As Figure 9c clearly shows, old (>85 Ma) oceanic seafloor subsides at a rate that smoothly decreases from being proportional to the square root of crustal age, or flattens. Previously suggested mechanisms for the flattening (some originally to explain gravity and heat flow data) include an increase in crustal thickness [Humler et al., 1999], phase changes [Wood and Yuen, 1983], a buoyant mantle residuum left over from melt extraction [Johnson and Carlson, 1992; Phipps Morgan et al., 1995], dynamic upwelling [Davies and Pribac, 1993; Carlson and Johnson, 1994] perhaps including internal radiogenic heating [Jarvis and Peltier, 1980], return trench-to-axis sublithospheric flow [Schubert and Turcotte, 1972; Schubert et al., 1978; Morgan and Smith, 1992], thermal rejuvenation of the lithosphere by random hot spot reheating events [Crough, 1979; Heestand and Crough, 1981; Nagihara et al., 1996; Smith and Sandwell, 1997], and generalized thermal input into older lithosphere [Langseth et al., 1966; McKenzie, 1967; Doin and Fleitout, 1996] perhaps by intense small-scale (λ ∼400 km) convection and significant internal heating in the sublithospheric mantle [e.g., Parsons and McKenzie, 1978; O‘Connell and Hager, 1980; Huang and Zhong, 2005]. McNutt  gives, in our view, an unbiased review of many of these possibilities. However, we limit our analysis to the implications of our empirical depth-age curve for the cooling plate model.
 The subsidence, w, of a cooling plate [Langseth et al., 1966] at a given age, t, is described by eight independent parameters [McKenzie, 1967]: plate thickness L, plate basal temperature Tb, water temperature Tw, water density ρw, coefficient of thermal expansion α, zero-temperature mantle density ρ0, thermal conductivity k, and heat capacity Cp. Thermal diffusivity κ is equal to k/πρ0 (i.e., density in this relation is assumed independent of temperature) [Stein and Stein, 1992]. For comparison to measured depths a zero-age depth, Zr, is also required. These are related by equations given by Turcotte and Schubert  for the subsidence, w,
and the surface heat flow, q,
Of the nine parameters, Tw, ρ0, ρw, and Cp are assumed to be relatively well known [Parsons and Sclater, 1977; Stein and Stein, 1992] (Table 2) and Zr may be deduced from young seafloor (e.g., Figure 9a), which leaves four independent parameters undetermined. The equations describing seafloor subsidence and surface heat flow, however, contain only three independent relationships [Parsons and Sclater, 1977], and so only three parameters may be evaluated from that data. Since thermal conductivity is comparatively better known [Parsons and Sclater, 1977], it is assumed to be 3.138 mW m−2, and the remaining three parameters L, α, and Tb investigated [Parsons and Sclater, 1977; Johnson and Carlson, 1992; Stein and Stein, 1992].
 When numerically fitting data, it is necessary to first estimate a zero-age depth [Stein and Stein, 1992]. Such a depth is distinct from a ridge depth [Malinverno, 1990] in that it does not include axial morphology such as axial horsts [e.g., Madsen et al., 1984; Wang and Cochran, 1993; Buck, 2001] or median valleys and flank uplifts [e.g., Sleep and Biehler, 1970; Phipps Morgan et al., 1987; Malinverno, 1990]. To estimate plate parameters for their North Pacific model NPC1, Stein and Stein  simply used the same value of 2600 m as for their global model GDH1. We, however, use an estimate from fitting the square root of time curves to young seafloor. Specifically, curve iv of the data set of Smith and Sandwell  indicates that 3010 ± 165 m (error from F test; see Table 1) is more appropriate (details in section 5.1). This is similar to previous estimates for the Pacific (shaded symbols on Figure 9a) and agrees well with estimates of 2970–3000 m from other bathymetric data sets (Table 1). However, it is less than one estimate of 3163 m [Marty and Cazenave, 1989], which is deeper because this square root of time curve was fitted to data of all ages of data, including older seafloor that is flattening and not best described as a square root of time curve. The arrows on Figures 9a–9c illustrate the effect of including older, flattening seafloor on zero-age depth estimates.
