2.2. Sampling Criteria for the Argo Profile Pairs
 For the tropical and subtropical region of the western North Pacific (0°∼30°N, 120°E∼180°), there were about 42,600 quality controlled Argo temperature profiles from the Global Argo Data Center for the period 2000 to 2008 (Figure 3). We selected the Argo data only with “good” and “probably good” flag and then manually removed the leftover erroneous data. (Details about the delayed mode QC and Argo data flags can be found in work by T. Carval et al. (Argo data user manual version 2.3, 2010, http://www.argo.net) and Park and Kim .) The Argo profiles before and after a TC event were then paired to calculate the heat content changes induced by TC. This data set was sorted to retain only those profile pairs that fit four sampling criteria appropriate to a TC response study: (1) For water depth, we have retained data only where the water depth was greater than 1000 m in order to avoid coastal regions. (2) For the distance from a TC track, both of the Argo profiles must have been within the search radius, R17, to be retained (Figure 2b). This histogram of R17 is comparable to that from North Atlantic TCs in terms of its shape and range [Dean et al., 2009]. The results we describe here are not sensitive to the choice of spatial search radius (Figure S1 in the auxiliary material versus Figure 6). In addition, the distance between the paired profiles must have been less than 200 km to minimize the anomalies due to background spatial ocean variability. (3) For the temporal window, the pre-TC profile must have been obtained no more than 10 days before the TC passage. We have experimented with three different criteria for choosing the post-TC profile, 0–10 days, 10–20 days, and 20–30 days after the TC passage. The difference among these windows is used to estimate the time evolution of heat content changes. (4) Finally, for the float identity, the profile pairs must be from the same float to avoid duplicate usage of profiles and float-to-float calibration differences.
Figure 3. (a) Number of TCs with maximum sustained wind stronger than 17.5 m/s in each 2° × 2° region during 2000–2008. R17 (radius of 34 knot wind) swath is assumed from each storm track. (b) Number of Argo profile pairs for TC event. (c) Number of Argo profile pairs without TC event (apart from more than 15 days before and 60 days after TC events and outside of 400 km swath away from the tracks).
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 Despite all of these (necessary) sampling criteria, a significant number of Argo profile pairs were found (Figure 3b). For comparison, the spatial distribution of TCs alone is shown in Figure 3a, and the distribution of non-TC Argo profile data is in Figure 3c. As expected, the northwestern part of the western North Pacific domain is most frequently affected by TCs. It is fortunate that the profile pairs from TC and non-TC periods have very similar geographical distributions, and so we have assumed that statistics of background variability in non-TC periods is the same as that in TC periods (more on this in section 2.3.). In addition, the sampled Argo profile pair distributions have similar spatial patterns to that of the TC distribution. From this we infer that the Argo profiles analyzed here are likely to be representative of the basin-scale averaged TC response in the western North Pacific.
2.3. Statistical Approach
 Even with the above search criteria in place, the difference between the pre-TC and post-TC temperature profiles inevitably contains not only the TC-induced responses but also significant background variability that is independent of the TC passage, e.g., the seasonal cycle (which has a large signal in this analysis), internal waves, mesoscale fronts, and eddies. This “background” variability is significant even when the TC response has a larger spatial scale than the spatial search radius. We are not aware of any method that would be suitable for removing this background variability (noise for our purpose) directly from profile pairs to leave only the TC-induced response. Hence, we have attempted to separate the TC-induced signal using a statistical approach.
 Examples of this background variability and the resulting contamination of the TC-induced signal are shown in Figure 4 for a strong (Figures 4a–4c) and a weak (Figures 4d–4f) TC event. These particular floats measured temperature profiles once per day, while typical Argo floats sample one profile every 10 days. In such high temporal resolution data, we may separate TC-induced heat content changes, near-inertial oscillation, and ocean background variability, on the basis of their distinctive time scales [e.g., Sanford et al., 2007]. The expected near-surface cooling and subsurface warming are evident during both of these TC events. However, there is also clearly complex background variability and an inertia-gravity wave response to TC (Figures 4b and 4e). In the much more typical 10 day sampled Argo profiles, as mimicked by subsampling the high-frequency samples in Figures 4c and 4f, the various kinds of background variability are practically inseparable. Therefore, a robust statistical approach is necessary to extract the TC response.
