[56] 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 trenchtoaxis 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 smallscale (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 [1995] gives, in our view, an unbiased review of many of these possibilities. However, we limit our analysis to the implications of our empirical depthage curve for the cooling plate model.
[57] 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 T_{b}, water temperature T_{w}, water density ρ_{w}, coefficient of thermal expansion α, zerotemperature mantle density ρ_{0}, thermal conductivity k, and heat capacity C_{p}. Thermal diffusivity κ is equal to k/πρ_{0} (i.e., density in this relation is assumed independent of temperature) [Stein and Stein, 1992]. For comparison to measured depths a zeroage depth, Z_{r}, is also required. These are related by equations given by Turcotte and Schubert [2002] for the subsidence, w,
and the surface heat flow, q,
Of the nine parameters, T_{w}, ρ_{0}, ρ_{w}, and C_{p} are assumed to be relatively well known [Parsons and Sclater, 1977; Stein and Stein, 1992] (Table 2) and Z_{r} 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 T_{b} investigated [Parsons and Sclater, 1977; Johnson and Carlson, 1992; Stein and Stein, 1992].
[58] When numerically fitting data, it is necessary to first estimate a zeroage 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 [1993] 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 [1997] 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 zeroage depth estimates.
[59] Using the estimated Z_{r} and keeping T_{w}, ρ_{0}, ρ_{w}, k, and C_{p} the same as Parsons and Sclater [1977] and Stein and Stein [1992], the values of the remaining parameters (L, T_{b}, and α) that fit the data best can then be determined by minimizing the misfit. Misfit, s^{2}, is
where n and m are the number of mean depth and heat flow data, z and q are depth and heat flow, and σ_{z} and σ_{q} 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.
[60] 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 [1993], gives L = 93.0 km, α = 3.12 × 10^{−5} °C^{−1} and T_{b} = 1447°C. These agree with GDH1's parameters (L = 95.0 km, α = 3.1 × 10^{−5} °C^{−1} and T_{b} = 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
Table 3. Parameters for the Best Fitting Cooling Plate Models^{a}Model  Data Set  Curve  Parameter  Note^{b}  Fit 

T_{b}  α  L  k  z  q  s^{2} 


A  SS97  i  1437_{−233}^{+191}  2.80_{−0.47}^{+0.85}  93_{−17}^{+25}  3.138   187.9  4.56  0.399 
B  SS97  ii  1468_{−200}^{+191}  2.69_{−0.43}^{+0.64}  97_{−15}^{+19}  3.138   117.6  5.03  0.441 
C  SS97  iii  1524_{−177}^{+189}  2.53_{−0.35}^{+0.44}  116_{−16}^{+24}  3.138   87.1  6.60  0.418 
D  SS97  iv  1522_{−175}^{+184}  2.57_{−0.36}^{+0.43}  115_{−15}^{+22}  3.138   74.6  6.55  0.414 
E  SS97  v  1526_{−171}^{+165}  2.57_{−0.31}^{+0.42}  117_{−14}^{+21}  3.138   67.8  6.66  0.377 
F  GEBCO  iv  1521_{−188}^{+191}  2.57_{−0.36}^{+0.49}  112_{−14}^{+19}  3.138   71.0  6.32  0.458 
G  ETOPO  iv  1515_{−198}^{+194}  2.60_{−0.36}^{+0.51}  108_{−14}^{+20}  3.138   87.0  6.05  0.464 
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 
J  SS97  iv  1368_{−5}^{+6}  2.56_{−0.30}^{+0.39}  129_{−21}^{+17}  3.90_{−0.87}^{+0.87}  1  75.6  6.59  0.414 
K  SS97  iv  1363_{−7}^{+7}  2.57_{−0.30}^{+0.42}  130_{−25}^{+23}  3.92_{−0.84}^{+0.89}  1  74.5  6.59  0.414 
L  SS97  iv  1903_{−192}^{+224}  2.58_{−0.36}^{+0.41}  92_{−12}^{+17}  2.0   74.7  6.58  0.414 
M  SS97  iv  1704_{−181}^{+203}  2.58_{−0.35}^{+0.42}  103_{−13}^{+19}  2.5   74.3  6.55  0.414 
N  SS97  iv  1557_{−183}^{+172}  2.58_{−0.35}^{+0.40}  113_{−14}^{+23}  3   74.7  6.56  0.414 
P  SS97  iv  1440_{−153}^{+175}  2.58_{−0.35}^{+0.42}  122_{−15}^{+24}  3.5   75.2  6.54  0.414 
Q  SS97  iv  1344_{−139}^{+168}  2.58_{−0.36}^{+0.43}  130_{−16}^{+20}  4.0   75.0  6.56  0.414 
R  SS97  iv  1271_{−147}^{+155}  2.58_{−0.36}^{+0.41}  138_{−17}^{+12}  4.5   74.9  6.54  0.414 
S  SS97  iv  1212_{−112}^{+128}  2.56_{−0.29}^{+0.44}  146_{−18}^{+4}  5.0   75.3  6.56  0.414 
T  SS97  iv  1490_{−176}^{+189}  3.21_{−0.43}^{+0.57}  104_{−12}^{+14}  3.138  2  73.3  5.61  0.433 
U  SS97  i  1398_{−220}^{+184}  3.36_{−0.57}^{+0.44}  87_{−15}^{+19}  3.138  2  188.7  5.47  0.363 