 Using the estimated Zr and keeping Tw, ρ0, ρw, k, and Cp the same as Parsons and Sclater  and Stein and Stein , the values of the remaining parameters (L, Tb, and α) that fit the data best can then be determined by minimizing the misfit. Misfit, s2, is
where n and m are the number of mean depth and heat flow data, z and q are depth and heat flow, and σz and σq are standard deviations for the depth and heat flow data [Stein and Stein, 1992]. We use a search routine based on a regular mesh of evaluated points across a volume of parameter space and a series of increasing “accuracy levels,” which are simply decreasing multiples of the minimum misfit. Initially a broad mesh is used, which contracts until the volume of parameter space within an accuracy contour is well sampled, and then a higher accuracy level is used. This is robust and more efficient than searches with a single resolution mesh.
 The search routine for the inversion was tested by assessing its recovery of synthetic curves and reproduction of published results. Parameters used to create theoretical curves are retrieved with accuracies better than 0.2 km, 2°C, and 0.003°C−1 for L, Tb, and α, respectively, or ∼0.1%. Application of the routine to the data used to deduce GDH1, as tabulated by Stein and Stein , gives L = 93.0 km, α = 3.12 × 10−5 °C−1 and Tb = 1447°C. These agree with GDH1's parameters (L = 95.0 km, α = 3.1 × 10−5 °C−1 and Tb = 1450°C) to within the accuracy (5 km by 25°C by 0.05 × 10−5 °C−1) of their grid search.
5.2.1. Goodness of Fit to Geophysical Data
Figure 10. Fit of cooling plate models to depth and heat flow data. Depths (irregular, thick black line) are means for 1 Myr intervals of the bathymetry of Smith and Sandwell  after all small- and medium-scale features (Figures 5b and 5c) have been removed using the 500 m contour, curve iv. Gray shading is ±2σ (standard deviations) about the means. Smooth, solid black line is the subsidence of the plate model of Parsons and Sclater . Dotted and dashed lines are the subsidence of global and North Pacific models GDH1 [Stein and Stein, 1992] and NPC1 [Stein and Stein, 1993]. Gray line is the subsidence of plate models (D, J, K, L–S, T) that best fit these data. Inset (age axis aligned with main plot) is heat flow data [Pollack et al., 1993] of ages >50 Ma [Stein and Stein, 1992] used in the inversions. Data (thick black line) are averages for every 10 Myr interval. Gray shading and other lines are heat flow equivalents of the subsidence curves on the main plot.
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Table 3. Parameters for the Best Fitting Cooling Plate Modelsa
|H||T = 0||iv||1528−172+153||2.55−0.32+0.41||117−14+24||3.138|| ||74.1||6.64||0.384|
|I||T = 1||iv||1531−187+151||2.55−0.31+0.42||116−16+27||3.138|| ||74.9||6.58||0.398|
 The subsidence of this best fit model explains curve iv well, differing from it on average by 75 ± 54 m (1σ). The fit is a significant improvement on either that of Parsons and Sclater  (PS77) or (NPC1 and GDH1) of [Stein and Stein, 1993]. The fit of our models to the heat flow is also good, differing by 6.55 ± 5.47 mW m−2 (1σ). This is better than the mean difference of 9.56 ± 6.59 mW m−2 (1σ) of PS77 but worse than 4.51 ± 4.33 and 4.76 ± 4.41 for NPC1 and GDH1, respectively. The worse fit is probably because more weight is effectively given to bathymetry in our inversion because of its much lower variances (variance of curve iv is approximately half of the unprocessed data). Asymptotic depth and surface heat flow for our best fit model are ∼6230 m and ∼41.5 mW m−2, respectively, intermediate between PS77 (6400 m, 34 mW m−2) and GDH1 (5650 m, 48 mW m−2), and in excellent agreement with the best available z-t and q-t data of Nagihara et al.  from basins containing old seafloor in the North Atlantic and North Pacific (e.g., Pigafetta Basin).
 The subsidence and surface heat flow data, however, can be equally well produced by cooling plates that have constants and variables different from those of model D. Indeed, 12 models, 7 of which for example have differing fixed values of k (see section 5.2.2), fit the data and overlie each other on the gray line on Figure 10. This occurs because of the number of independent relationships in the z and q equations (see earlier). Namely, inverting z-t and q-t data for three of the variables α, k, Tb, and L can find a plate that well describes the subsidence largely irrespective of the values of the other constants (i.e., Tw, ρ0, ρw, Cp, and the remaining variable). Equally, for the same reason, one cannot uniquely invert for the four variables simultaneously unless there is some constraint from additional data, crustal thickness for example. Furthermore, investigation of the different bathymetric data sets, models F–I (Table 3), demonstrates similar results to those described above for the data set of Smith and Sandwell .