Figure 4. Upper ocean temperature response to TCs measured by Argo floats with unusually short profiling interval. (a–c) Float ID 5901578 (profiling interval of 24.0 h) under a strong TC (maximum sustained wind of 69 m/s, category 4). (d–f) Float ID 290240 (profiling interval of 20.5 h) under a weak TC (maximum sustained wind of 25 m/s, category TS). Figures 4a and 4d are the locations of profiles used in Figures 4b and 4e (red dots), with the locations of profiles in Figures 4c and 4f (black dots). The solid lines denote the TC tracks. Figures 4b and 4e are the temporal section of temperature anomalies (T − Tref). Black lines indicate timings of the reference temperatures (Tref) regarded as before-TC profiles, and all the colored lines including black ones indicate timings of profiles shown in Figures 4c and 4f. Dotted lines denote the timings of TC passages. Figures 4e and 4f are the temperature profiles 2 days before (black), 5 days after (red), 10 days after (green), and 15 days after (blue) the TC passages.
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 Under the assumption that the profile pairs are randomly sampled and that instrumental noise is homogeneous, we have estimated the probability density function, P, of the upper ocean heat content change, x, in response to TC. The observed x is in practice a combination of TC-induced response, xTC, and background variability, xBG, that is presumed to be independent of TC. That is, the measured x is presumed to be a linear sum of xTC and xBG. The probability density function P(x) can then be written as P(x) = P(xTC + xBG) = ∫ PTC(c)PBG(x − c)dc, according to probability theory, provided that xTC and xBG are independent variables, and PTC ≡ P(xTC), and PBG ≡ P(xBG) [Hirschman and Widder, 1955]. For instance, let's consider that xTC and xBG are positive integers, and the case xTC + xBG = 3. The probability PTC+BG can be written as the sum PTC (1)PBG (2) + PTC (2)PBG (1) + PTC (3)PBG (0) + PTC (0)PBG (3). When xTC and xBG are extended to real numbers, the probability PTC+BG (3) can be expressed as ∫ PTC (c)PBG (3 − c)dc. Therefore, the probability density function (PDF) obtained from Argo profile pairs sampled during TC events (PTC+BG) can be written as a convolution between the PDF of the storm-induced response (PTC) that we seek, and the PDF of the background variability (PBG) that we can estimate by using the (very large number of) Argo profile pairs that fall well outside the TC response sampling windows noted above. The estimated PBG is assumed to be the background variability during TC events on the basis of similarity of the data distributions shown in Figures 3b and 3c. The mean value of xTC may be computed by straightforward averaging; by solving for the probability density function PTC(x), we can also make an estimate of the statistical significance of the mean.
 The deconvolution is solved by the Lucy–Richardson algorithm, an iterative procedure often used in image processing to recover a blurred image [Biggs and Andrews, 1997]. The algorithm is tested here using artificially generated Gaussian PDFs (Figure 5). This example suggests not only the robustness of the algorithm, but also small uncertainty in the method itself. Assuming PDFs with Gaussian shapes, the error of PTC* (an estimate of PTC) depends upon the ratio of widths of PBG and PTC+BG; the error is independent of the mean distance between the PDFs. If the width of PTC+BG is similar to that of PBG, PTC would be close to a Delta function and the error of the deconvolved PTC* becomes greater. As the ratio gets larger than 1, error of the deconvolution rapidly decreases, and is ∼10% for a ratio of 1.1 (see Figure 5c). The ratio of widths between PTC+BG and PBG obtained from Argo observation typically ranges between 1.15 and 1.35, implying that the overestimation of the width of PTC* should be less than 5%. It is notable that the accuracy of the deconvolution is not sensitive to PDF skewness (∫((x − m)3/σ3)P(x)dx, where m is mean and σ is standard deviation) within the range of the observational PDFs (from −89 to +10) (not shown).
Figure 5. Sensitivity test of Lucy-Richardson deconvolution method using Gaussian PDFs. (a) An example of the test. Black line is a PDF of background (PBG). Green line is a target PDF for the response to TC (PTC). Blue line is the convolution of PBG and PTC, i.e., PBG+TC. Red line with dots is the estimated target PDF (PTC*) using the deconvolution of PBG+TC and PBG. (b) Difference between the true (PTC) and the estimated (PTC*). (c) Dependency of error in the width (standard deviation σ) of deconvolved PDF as a function of the ratio between the input widths of PBG+TC and PBG. Open red circle presents where the examples (Figures 5a and 5b) are placed (σTC+BG/σBG. = 1.12, σTC* − σTC /σTC = 0.07).
