained (Figure 3c). The BSE composition is meaningful only with whole mantle convection (Figure 3a), and it cannot be used to estimate the composition of a hidden reservoir.
2.1.5. Isotopic Mass Balance and Two Versions of Chondrite Assumptions
 The final kinds of mass balance to be discussed are the ones that involve isotopic ratios such as 143Nd/144Nd, 176Hf/177Hf, and, more recently, 142Nd/144Nd. Though similar to other mass balances discussed so far, isotopic mass balance arguments involve a much more strict definition for the chondrite assumption, warranting a separate treatment.
 The 143Nd/144Nd ratio, for example, reflects the long-term evolution of 147Sm/144Nd (or Sm/Nd) as 143Nd is a decay product of 147Sm. Chondritic meteorites exhibit a tight distribution of 147Sm/144Nd (0.196 ± 0.002), which corresponds to 143Nd/144Nd (0.51263 ± 0.00005) [Patchett et al., 2004, Figure 1] (Note that the ∼1% deviation in 147Sm/144Nd reduces to only ∼0.01% deviation in 143Nd/144Nd because of the long half-life involved.) The 143Nd/144Nd ratio of terrestrial samples is usually compared to this well-defined (present day) chondritic value or its value in the past, and the difference is expressed in the ε143Nd notation defined as
where t is the age of a sample and CHUR refers to a chondritic uniform reservoir. A common argument is that the BSE ε143Nd would be ∼5 if CC contains 20 ppm Nd with ε143Nd of −16 and DM contains 0.7 ppm Nd with ε143Nd of 9 [e.g., Lassiter, 2004]. A major source of uncertainty is the depleted mantle, which stores more than 80% of total Nd in this mass balance. Though both ε143Nd and Nd content of MORB vary from sample to sample by ∼30%, they tend to be negatively correlated (i.e., a more enriched sample has a lower ε143Nd) [e.g., Hofmann, 2003, Figure 14], so it is difficult to lower the BSE ε143Nd below ∼3 even if we take into account all of the likely uncertainties (though a more complete statistical study of MORB chemistry may indicate otherwise in the future). Because chondritic samples show ε143Nd of ±1 as mentioned above, this relatively high BSE value seems to require a hidden reservoir with low Sm/Nd (i.e., enriched in incompatible elements because Nd is more incompatible than Sm).
 Note that the chondrite assumption for BSE is more strict here than in the case of estimating the BSE composition. Requiring the BSE ε143Nd to be within ±1 is equivalent to assuming that the ratio Sm/Nd is within 1% deviation from the chondrite average, but the Sm-Nd isotopic evolution recorded by terrestrial samples does not place such a tight constraint [Lyubetskaya and Korenaga, 2007a, 2007b]. Up to a few percent deviation is acceptable because the earliest record for the depleted mantle at ∼4 Ga ago shows ε143Nd of as high as ∼4 [e.g., Bennett, 2003]. Thus, requiring the RLE ratios such as Sm/Nd to be within a few percent deviation from the chondritic value is justified by existing data, whereas imposing the ±1% deviation is not. It is important to distinguish between these weak and strong versions of the chondrite assumptions. When the strong version is not satisfied by a certain mass balance, it is often called “nonchondritic,” and this might imply that refusing the presence of a hidden reservoir violates cosmochemical considerations. We do not understand well, however, how efficiently elements and isotopes were homogenized in the presolar nebula and how much they were later modified by planetary accretion processes [e.g., Halliday and Porcelli, 2001].
 As already mentioned, even 147Sm/144Nd in chondrites has as much as ±1% variation (and 176Lu/177Hf has more than ±5% variation [Patchett et al., 2004]). Furthermore, chondrites are believed to originate mostly from the asteroid belt [e.g., Scott and Krot, 2003], so even though the number of chondrite samples is large, we are essentially looking at the highly restricted portion of the solar system; the mass of the asteroid belt is less than 0.1% of that of Earth. The stable statistics of chondrite compositions may be just due to more or less repetitive sampling. Earth is the best sampled planet, and yet we cannot measure its bulk isotope characteristics. We have only a few dozen meteorites from Mars, and no sample for Venus and Mercury, so their bulk isotopic compositions are even more uncertain than Earth's. Note that the spectroscopic measurement of the photospheric composition of the Sun may have ∼25% uncertainty regarding Sm/Nd [Grevesse and Sauval, 1998] (assuming that reported errors are not correlated). A remarkable correlation between the photospheric composition and the chondrite composition exists for a number of elements, but this correlation in the logarithmic space does not impose a tight constraint on the ratios of trace elements. Some authors are well aware of this possibility of limited sampling by chondrites [e.g., Wasson and Kallemeyn, 1988; Drake and Righter, 2002], but the strong version of the chondrite assumption has long been a part of central dogma in geochemistry.
 Recently, Boyet and Carlson  proposed a new kind of argument for a hidden reservoir based on the now extinct isotope 146Sm, which decayed into 142Nd. So far, measured terrestrial samples show systematically higher 142Nd/144Nd than chondrites by ∼0.2 ± 0.14 ε unit. To have the BSE ε142Nd be (strictly) chondritic, therefore, there must be an unobserved reservoir, whose ε142Nd is more negative than chondritic. Also, because of the short half-life of 146Sm, this reservoir has to have formed within the first 100 Ma of Earth's history. This presumed reservoir does not seem to be sampled even by hot spot magmatism, so if its volume is substantial, it must somehow be able to avoid entrainment by the convecting mantle. Alternatively, the reservoir could be trivially small or even nonexistent if the BSE ε142Nd (or Sm/Nd) does not have to exactly equal the chondritic average. With igneous differentiation, Sm/Nd in the outer portions of planetesimal bodies would be low, and impact erosion could preferentially remove major portions of the outer silicate portions of accreting planetesimals, resulting in a slightly elevated Sm/Nd ratio in the growing Earth [Halliday, 2003]. As Lyubetskaya and Korenaga [2007b] argued, both ε143Nd and ε142Nd arguments rely heavily on the strong version of the chondrite assumption. Andreasen and Sharma  showed that ordinary chondrites have systematically higher ε142Nd than carbonaceous chondrites, and the bulk Earth may be located somewhere along this trend. To summarize, given the resolution of available observational constraints on the BSE composition, existing mass balance arguments, whether elemental or isotopic, do not necessarily contradict the notion of whole mantle convection.