 To summarize, in contrast to the conclusions of Carlson and Johnson , the bathymetry and heat flow for all ages of seafloor in the North Pacific can, we believe, be approximately described by a single cooling plate model. This is not dependent on the assumed value of k or the bathymetric data set used. It is premature, however, to conclude that the North Pacific oceanic lithosphere behaves as a conductively cooling plate. We first need to assess how physically reasonable the values of the parameters required by the best fit models are.
5.2.2. Consistency With Experimental Determinations of Physical Properties
 Model D (Table 3) assumes the same constants (Tw, ρ0, ρw, k, and Cp) and inverts for the same parameters (L, Tb, and α) as Parsons and Sclater  and Stein and Stein . Figures 11a–11c show the parameters that best fit curve iv (L = 115 ± 16 km, α = 2.57 ± 0.40 × 10−5 °C−1, and Tb = 1522 ± 180°C), indicated by a white star. With best fits spanning the ranges L = 108–117 km, α = 2.55–2.60 × 10−5 °C−1, and Tb = 1515–1531 °C, the other bathymetric data sets agree well (models F–I, Table 3).
Figure 11. Parameters of best fitting plate models. (a)–(c) A single inversion with constants (Tw, ρ0, ρw, k, and Cp) and variable parameters (L, Tb, and α) same as Parsons and Sclater  and Stein and Stein  but fit to data in Figure 10. This is model D. Zr is 3010 m (Figure 9a). Best fit is a white star at L = 115 ± 16 km, α = 2.57 ± 0.40 × 10−5 °C−1, and Tb = 1522 ± 180°C. Other symbols are previous estimates: white circle and dashed error box, Parsons and Sclater ; white square and white ×1.25 misfit error ellipse, Stein and Stein ; and triangle, Johnson and Carlson . Shading (calculated every 5°C, 0.05 × 10−5 °C−1, and 1 km) and contours are multiples of the minimum misfit. Figures 11a and 11c show the previously observed inverse relationships that occur because the depths for young and old ages depend on the products of α Tb and αTbL, respectively [Stein and Stein, 1992]. (d)–(f) Illustration of the effect of two trends: (1) effect of progressively including bathymetric features then forcing a 2600 m [Stein and Stein, 1992] ridge depth (models E, D, C, B, A, then U) (shown as line with black squares moving away from star; cross is North Pacific model NPC1 [Stein and Stein, 1993]) and (2) effect of increasing k (models L to S). Line with black circles (arrow indicates direction of increase). Also shown is effect of constraining inversion with crustal thickness, shown as gray diamond (model K) largely hiding open diamond (model J). Black circle is parameters of model X, our preferred values.
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 More notable, however, is the basal temperature of about 1520°C. This is hotter than the hot and thin (1450°C, 95 km) plate GDH1 and, importantly, much hotter than estimates of normal mantle temperature from mid-ocean ridge basalts. The latter indicate a potential temperature Tp of ∼1300°C (Tp = 1280°C [McKenzie and Bickle, 1988], Tp = <1395°C [Kinzler and Grove, 1992], Tp = 1260°C [Kojitani and Akaogi, 1997], which (e.g., via equation (5) [McKenzie and Bickle, 1988]) is consistent with a sublithospheric mantle temperature of about 1350°C [McKenzie, 1984].
 These differences from previous analyses, especially NPC1 of Stein and Stein , may in part be explained by the depths considered to represent the current plate-scale thermal state of the lithosphere. Plates fitted to data that progressively include more bathymetric features (e.g., models E–A in Table 3) become progressively thinner, cooler, and more expansive. These values are plotted as solid squares on a solid line in Figures 11d–11f. Model E is hidden under the white star (from earlier) of model D, models C to A plot progressively further away from the star. The final point on the line, however, is model U, which is like A but is forced to have a zero-age depth of 2600 m [Stein and Stein, 1992, 1993]. For model U, α is dramatically increased (∼0.7 °C−1), Tb lowered (∼30°C), and the plate thinned (∼6 km) to values very close to those of NPC1 [Stein and Stein, 1993] (marked by cross). Other factors, for example, the assumed k, however, also have a large effect.