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2.4. Definition of Near-Surface and Subsurface Heat Content Changes
 Our focus in this paper is on the near-surface heat content change ΔHA and subsurface heat content change ΔHB (Figure 1). These are calculated from the individual profile pair as ΔHA = ∆T(z)dz and ΔHB = ∆T(z)dz, where ΔT(z) = T2(z) − T1(z). T1(z) and T2(z) are temperature profiles before and after TC event, respectively, ρ0 is the water density ( = 1024 kg/m3), and cp is the heat capacity ( = 4186 J/kg°C). zc is the depth at which pre-TC and post-TC profiles intersect, i.e., T2(z) − T1(z) = 0, when an intersection is found between the mixed layer depths (MLDs) of pre-TC and post-TC profiles, i.e., if MLD2 < zc < MLD1. This is expected to hold when vertical mixing is the dominant process causing temperature change, as shown schematically in Figure 1a (see also an example from an actual Argo profile pair in Figures 4c, 4f and 8c). This case has been assumed in all of the previous studies on the TC-induced OHU.
 In fact, however, many of the Argo profile pairs (roughly 55%) do not show the expected intersection between pre-TC and post-TC profiles (as in Figures 8f, S2b, and S2c). In those cases, we have estimated zc as the max(MLD2,MLD1), where MLD1 and MLD2 are MLDs before and after TC passages, respectively, defined by the density-based criterion equivalent to the SST-T(Z = MLD) ≥ 0.5 °C [Glover and Brewer, 1988; Kara et al., 2000]. There is no significant difference in our results when choosing zc as the min(MLD2,MLD1). Our definition of zc, i.e., max(MLD2,MLD1), avoids producing negative bias in ΔHA and positive bias in ΔHB because of cancellation of samples from various random state of the background variability, e.g., phase of internal gravity wave. For example, suppose the given temperature profile is the black line in Figure 4f and that there is a thermocline depth fluctuation as 20 m [e.g., Bond et al., 2011]. The resultant bias would then be 0.05 GJ/m2 if selecting zc as either MLD1 or MLD2 for the nonintersecting pairs (see Figure S3). However, by choosing zc as max(MLD2,MLD1), this bias becomes negligible.
 The deviation from the ideal, vertical mixing dominated case is likely due in part to background variability, which must be taken into account when using paired profiles from Argo floats to study TC response. The other reason of the deviation may be upwelling (upward vertical advection) near the track of a TC, which would be a case of cooling and a decrease of ΔHB. Around 17% of the total samples were within 2 × RMW of the track, where upwelling may be important. The PDFs separated by upwelling and nonupwelling region are shown in Figure S4, although the analysis of basin-averaged heat content change includes all the samples. We note too that the results presented in section 3 are not sensitive to the lower depth (i.e., 400 m) chosen to estimate ΔHB (discussed further in section 3).
 In order to test the significance of mean values of ΔHA and ΔHB, the standard error at 95% significance level (= 1.96σ/ ) has been estimated, where mean (m) and standard deviations (σ) can be formulated by m = ∫ xP(x)dx and σ = ∫(x − m)2P(x)dx. Table 1 shows the mean values and standard errors of ΔHA and ΔHB.
Table 1. Statistics of Heat Content Change Solely Induced by TCsa
|Warming Season||Cooling Season||Warming Season||Cooling Season|
|0–10 (day)||−0.10 ± 0.01||−0.02 ± 0.01||−0.04 ± 0.04||−0.04 ± 0.05|
|10–20 (day)||−0.10 ± 0.01||−0.02 ± 0.01||0.04 ± 0.06||−0.08 ± 0.10|
|20–30 (day)||−0.08 ± 0.01||−0.02 ± 0.01||0.04 ± 0.06||−0.05 ± 0.09|
2.5. Limitation of Data and Method
 There are three signals that the Argo profile pairs during TC periods measure: (1) Pure TC response is the heat content changes induced by TCs which we want to extract from the data. (2) TC-induced noise indicates the noise of the heat content changes generated by TC-induced high-frequency phenomenon, such as near-inertial oscillation and upwelling/downwelling. (3) Background variability is the noise due to background variability irrelevant to TC responses, such as seasonal cycle, internal tides, mesoscale eddies, and internal waves. The background variability, PBG, is presumably independent of the TC-induced heat content change, PTC, and can be separated from the observed PTC+BG using the deconvolution method as already discussed in section 2.3. Nevertheless, analysis is done separately for the warm and cold season because both the TC statistics and the background variability exhibit apparent seasonal dependence.