V  SS97  iv  1409_{−162}^{+171}  2.77_{−0.37}^{+0.47}  116_{−14}^{+22}  3.138  3  75.3  6.08  0.390 
W  SS97  iv  1487  2.79  112  3.138  4  70.2  6.23   
X  SS97  iv  1363_{−7}^{+7}  2.77_{−0.32}^{+0.47}  120_{−21}^{+21}  3.37_{−0.71}^{+0.86}  1, 3  75.5  6.09  0.390 
[62] 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 [1977] (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 zt and qt data of Nagihara et al. [1996] from basins containing old seafloor in the North Atlantic and North Pacific (e.g., Pigafetta Basin).
[63] 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 zt and qt data for three of the variables α, k, T_{b}, and L can find a plate that well describes the subsidence largely irrespective of the values of the other constants (i.e., T_{w}, ρ_{0}, ρ_{w}, C_{p}, 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 [1997].
[64] To summarize, in contrast to the conclusions of Carlson and Johnson [1994], 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
[65] Model D (Table 3) assumes the same constants (T_{w}, ρ_{0}, ρ_{w}, k, and C_{p}) and inverts for the same parameters (L, T_{b}, and α) as Parsons and Sclater [1977] and Stein and Stein [1992]. 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 T_{b} = 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 T_{b} = 1515–1531 °C, the other bathymetric data sets agree well (models F–I, Table 3).
[67] 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 midocean ridge basalts. The latter indicate a potential temperature T_{p} of ∼1300°C (T_{p} = 1280°C [McKenzie and Bickle, 1988], T_{p} = <1395°C [Kinzler and Grove, 1992], T_{p} = 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].
[69] These differences from previous analyses, especially NPC1 of Stein and Stein [1993], may in part be explained by the depths considered to represent the current platescale 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 zeroage depth of 2600 m [Stein and Stein, 1992, 1993]. For model U, α is dramatically increased (∼0.7 °C^{−1}), T_{b} 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.
[70] 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 [1977] is possible. Interestingly, this proximity is not entirely arbitrary. Closer estimates occur when basal temperature, T_{b}, is constrained by crustal thickness, c, and L, α, and k are all inverted for (i.e. also constrained). T_{b} is constrained at about 1360°C by the c values of White [1992] (7.1 ± 0.8 km) using the geochemical relations of McKenzie [1984] or White and McKenzie [1995], to give models J and K, respectively. These are shown as diamonds on Figures 11d–11f. The misfit criterion, s^{2}, now used is
where c_{data} and σ_{c}^{2} are from White et al. [1992] and c_{model} 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 pressuretemperature dependencies involved are considered [Doin and Fleitout, 1996].
[71] 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 T_{b} by over 100°C, raises α (model V), and reduces s^{2}. The misfit reduction should not be overinterpreted, however, as it continues reducing with increasing amounts of q assigned to being radiogenic simply because flatter model qt curves are possible.
[72] 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 fourparameter 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 T_{b} = 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.
[73] 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 “platescale” thermal behavior of the lithosphere. However, sublithospheric radiogenic heating and smallscale 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 largerscale endmember 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 platescale 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.
[74] 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., C_{p}, k, and α) in the plate model are currently representative averages of temperature and pressuredependent quantities. Incorporating these dependencies has little effect on observables (i.e., the shape of zt and qt 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 temperaturedependent plate that best fits the data of Parsons and Sclater [1977] 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.