 Starting from model D and varying k (models L–S, solid line with circles on Figures 11d–11f) has a different effect, and an estimate closer to that of Parsons and Sclater  is possible. Interestingly, this proximity is not entirely arbitrary. Closer estimates occur when basal temperature, Tb, is constrained by crustal thickness, c, and L, α, and k are all inverted for (i.e. also constrained). Tb is constrained at about 1360°C by the c values of White  (7.1 ± 0.8 km) using the geochemical relations of McKenzie  or White and McKenzie , to give models J and K, respectively. These are shown as diamonds on Figures 11d–11f. The misfit criterion, s2, now used is
where cdata and σc2 are from White et al.  and cmodel is from the geochemical relations. The values of k required for models J and K are 3.9 W m−1 °C−1, which is not unreasonably different from experimental constraints. In ambient condition silicates have a thermal conductivity of 4.5–5.5 W m−1 °C−1, which reduces to 2.7 or possibly 2.0 W m−1 °C−1 at the base of the lithosphere [Hofmeister, 1999]. Thus values of k representative of the lithosphere between 3 and 4 W m−1 °C−1 do not appear unreasonable, even when the pressure-temperature dependencies involved are considered [Doin and Fleitout, 1996].
 Another strong effect on the values of the best fit parameters may come from limits in our understanding of the surface heat flow data. For example, these data may also contain a 4 mW m−2 contribution because of radioactive heating in the lithosphere [Parsons and Sclater, 1977], which reduces Tb by over 100°C, raises α (model V), and reduces s2. The misfit reduction should not be over-interpreted, however, as it continues reducing with increasing amounts of q assigned to being radiogenic simply because flatter model q-t curves are possible.
 In order to determine if a model less hot and inexpansive can be found, our final model, X (Table 3), combined several patterns observed above and is a four-parameter inversion in which both the crustal thickness constrains the basal temperature and 4 mW m−2 of surface heat flow is allowed to be radiogenic. Then Tb = 1363°C, k = 3.371 W m−1 °C−1, α = 2.77 ± 0.40 × 10−5 °C−1, and L = 120 km, all within experimental constraints although α remains a little low. Thus, being the most physically reasonable, these are our preferred parameters for the conventional plate model. We caution, however, against the literal use of these parameters, especially without consideration first being given to a physical mechanism for the plate model.
 In section 1, we assert a dichotomy of scales. Specifically, that hot spot swells, which may have formed by thermal rejuvenation of the lithosphere, are distinct from and superimposed on a “plate-scale” thermal behavior of the lithosphere. However, sublithospheric radiogenic heating and small-scale convection are argued to be a dynamically viable mechanism for providing heat at the base of a cooling plate [Huang and Zhong, 2005]. This raises the possibility that the mechanism supplying heat to the base of a plate and hot spots are related, with the hot spot just the larger-scale end-member manifestation of a size spectrum of mantle convection. If this is so, all sizes and ages of thermal perturbation should be retained as part of the plate-scale trend. In this case, the effect of compositional buoyancy within the lithosphere (i.e., surface volcanism, magmatic underplating, and the buoyant depleted residuum of melting [Johnson and Carlson, 1992]) should be isolated from the thermal buoyancy, otherwise the thermal subsidence will be incorrectly estimated. This is clearly difficult to do at the present time.
 Future work, we believe, must combine careful data analysis with model refinements. Bathymetry data, we suggest, should be analyzed using techniques similar to MiMIC, and inversions for parameters of the plate model should include constraints from surface heat flow, probably crustal thickness and possibly the geoid. With regard to the model, material parameters (i.e., Cp, k, and α) in the plate model are currently representative averages of temperature- and pressure-dependent quantities. Incorporating these dependencies has little effect on observables (i.e., the shape of z-t and q-t curves) for either CHABLIS [Doin and Fleitout, 1996] or conventional plates (D. McKenzie, personal communication, 2004). However, some other parameters of conventional plates, at least, are notably affected. For instance, a temperature-dependent plate that best fits the data of Parsons and Sclater  is 106 km thick, ∼15% thinner than the model with constant coefficients (D. McKenzie, personal communication, 2004). So, this should also be investigated. Additionally, we suggest that radiogenic heating in the lithosphere may be an important influence on plate parameters.