 Differently from the background variability, due to limited temporal resolution of Argo data, the TC-induced noise cannot be directly removed from the observed PTC+BG or the deconvoluted PTC*. However, the unresolved TC-induced high-frequency phenomena are unbiased, i.e., that they do not affect the estimated mean values but will increase the uncertainty of the estimated mean. One may concern that the upwelling expected along a TC track could induce a mean bias in the PDFs. For example the insignificant mean cooling in the profile of Figure 6f may be attributed to upwelling. Upwelling possibly causes a negative heat content change especially for the vertically integrated heat content, which may dominate over the heat content changes induced by the entrainment, especially at locations near the TC track (see Figure S4b). If upwelling dominates the heat content changes, especially the ΔHB, the cooling signal should be greater with greater depth range of vertical integration. For example, in the single float data in the later half in Figure 4b, the integration depth dependency of ΔHB appears to be significant.
Figure 6. Probability density functions of (left) near-surface (ΔHA) and (right) subsurface (ΔHB) heat content changes observed from the Argo profile pairs. The warm colors are for the warming season (July–September), and cool colors are for the cooling season (October–December). In addition, different colors indicate different time windows after TC passage for the after-TC profiles. The dots and horizontal lines show means and 25%–75% percentile ranges of corresponding PDFs. (a and b) PDFs of ΔHA and ΔHB from the profile pairs during TC event which contain not only TC-induced responses but also background variability. (c and d) PDFs of ΔHA and ΔHB from the profile pairs during non-TC periods in which only the background variability is present. (e and f) PDFs of ΔHA and ΔHB obtained from deconvolution of Figures 6a and 6b and Figures 6c and 6d, which are estimated TC-induced responses.
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 However, even when a substantially smaller depth range is used for the integration, i.e., zc = 200 m instead of 400 m, the major results are unchanged, though the width of P(ΔHB) becomes smaller (see Figure S5). Also, the upwelling should be balanced by downwelling over the greater area outside of RMW. The search radius, R17 encompasses the RMW of which the mean is ∼40 km. Most importantly, even considering just the samples outside of the likely upwelling region, the mean values of ΔHB are still insignificant (see Figure S4d). Therefore, we expect that the negative (cooling) biases in the PDFs associated with upwelling near a track are not large enough to change the main conclusions reached in this analysis.
 The Argo profile pairing method employed here has a limitation in the analysis of longer-term variation, >30 days, because the number of samples meeting the sampling criteria becomes dramatically reduced. Most of the Argo floats considered in this study moved slowly enough to stay within 200 km window within 30 days, especially in the subtropical gyre. However, this is not necessarily true for a longer time window. Hence, we limit our analysis to the periods shorter than 30 days from the TC passages.
 Last, the PDF-based approach assumes that the sampled PDF is close to the true PDF. Assuming that there are enough data in regularly spaced grids, a PDF from randomly sampled data like Argo floats could approach the true PDF as the number of samples grows. Monte Carlo experiments with random samples using numerical simulations of the 3-D Price-Weller-Pinkel (PWP) model [Price et al., 1986] suggest that more than 600 samples are required to achieve less than 20% of RMS differences in each PDF (not shown). All the PDFs shown here were constructed from more than 600 samples, and hence are believed to be fairly robust in this statistical sense. However, the 3-D PWP sampling exercise only considers the effect of the TC-induced noise on the PDF, but not that of the background variability. It is noted that about 6100 samples on average are used for estimating each PDF of background variability during non-TC events. Experiments of random resampling the above samples suggest that 600 samples also provide robust PDFs of background variability within about 5% of RMS difference from the PDFs obtained from the entire full sample. This apparent difference in the RMS resulting from 600 samples implies that an accurate estimation of the TC-induced noise requires more samples than that of the background.